International Nuclear Information System (INIS)
Prakash, M.
1985-01-01
The theory of supergravity has attracted increasing attention in the recent years as a unified theory of elementary particle interactions. The superspace formulation of the theory is highly suggestive of an underlying geometrical structure of superspace. It also incorporates the beautifully geometrical general theory of relativity. It leads us to believe that a better understanding of its geometry would result in a better understanding of the theory itself, and furthermore, that the geometry of superspace would also have physical consequences. As a first step towards that goal, we develop here a theory of super Lie groups. These are groups that have the same relation to a super Lie algebra as Lie groups have to a Lie algebra. More precisely, a super Lie group is a super-manifold and a group such that the group operations are super-analytic. The super Lie algebra of a super Lie group is related to the local properties of the group near the identity. This work develops the algebraic and super-analytical tools necessary for our theory, including proofs of a set of existence and uniqueness theorems for a class of super-differential equations
Introduction to the theory of Lie groups
Godement, Roger
2017-01-01
This textbook covers the general theory of Lie groups. By first considering the case of linear groups (following von Neumann's method) before proceeding to the general case, the reader is naturally introduced to Lie theory. Written by a master of the subject and influential member of the Bourbaki group, the French edition of this textbook has been used by several generations of students. This translation preserves the distinctive style and lively exposition of the original. Requiring only basics of topology and algebra, this book offers an engaging introduction to Lie groups for graduate students and a valuable resource for researchers.
Lie groups and grand unified theories
International Nuclear Information System (INIS)
Gubitoso, M.D.
1987-01-01
This work presents some concepts in group theory and Lie algebras and, at same time, shows a method to study and work with semisimple Lie groups, based on Dynkin diagrams. The aproach taken is not completely formal, but it presents the main points of the elaboration of the method, so its mathematical basis is designed with the purpose of making the reading not so cumbersome to those who are interested only in a general picture of the method and its usefulness. At the end it is shown a brief review of gauge theories and two grand-unification models based on SO(13) and E 7 gauge groups. (author) [pt
Lie groups and Lie algebras for physicists
Das, Ashok
2015-01-01
The book is intended for graduate students of theoretical physics (with a background in quantum mechanics) as well as researchers interested in applications of Lie group theory and Lie algebras in physics. The emphasis is on the inter-relations of representation theories of Lie groups and the corresponding Lie algebras.
Hsiang, Wu-Yi
2017-01-01
This volume consists of nine lectures on selected topics of Lie group theory. We provide the readers a concise introduction as well as a comprehensive 'tour of revisiting' the remarkable achievements of S Lie, W Killing, É Cartan and H Weyl on structural and classification theory of semi-simple Lie groups, Lie algebras and their representations; and also the wonderful duet of Cartans' theory on Lie groups and symmetric spaces.With the benefit of retrospective hindsight, mainly inspired by the outstanding contribution of H Weyl in the special case of compact connected Lie groups, we develop the above theory via a route quite different from the original methods engaged by most other books.We begin our revisiting with the compact theory which is much simpler than that of the general semi-simple Lie theory; mainly due to the well fittings between the Frobenius-Schur character theory and the maximal tori theorem of É Cartan together with Weyl's reduction (cf. Lectures 1-4). It is a wonderful reality of the Lie t...
Lie groups, lie algebras, and representations an elementary introduction
Hall, Brian
2015-01-01
This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject. In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including: a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C) an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebras a self-contained construction of the representations of compac...
Chevalley, Claude
2018-01-01
The standard text on the subject for many years, this introductory treatment covers classical linear groups, topological groups, manifolds, analytic groups, differential calculus of Cartan, and compact Lie groups and their representations. 1946 edition.
Lie groups, Lie algebras, and some of their applications
Gilmore, Robert
1974-01-01
Lie group theory plays an increasingly important role in modern physical theories. Many of its calculations remain fundamentally unchanged from one field of physics to another, altering only in terms of symbols and the language. Using the theory of Lie groups as a unifying vehicle, concepts and results from several fields of physics can be expressed in an extremely economical way. With rigor and clarity, this text introduces upper-level undergraduate students to Lie group theory and its physical applications.An opening discussion of introductory concepts leads to explorations of the classical
Galois Theory of Differential Equations, Algebraic Groups and Lie Algebras
Put, Marius van der
1999-01-01
The Galois theory of linear differential equations is presented, including full proofs. The connection with algebraic groups and their Lie algebras is given. As an application the inverse problem of differential Galois theory is discussed. There are many exercises in the text.
Exceptional Lie groups, E-infinity theory and Higgs Boson
International Nuclear Information System (INIS)
El-Okaby, Ayman A.
2008-01-01
In this paper we study the correlation between El-Naschie's exceptional Lie groups hierarchies and his transfinite E-infinity space-time theory. Subsequently this correlation is used to calculate the number of elementary particles in the standard model, mass of the Higgs Bosons and some coupling constants
The structure of complex Lie groups
Lee, Dong Hoon
2001-01-01
Complex Lie groups have often been used as auxiliaries in the study of real Lie groups in areas such as differential geometry and representation theory. To date, however, no book has fully explored and developed their structural aspects.The Structure of Complex Lie Groups addresses this need. Self-contained, it begins with general concepts introduced via an almost complex structure on a real Lie group. It then moves to the theory of representative functions of Lie groups- used as a primary tool in subsequent chapters-and discusses the extension problem of representations that is essential for studying the structure of complex Lie groups. This is followed by a discourse on complex analytic groups that carry the structure of affine algebraic groups compatible with their analytic group structure. The author then uses the results of his earlier discussions to determine the observability of subgroups of complex Lie groups.The differences between complex algebraic groups and complex Lie groups are sometimes subtle ...
Essays in the history of Lie groups and algebraic groups
Borel, Armand
2001-01-01
Lie groups and algebraic groups are important in many major areas of mathematics and mathematical physics. We find them in diverse roles, notably as groups of automorphisms of geometric structures, as symmetries of differential systems, or as basic tools in the theory of automorphic forms. The author looks at their development, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. Starting from Lie's theory of local analytic transformation groups and early work on Lie algebras, he follows the process of globalization in its two main frameworks: differential geometry and topology on one hand, algebraic geometry on the other. Chapters II to IV are devoted to the former, Chapters V to VIII, to the latter. The essays in the first part of the book survey various proofs of the full reducibility of linear representations of \\mathbf{SL}_2{(\\mathbb{C})}, the contributions of H. Weyl to representations and invariant theory for semisimple Lie groups, and con...
Pro-Lie Groups: A Survey with Open Problems
Directory of Open Access Journals (Sweden)
Karl H. Hofmann
2015-07-01
Full Text Available A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity component and, thus, in particular, each compact and each connected locally-compact group; it also includes all locally-compact Abelian groups. This paper provides an overview of the structure theory and the Lie theory of pro-Lie groups, including results more recent than those in the authors’ reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly-complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function that links the two. (A topological vector space is weakly complete if it is isomorphic to a power RX of an arbitrary set of copies of R. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups. The article also lists 12 open questions connected to pro-Lie groups.
Reflection Positive Stochastic Processes Indexed by Lie Groups
Jorgensen, Palle E. T.; Neeb, Karl-Hermann; Ólafsson, Gestur
2016-06-01
Reflection positivity originates from one of the Osterwalder-Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symmetric Lie groups (Lie groups with an involution) and results in a transformation of a unitary representation of a symmetric Lie group to a unitary representation of its Cartan dual. In this article we continue our investigation of representation theoretic aspects of reflection positivity by discussing reflection positive Markov processes indexed by Lie groups, measures on path spaces, and invariant gaussian measures in spaces of distribution vectors. This provides new constructions of reflection positive unitary representations.
Quantum spaces, central extensions of Lie groups and related quantum field theories
Poulain, Timothé; Wallet, Jean-Christophe
2018-02-01
Quantum spaces with su(2) noncommutativity can be modelled by using a family of SO(3)-equivariant differential *-representations. The quantization maps are determined from the combination of the Wigner theorem for SU(2) with the polar decomposition of the quantized plane waves. A tracial star-product, equivalent to the Kontsevich product for the Poisson manifold dual to su(2) is obtained from a subfamily of differential *-representations. Noncommutative (scalar) field theories free from UV/IR mixing and whose commutative limit coincides with the usual ϕ 4 theory on ℛ3 are presented. A generalization of the construction to semi-simple possibly non simply connected Lie groups based on their central extensions by suitable abelian Lie groups is discussed. Based on a talk presented by Poulain T at the XXVth International Conference on Integrable Systems and Quantum symmetries (ISQS-25), Prague, June 6-10 2017.
Lipkin, Harry J
2002-01-01
According to the author of this concise, high-level study, physicists often shy away from group theory, perhaps because they are unsure which parts of the subject belong to the physicist and which belong to the mathematician. However, it is possible for physicists to understand and use many techniques which have a group theoretical basis without necessarily understanding all of group theory. This book is designed to familiarize physicists with those techniques. Specifically, the author aims to show how the well-known methods of angular momentum algebra can be extended to treat other Lie group
A Lie based 4-dimensional higher Chern-Simons theory
Zucchini, Roberto
2016-05-01
We present and study a model of 4-dimensional higher Chern-Simons theory, special Chern-Simons (SCS) theory, instances of which have appeared in the string literature, whose symmetry is encoded in a skeletal semistrict Lie 2-algebra constructed from a compact Lie group with non discrete center. The field content of SCS theory consists of a Lie valued 2-connection coupled to a background closed 3-form. SCS theory enjoys a large gauge and gauge for gauge symmetry organized in an infinite dimensional strict Lie 2-group. The partition function of SCS theory is simply related to that of a topological gauge theory localizing on flat connections with degree 3 second characteristic class determined by the background 3-form. Finally, SCS theory is related to a 3-dimensional special gauge theory whose 2-connection space has a natural symplectic structure with respect to which the 1-gauge transformation action is Hamiltonian, the 2-curvature map acting as moment map.
Energy Technology Data Exchange (ETDEWEB)
Salam, A. [Imperial College of Science and Technology, London (United Kingdom)
1963-01-15
Throughout the history of quantum theory, a battle has raged between the amateurs and professional group theorists. The amateurs have maintained that everything one needs in the theory of groups can be discovered by the light of nature provided one knows how to multiply two matrices. In support of this claim, they of course, justifiably, point to the successes of that prince of amateurs in this field, Dirac, particularly with the spinor representations of the Lorentz group. As an amateur myself, I strongly believe in the truth of the non-professionalist creed. I think perhaps there is not much one has to learn in the way of methodology from the group theorists except caution. But this does not mean one should not be aware of the riches which have been amassed over the course of years particularly in that most highly developed of all mathematical disciplines - the theory of Lie groups. My lectures then are an amateur's attempt to gather some of the fascinating results for compact simple Lie groups which are likely to be of physical interest. I shall state theorems; and with a physicist's typical unconcern rarely, if ever, shall I prove these. Throughout, the emphasis will be to show the close similarity of these general groups with that most familiar of all groups, the group of rotations in three dimensions.
Quasi-Lie algebras and Lie groups
International Nuclear Information System (INIS)
Momo Bangoura
2006-07-01
In this work, we define the quasi-Poisson Lie quasigroups, dual objects to the quasi-Poisson Lie groups and we establish the correspondence between the local quasi-Poisson Lie quasigoups and quasi-Lie bialgebras (up to isomorphism). (author) [fr
Expansion in finite simple groups of Lie type
Tao, Terence
2015-01-01
Expander graphs are an important tool in theoretical computer science, geometric group theory, probability, and number theory. Furthermore, the techniques used to rigorously establish the expansion property of a graph draw from such diverse areas of mathematics as representation theory, algebraic geometry, and arithmetic combinatorics. This text focuses on the latter topic in the important case of Cayley graphs on finite groups of Lie type, developing tools such as Kazhdan's property (T), quasirandomness, product estimates, escape from subvarieties, and the Balog-Szemerédi-Gowers lemma. Applications to the affine sieve of Bourgain, Gamburd, and Sarnak are also given. The material is largely self-contained, with additional sections on the general theory of expanders, spectral theory, Lie theory, and the Lang-Weil bound, as well as numerous exercises and other optional material.
An introduction to Lie groups and the geometry of homogeneous spaces
Arvanitoyeorgos, Andreas
2003-01-01
It is remarkable that so much about Lie groups could be packed into this small book. But after reading it, students will be well-prepared to continue with more advanced, graduate-level topics in differential geometry or the theory of Lie groups. The theory of Lie groups involves many areas of mathematics. In this book, Arvanitoyeorgos outlines enough of the prerequisites to get the reader started. He then chooses a path through this rich and diverse theory that aims for an understanding of the geometry of Lie groups and homogeneous spaces. In this way, he avoids the extra detail needed for a thorough discussion of other topics. Lie groups and homogeneous spaces are especially useful to study in geometry, as they provide excellent examples where quantities (such as curvature) are easier to compute. A good understanding of them provides lasting intuition, especially in differential geometry. The book is suitable for advanced undergraduates, graduate students, and research mathematicians interested in differenti...
Lie families: theory and applications
International Nuclear Information System (INIS)
Carinena, Jose F; Grabowski, Janusz; De Lucas, Javier
2010-01-01
We analyze the families of non-autonomous systems of first-order ordinary differential equations admitting a common time-dependent superposition rule, i.e. a time-dependent map expressing any solution of each of these systems in terms of a generic set of particular solutions of the system and some constants. We next study the relations of these families, called Lie families, with the theory of Lie and quasi-Lie systems and apply our theory to provide common time-dependent superposition rules for certain Lie families.
Introduction to quantized LIE groups and algebras
International Nuclear Information System (INIS)
Tjin, T.
1992-01-01
In this paper, the authors give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups the authors study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then the authors explain in detail the concept of quantization for them. As an example the quantization of sl 2 is explicitly carried out. Next, the authors show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction, the authors explicitly construct the universal R matrix for the quantum sl 2 algebra. In the last section, the authors deduce all finite-dimensional irreducible representations for q a root of unity. The authors also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory
Harmonic analysis on exponential solvable Lie groups
Fujiwara, Hidenori
2015-01-01
This book is the first one that brings together recent results on the harmonic analysis of exponential solvable Lie groups. There still are many interesting open problems, and the book contributes to the future progress of this research field. As well, various related topics are presented to motivate young researchers. The orbit method invented by Kirillov is applied to study basic problems in the analysis on exponential solvable Lie groups. This method tells us that the unitary dual of these groups is realized as the space of their coadjoint orbits. This fact is established using the Mackey theory for induced representations, and that mechanism is explained first. One of the fundamental problems in the representation theory is the irreducible decomposition of induced or restricted representations. Therefore, these decompositions are studied in detail before proceeding to various related problems: the multiplicity formula, Plancherel formulas, intertwining operators, Frobenius reciprocity, and associated alge...
Carnovale, Giovanna; Caselli, Fabrizio; Concini, Corrado; Sole, Alberto
2017-01-01
Lie theory is a mathematical framework for encoding the concept of symmetries of a problem, and was the central theme of an INdAM intensive research period at the Centro de Giorgi in Pisa, Italy, in the academic year 2014-2015. This book gathers the key outcomes of this period, addressing topics such as: structure and representation theory of vertex algebras, Lie algebras and superalgebras, as well as hyperplane arrangements with different approaches, ranging from geometry and topology to combinatorics.
Jacobson, Nathan
1979-01-01
Lie group theory, developed by M. Sophus Lie in the 19th century, ranks among the more important developments in modern mathematics. Lie algebras comprise a significant part of Lie group theory and are being actively studied today. This book, by Professor Nathan Jacobson of Yale, is the definitive treatment of the subject and can be used as a textbook for graduate courses.Chapter I introduces basic concepts that are necessary for an understanding of structure theory, while the following three chapters present the theory itself: solvable and nilpotent Lie algebras, Carlan's criterion and its
Discrete finite nilpotent Lie analogs: New models for unified gauge field theory
International Nuclear Information System (INIS)
Kornacker, K.
1978-01-01
To each finite dimensional real Lie algebra with integer structure constants there corresponds a countable family of discrete finite nilpotent Lie analogs. Each finite Lie analog maps exponentially onto a finite unipotent group G, and is isomorphic to the Lie algebra of G. Reformulation of quantum field theory in discrete finite form, utilizing nilpotent Lie analogs, should elminate all divergence problems even though some non-Abelian gauge symmetry may not be spontaneously broken. Preliminary results in the new finite representation theory indicate that a natural hierarchy of spontaneously broken symmetries can arise from a single unbroken non-Abelian gauge symmetry, and suggest the possibility of a new unified group theoretic interpretation for hadron colors and flavors
Group theory and its applications
Loebl, Ernest M
1975-01-01
Group Theory and its Applications, Volume III covers the two broad areas of applications of group theory, namely, all atomic and molecular phenomena, as well as all aspects of nuclear structure and elementary particle theory.This volume contains five chapters and begins with an introduction to Wedderburn's theory to establish the structure of semisimple algebras, algebras of quantum mechanical interest, and group algebras. The succeeding chapter deals with Dynkin's theory for the embedding of semisimple complex Lie algebras in semisimple complex Lie algebras. These topics are followed by a rev
Applications of Lie Group Theory to the Modeling and Control of Multibody Systems
International Nuclear Information System (INIS)
Mladenova, Clementina D.
1999-01-01
This paper reviews our research activities concerning the modeling and control of rigid and elastic joint multibody mechanical systems, including some investigations into nonholonomic systems. Bearing in mind the different parameterizations of the rotation group in three-dimensional space SO(3), and the fact that the properties of the parameterization more or less influence the efficiency of the dynamics model, here the so-called vector parameter is used for parallel considerations of rigid body motion and of rigid and elastic joint multibody mechanical systems. Besides the fundamental role of this study, the vector-parameter approach is efficient in its computational aspect and quite convenient for real time simulation and control. The consideration of the mechanical system on the configuration space of pure vector parameters with a group structure opens the possibilities for the Lie group theory to be applied in problems of dynamics and control
Lie transforms and their use in Hamiltonian perturbation theory
International Nuclear Information System (INIS)
Cary, J.R.
1978-06-01
A review is presented of the theory of Lie transforms as applied to Hamiltonian systems. We begin by presenting some general background on the Hamiltonian formalism and by introducing the operator notation for canonical transformations. We then derive the general theory of Lie transforms. We derive the formula for the new Hamiltonian when one uses a Lie transform to effect a canonical transformation, and we use Lie transforms to prove a very general version of Noether's theorem, or the symmetry-equals-invariant theorem. Next we use the general Lie transform theory to derive Deprit's perturbation theory. We illustrate this perturbation theory by application to two well-known problems in classical mechanics. Finally we present a chapter on conventions. There are many ways to develop Lie transforms. The last chapter explains the reasons for the choices made here
Lie groups and algebraic groups
Indian Academy of Sciences (India)
We give an exposition of certain topics in Lie groups and algebraic groups. This is not a complete ... of a polynomial equation is equivalent to the solva- bility of the equation ..... to a subgroup of the group of roots of unity in k (in particular, it is a ...
van der Noort, V.
2009-01-01
This thesis is written in the subfield of mathematics known as representation theory of real reductive Lie groups. Let G be a Lie group in the Harish-Chandra class with maximal compact subgroup K and Lie algebra g. Let Omega be a connected complex manifold. By a family of G-representations
Directory of Open Access Journals (Sweden)
Decio Levi
2013-10-01
Full Text Available We briefly review two different methods of applying Lie group theory in the numerical solution of ordinary differential equations. On specific examples we show how the symmetry preserving discretization provides difference schemes for which the “first differential approximation” is invariant under the same Lie group as the original ordinary differential equation.
International Nuclear Information System (INIS)
Guenaydin, M.; Saclioglu, C.
1981-06-01
We give a construction of the Lie algebras of the non-compact groups appearing in four dimensional supergravity theories in terms of boson operators. Our construction parallels very closely their emergence in supergravity and is an extension of the well-known construction of the Lie algebras of the non-compact groups Sp(2n,IR) and SO(2n) from boson operators transforming like a fundamental representation of their maximal compact subgroup U(n). However this extension is non-trivial only for n >= 4 and stops at n = 8 leading to the Lie algebras of SU(4) x SU(1,1), SU(5,1), SO(12) and Esub(7(7)). We then give a general construction of an infinite class of unitary irreducible representations of the respective non-compact groups (except for Esub(7(7)) and SO(12) obtained from the extended construction). We illustrate our construction with the examples of SU(5,1) and SO(12). (orig.)
A representation independent propagator. Pt. 1. Compact Lie groups
International Nuclear Information System (INIS)
Tome, W.A.
1995-01-01
Conventional path integral expressions for propagators are representation dependent. Rather than having to adapt each propagator to the representation in question, it is shown that for compact Lie groups it is possible to introduce a propagator that is representation independent. For a given set of kinematical variables this propagator is a single function independent of any particular choice of fiducial vector, which monetheless, correctly propagates each element of the coherent state representation associated with these kinematical variables. Although the configuration space is in general curved, nevertheless the lattice phase-space path integral for the representation independent propagator has the form appropriate to flat space. To illustrate the general theory a representation independent propagator is explicitly constructed for the Lie group SU(2). (orig.)
Lie groups, differential equations, and geometry advances and surveys
2017-01-01
This book collects a series of contributions addressing the various contexts in which the theory of Lie groups is applied. A preliminary chapter serves the reader both as a basic reference source and as an ongoing thread that runs through the subsequent chapters. From representation theory and Gerstenhaber algebras to control theory, from differential equations to Finsler geometry and Lepage manifolds, the book introduces young researchers in Mathematics to a wealth of different topics, encouraging a multidisciplinary approach to research. As such, it is suitable for students in doctoral courses, and will also benefit researchers who want to expand their field of interest.
On a Lie-isotopic theory of gravity
International Nuclear Information System (INIS)
Gasperini, M.
1984-01-01
Starting from the isotopic lifting of the Poincare algebra, a Lie-isotopic theory of gravity is formulated, its physical interpretation is given in terms of a generalized principle of equivalence, and it is shown that a local Lorentz-isotopic symmetry motivates the introduction of a generalized metric-affine geometrical structure. Finally, possible applications of a Lie-isotopic theory to the problem of unifying gravity with internal symmetries, in four and more than four dimensions, are discussed
Lie-algebraic classification of effective theories with enhanced soft limits
Bogers, Mark P.; Brauner, Tomáš
2018-05-01
A great deal of effort has recently been invested in developing methods of calculating scattering amplitudes that bypass the traditional construction based on Lagrangians and Feynman rules. Motivated by this progress, we investigate the long-wavelength behavior of scattering amplitudes of massless scalar particles: Nambu-Goldstone (NG) bosons. The low-energy dynamics of NG bosons is governed by the underlying spontaneously broken symmetry, which likewise allows one to bypass the Lagrangian and connect the scaling of the scattering amplitudes directly to the Lie algebra of the symmetry generators. We focus on theories with enhanced soft limits, where the scattering amplitudes scale with a higher power of momentum than expected based on the mere existence of Adler's zero. Our approach is complementary to that developed recently in ref. [1], and in the first step we reproduce their result. That is, as far as Lorentz-invariant theories with a single physical NG boson are concerned, we find no other nontrivial theories featuring enhanced soft limits beyond the already well-known ones: the Galileon and the Dirac-Born-Infeld (DBI) scalar. Next, we show that in a certain sense, these theories do not admit a nontrivial generalization to non-Abelian internal symmetries. Namely, for compact internal symmetry groups, all NG bosons featuring enhanced soft limits necessarily belong to the center of the group. For noncompact symmetry groups such as the ISO( n) group featured by some multi-Galileon theories, these NG bosons then necessarily belong to an Abelian normal subgroup. The Lie-algebraic consistency constraints admit two infinite classes of solutions, generalizing the known multi-Galileon and multi-flavor DBI theories.
Quantum Lie theory a multilinear approach
Kharchenko, Vladislav
2015-01-01
This is an introduction to the mathematics behind the phrase “quantum Lie algebra”. The numerous attempts over the last 15-20 years to define a quantum Lie algebra as an elegant algebraic object with a binary “quantum” Lie bracket have not been widely accepted. In this book, an alternative approach is developed that includes multivariable operations. Among the problems discussed are the following: a PBW-type theorem; quantum deformations of Kac--Moody algebras; generic and symmetric quantum Lie operations; the Nichols algebras; the Gurevich--Manin Lie algebras; and Shestakov--Umirbaev operations for the Lie theory of nonassociative products. Opening with an introduction for beginners and continuing as a textbook for graduate students in physics and mathematics, the book can also be used as a reference by more advanced readers. With the exception of the introductory chapter, the content of this monograph has not previously appeared in book form.
The representations of Lie groups and geometric quantizations
International Nuclear Information System (INIS)
Zhao Qiang
1998-01-01
In this paper we discuss the relation between representations of Lie groups and geometric quantizations. A series of representations of Lie groups are constructed by geometric quantization of coadjoint orbits. Particularly, all representations of compact Lie groups, holomorphic discrete series of representations and spherical representations of reductive Lie groups are constructed by geometric quantizations of elliptic and hyperbolic coadjoint orbits. (orig.)
International Nuclear Information System (INIS)
Ton-That, Tuong
2005-01-01
In a previous paper we gave a generalization of the notion of Casimir invariant differential operators for the infinite-dimensional Lie groups GL ∞ (C) (or equivalently, for its Lie algebra gj ∞ (C)). In this paper we give a generalization of the Casimir invariant differential operators for a class of infinite-dimensional Lie groups (or equivalently, for their Lie algebras) which contains the infinite-dimensional complex classical groups. These infinite-dimensional Lie groups, and their Lie algebras, are inductive limits of finite-dimensional Lie groups, and their Lie algebras, with some additional properties. These groups or their Lie algebras act via the generalized adjoint representations on projective limits of certain chains of vector spaces of universal enveloping algebras. Then the generalized Casimir operators are the invariants of the generalized adjoint representations. In order to be able to explicitly compute the Casimir operators one needs a basis for the universal enveloping algebra of a Lie algebra. The Poincare-Birkhoff-Witt (PBW) theorem gives an explicit construction of such a basis. Thus in the first part of this paper we give a generalization of the PBW theorem for inductive limits of Lie algebras. In the last part of this paper a generalization of the very important theorem in representation theory, namely the Chevalley-Racah theorem, is also discussed
International Nuclear Information System (INIS)
Berezin, F.A.
1977-01-01
Generalization of the Laplace-Casimir operator theory on the Lie supergroups is considered. The main result is the formula for radial parts of the Laplace operators under some general assumptions about the Lie supergroup. In particular these assumptions are valid for the Lie suppergroups U(p,g) and C (m,n). The first one is the analogue of the unitary group, the second one is the analogue of the linear group of canonical transformations
Elementary construction of graded lie groups
International Nuclear Information System (INIS)
Scheunert, M.; Rittenberg, V.
1977-06-01
We show how the definitions of the classical Lie groups have to be modified in the case where Grassmann variables are present. In particular, we construct the general linear, the special linear and the orthosymplectic graded Lie groups. Special attention is paid to the question of how to formulate an adequate 'unitarity condition'. (orig.) [de
Topological Poisson Sigma models on Poisson-Lie groups
International Nuclear Information System (INIS)
Calvo, Ivan; Falceto, Fernando; Garcia-Alvarez, David
2003-01-01
We solve the topological Poisson Sigma model for a Poisson-Lie group G and its dual G*. We show that the gauge symmetry for each model is given by its dual group that acts by dressing transformations on the target. The resolution of both models in the open geometry reveals that there exists a map from the reduced phase of each model (P and P*) to the main symplectic leaf of the Heisenberg double (D 0 ) such that the symplectic forms on P, P* are obtained as the pull-back by those maps of the symplectic structure on D 0 . This uncovers a duality between P and P* under the exchange of bulk degrees of freedom of one model with boundary degrees of freedom of the other one. We finally solve the Poisson Sigma model for the Poisson structure on G given by a pair of r-matrices that generalizes the Poisson-Lie case. The Hamiltonian analysis of the theory requires the introduction of a deformation of the Heisenberg double. (author)
Towards a structure theory for Lie-admissible algebras
International Nuclear Information System (INIS)
Wene, G.P.
1981-01-01
The concepts of radical and decomposition of algebras are presented. Following a discussion of the theory for associative algebras, examples are presented that illuminate the difficulties encountered in choosing a structure theory for nonassociative algebras. Suitable restrictions, based upon observed phenomenon, are given that reduce the class of Lie-admissible algebras to a manageable size. The concepts developed in the first part of the paper are then reexamined in the context of this smaller class of Lie-admissible algebras
S7 without any construction of Lie group
International Nuclear Information System (INIS)
Zhou Jian; Xu Senlin.
1988-12-01
It was proved that the sphere S n is a parallelizable manifold if and only if n = 1,3 or 7, and that S n is an H-space if and only if n = 0,1,3 or 7. Because a Lie group must necessarily be a parallelizable manifold and also an H-space, naturally one asks that S n is a Lie group for n = 0, 1,3 or 7? In this paper we prove that S 7 is not a Lie group, and it is not even a topological group. Therefore, S n is a Lie group (or a topological group) if and only if n = 0,1,3. (author). 11 refs
Enveloping algebras of Lie groups with descrete series
International Nuclear Information System (INIS)
Nguyen huu Anh; Vuong manh Son
1990-09-01
In this article we shall prove that the enveloping algebra of the Lie algebra of some unimodular Lie group having discrete series, when localized at some element of the center, is isomorphic to the tensor product of a Weyl algebra over the ring of Laurent polynomials of one variable and the enveloping algebra of some reductive Lie algebra. In particular, it will be proved that the Lie algebra of a unimodular solvable Lie group having discrete series satisfies the Gelfand-Kirillov conjecture. (author). 6 refs
Little strings, quasi-topological sigma model on loop group, and toroidal Lie algebras
Directory of Open Access Journals (Sweden)
Meer Ashwinkumar
2018-03-01
Full Text Available We study the ground states and left-excited states of the Ak−1 N=(2,0 little string theory. Via a theorem by Atiyah [1], these sectors can be captured by a supersymmetric nonlinear sigma model on CP1 with target space the based loop group of SU(k. The ground states, described by L2-cohomology classes, form modules over an affine Lie algebra, while the left-excited states, described by chiral differential operators, form modules over a toroidal Lie algebra. We also apply our results to analyze the 1/2 and 1/4 BPS sectors of the M5-brane worldvolume theory.
Little strings, quasi-topological sigma model on loop group, and toroidal Lie algebras
Ashwinkumar, Meer; Cao, Jingnan; Luo, Yuan; Tan, Meng-Chwan; Zhao, Qin
2018-03-01
We study the ground states and left-excited states of the Ak-1 N = (2 , 0) little string theory. Via a theorem by Atiyah [1], these sectors can be captured by a supersymmetric nonlinear sigma model on CP1 with target space the based loop group of SU (k). The ground states, described by L2-cohomology classes, form modules over an affine Lie algebra, while the left-excited states, described by chiral differential operators, form modules over a toroidal Lie algebra. We also apply our results to analyze the 1/2 and 1/4 BPS sectors of the M5-brane worldvolume theory.
Group covariance and metrical theory
International Nuclear Information System (INIS)
Halpern, L.
1983-01-01
The a priori introduction of a Lie group of transformations into a physical theory has often proved to be useful; it usually serves to describe special simplified conditions before a general theory can be worked out. Newton's assumptions of absolute space and time are examples where the Euclidian group and translation group have been introduced. These groups were extended to the Galilei group and modified in the special theory of relativity to the Poincare group to describe physics under the given conditions covariantly in the simplest way. The criticism of the a priori character leads to the formulation of the general theory of relativity. The general metric theory does not really give preference to a particular invariance group - even the principle of equivalence can be adapted to a whole family of groups. The physical laws covariantly inserted into the metric space are however adapted to the Poincare group. 8 references
Infinite-dimensional Lie algebras in 4D conformal quantum field theory
International Nuclear Information System (INIS)
Bakalov, Bojko; Nikolov, Nikolay M; Rehren, Karl-Henning; Todorov, Ivan
2008-01-01
The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of two-dimensional chiral conformal field theory, to a higher (even) dimensional spacetime. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields, V M (x, y), where the M span a finite dimensional real matrix algebra M closed under transposition. The associative algebra M is irreducible iff its commutant M' coincides with one of the three real division rings. The Lie algebra of (the modes of) the bilocal fields is in each case an infinite-dimensional Lie algebra: a central extension of sp(∞,R) corresponding to the field R of reals, of u(∞, ∞) associated with the field C of complex numbers, and of so*(4∞) related to the algebra H of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups O(N), U(N) and U(N,H)=Sp(2N), respectively
Unipotent and nilpotent classes in simple algebraic groups and lie algebras
Liebeck, Martin W
2012-01-01
This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras. These topics have been an important area of study for decades, with applications to representation theory, character theory, the subgroup structure of algebraic groups and finite groups, and the classification of the finite simple groups. The main focus is on obtaining full information on class representatives and centralizers of unipotent and nilpotent elements. Although there is a substantial literature on this topic, this book is the first single source where such information is presented completely in all characteristics. In addition, many of the results are new--for example, those concerning centralizers of nilpotent elements in small characteristics. Indeed, the whole approach, while using some ideas from the literature, is novel, and yields many new general and specific facts concerning the structure and embeddings of...
Theory of the unitary representations of compact groups
International Nuclear Information System (INIS)
Burzynski, A.; Burzynska, M.
1979-01-01
An introduction contains some basic notions used in group theory, Lie group, Lie algebras and unitary representations. Then we are dealing with compact groups. For these groups we show the problem of reduction of unitary representation of Wigner's projection operators, Clebsch-Gordan coefficients and Wigner-Eckart theorem. We show (this is a new approach) the representations reduction formalism by using superoperators in Hilbert-Schmidt space. (author)
Construction of Difference Equations Using Lie Groups
International Nuclear Information System (INIS)
Axford, R.A.
1998-01-01
The theory of prolongations of the generators of groups of point transformations to the grid point values of dependent variables and grid spacings is developed and applied to the construction of group invariant numerical algorithms. The concepts of invariant difference operators and generalized discrete sources are introduced for the discretization of systems of inhomogeneous differential equations and shown to produce exact difference equations. Invariant numerical flux functions are constructed from the general solutions of first order partial differential equations that come out of the evaluation of the Lie derivatives of conservation forms of difference schemes. It is demonstrated that invariant numerical flux functions with invariant flux or slope limiters can be determined to yield high resolution difference schemes. The introduction of an invariant flux or slope limiter can be done so as not to break the symmetry properties of a numerical flux-function
K-theory for discrete subgroups of the Lorentz groups
International Nuclear Information System (INIS)
Schwalbe, D.A.
1986-01-01
In the thesis, a conjecture on the structure of the topological K theory groups associated to an action of a discrete group on a manifold is verified in the special case when the group is a closed discrete subgroup of a Lorentz group. The K theory is the topological K theory of the reduced crossed product C algebra arising from the action of a countable discrete group acting by diffeomorphisms on a smooth, Hausdorf, and second and countable manifold. The proof uses the geometric K theory of Baum and Connes. In this situation, they have developed a geometrically realized K theory which they conjecture to be isomorphic to the analytic K theory. Work of Kasparov is used to show the geometric K groups and the analytic K groups are isomorphic for actions of the Lorentz groups on a manifold. Work of Marc Rieffel on Morita equivalence of C/sup */ algebras, shows the analytic K theory for a closed discrete subgroup of a Lie group acting on a manifold is isomorphic to the K theory of the Lie group itself, acting on an induced manifold
Uncertainty Principles on Two Step Nilpotent Lie Groups
Indian Academy of Sciences (India)
Abstract. We extend an uncertainty principle due to Cowling and Price to two step nilpotent Lie groups, which generalizes a classical theorem of Hardy. We also prove an analogue of Heisenberg inequality on two step nilpotent Lie groups.
Group and representation theory
Vergados, J D
2017-01-01
This volume goes beyond the understanding of symmetries and exploits them in the study of the behavior of both classical and quantum physical systems. Thus it is important to study the symmetries described by continuous (Lie) groups of transformations. We then discuss how we get operators that form a Lie algebra. Of particular interest to physics is the representation of the elements of the algebra and the group in terms of matrices and, in particular, the irreducible representations. These representations can be identified with physical observables. This leads to the study of the classical Lie algebras, associated with unitary, unimodular, orthogonal and symplectic transformations. We also discuss some special algebras in some detail. The discussion proceeds along the lines of the Cartan-Weyl theory via the root vectors and root diagrams and, in particular, the Dynkin representation of the roots. Thus the representations are expressed in terms of weights, which are generated by the application of the elemen...
Anti-Kählerian Geometry on Lie Groups
Fernández-Culma, Edison Alberto; Godoy, Yamile
2018-03-01
Let G be a Lie group of even dimension and let ( g, J) be a left invariant anti-Kähler structure on G. In this article we study anti-Kähler structures considering the distinguished cases where the complex structure J is abelian or bi-invariant. We find that if G admits a left invariant anti-Kähler structure ( g, J) where J is abelian then the Lie algebra of G is unimodular and ( G, g) is a flat pseudo-Riemannian manifold. For the second case, we see that for any left invariant metric g for which J is an anti-isometry we obtain that the triple ( G, g, J) is an anti-Kähler manifold. Besides, given a left invariant anti-Hermitian structure on G we associate a covariant 3-tensor 𝜃 on its Lie algebra and prove that such structure is anti-Kähler if and only if 𝜃 is a skew-symmetric and pure tensor. From this tensor we classify the real 4-dimensional Lie algebras for which the corresponding Lie group has a left invariant anti-Kähler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-Kähler structures).
Lie Algebras Associated with Group U(n)
International Nuclear Information System (INIS)
Zhang Yufeng; Dong Huanghe; Honwah Tam
2007-01-01
Starting from the subgroups of the group U(n), the corresponding Lie algebras of the Lie algebra A 1 are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.
Non-coboundary Poisson–Lie structures on the book group
International Nuclear Information System (INIS)
Ballesteros, Ángel; Blasco, Alfonso; Musso, Fabio
2012-01-01
All possible Poisson–Lie (PL) structures on the 3D real Lie group generated by a dilation and two commuting translations are obtained. Their classification is fully performed by relating these PL groups to the corresponding Lie bialgebra structures on the corresponding ‘book’ Lie algebra. By construction, all these Poisson structures are quadratic Poisson–Hopf algebras for which the group multiplication is a Poisson map. In contrast to the case of simple Lie groups, it turns out that most of the PL structures on the book group are non-coboundary ones. Moreover, from the viewpoint of Poisson dynamics, the most interesting PL book structures are just some of these non-coboundaries, which are explicitly analysed. In particular, we show that the two different q-deformed Poisson versions of the sl(2, R) algebra appear as two distinguished cases in this classification, as well as the quadratic Poisson structure that underlies the integrability of a large class of 3D Lotka–Volterra equations. Finally, the quantization problem for these PL groups is sketched. (paper)
2015-01-01
This modern translation of Sophus Lie's and Friedrich Engel's “Theorie der Transformationsgruppen Band I” will allow readers to discover the striking conceptual clarity and remarkably systematic organizational thought of the original German text. Volume I presents a comprehensive introduction to the theory and is mainly directed towards the generalization of ideas drawn from the study of examples. The major part of the present volume offers an extremely clear translation of the lucid original. The first four chapters provide not only a translation, but also a contemporary approach, which will help present day readers to familiarize themselves with the concepts at the heart of the subject. The editor's main objective was to encourage a renewed interest in the detailed classification of Lie algebras in dimensions 1, 2 and 3, and to offer access to Sophus Lie's monumental Galois theory of continuous transformation groups, established at the end of the 19th Century. Lie groups are widespread in mathematics, p...
Lie group structures on automorphism groups of real-analytic CR manifolds
ZAITSEV, DMITRI
2008-01-01
PUBLISHED Given any real-analytic CR manifold M, we provide general conditions on M guar- anteeing that the group of all its global real-analytic CR automorphisms AutCR(M) is a Lie group (in an appropriate topology). In particular, we obtain a Lie group structure for AutCR(M) when M is an arbitrary compact real-analytic hypersurface embedded in some Stein manifold. The first author was supported by the Austrian Science Fund FWF, Project P17111 and Project P19667. The second ...
Quantum algebras as quantizations of dual Poisson–Lie groups
International Nuclear Information System (INIS)
Ballesteros, Ángel; Musso, Fabio
2013-01-01
A systematic computational approach for the explicit construction of any quantum Hopf algebra (U z (g), Δ z ) starting from the Lie bialgebra (g, δ) that gives the first-order deformation of the coproduct map Δ z is presented. The procedure is based on the well-known ‘quantum duality principle’, namely the fact that any quantum algebra can be viewed as the quantization of the unique Poisson–Lie structure (G*, Λ g ) on the dual group G*, which is obtained by exponentiating the Lie algebra g* defined by the dual map δ*. From this perspective, the coproduct for U z (g) is just the pull-back of the group law for G*, and the Poisson analogues of the quantum commutation rules for U z (g) are given by the unique Poisson–Lie structure Λ g on G* whose linearization is the Poisson analogue of the initial Lie algebra g. This approach is shown to be a very useful technical tool in order to solve the Lie bialgebra quantization problem explicitly since, once a Lie bialgebra (g, δ) is given, the full dual Poisson–Lie group (G*, Λ) can be obtained either by applying standard Poisson–Lie group techniques or by implementing the algorithm presented here with the aid of symbolic manipulation programs. As a consequence, the quantization of (G*, Λ) will give rise to the full U z (g) quantum algebra, provided that ordering problems are appropriately fixed through the choice of certain local coordinates on G* whose coproduct fulfils a precise ‘quantum symmetry’ property. The applicability of this approach is explicitly demonstrated by reviewing the construction of several instances of quantum deformations of physically relevant Lie algebras such as sl(2,R), the (2+1) anti-de Sitter algebra so(2, 2) and the Poincaré algebra in (3+1) dimensions. (paper)
Graded-Lie-algebra cohomology and supergravity
International Nuclear Information System (INIS)
D'Auria, R.; Fre, P.; Regge, T.
1980-01-01
Detailed explanations of the cohomology invoked in the group-manifold approach to supergravity is given. The Chevalley cohomology theory of Lie algebras is extended to graded Lie algebras. The scheme of geometrical theories is enlarged so to include cosmological terms and higher powers of the curvature. (author)
On approximation of Lie groups by discrete subgroups
Indian Academy of Sciences (India)
2016-08-26
Aug 26, 2016 ... The notion of approximation of Lie groups by discrete subgroups was introduced by Tôyama in Kodai Math. Sem. Rep. 1 (1949) 36–37 and investigated in detail by Kuranishi in Nagoya Math. J. 2 (1951) 63–71. It is known as a theorem of Tôyama that any connected Lie group approximated by discrete ...
An introduction to Lie group integrators – basics, new developments and applications
International Nuclear Information System (INIS)
Celledoni, Elena; Marthinsen, Håkon; Owren, Brynjulf
2014-01-01
We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion of discrete gradient methods is generalised to Lie groups
An introduction to tensors and group theory for physicists
Jeevanjee, Nadir
2011-01-01
An Introduction to Tensors and Group Theory for Physicists provides both an intuitive and rigorous approach to tensors and groups and their role in theoretical physics and applied mathematics. A particular aim is to demystify tensors and provide a unified framework for understanding them in the context of classical and quantum physics. Connecting the component formalism prevalent in physics calculations with the abstract but more conceptual formulation found in many mathematical texts, the work will be a welcome addition to the literature on tensors and group theory. Part I of the text begins with linear algebraic foundations, follows with the modern component-free definition of tensors, and concludes with applications to classical and quantum physics through the use of tensor products. Part II introduces abstract groups along with matrix Lie groups and Lie algebras, then intertwines this material with that of Part I by introducing representation theory. Exercises and examples are provided throughout for go...
International Nuclear Information System (INIS)
Halpern, L.
1981-01-01
Invariant varieties of suitable semisimple groups of transformations can serve as models of the space-time of the universe. The metric is expressible in terms of the basis vectors of the group. The symmetry of the group is broken by introducing a gauge formalism in the space of the basis vectors with the adjoint group as gauge group. The gauge potentials are expressible in terms of the basis vectors for the case of the De Sitter group. The resulting gauge theory is equivalent to De Sitter covariant general relativity. Group covariant generalizations of gravitational theory are discussed. (Auth.)
String Topology for Lie Groups
DEFF Research Database (Denmark)
A. Hepworth, Richard
2010-01-01
In 1999 Chas and Sullivan showed that the homology of the free loop space of an oriented manifold admits the structure of a Batalin-Vilkovisky algebra. In this paper we give a direct description of this Batalin-Vilkovisky algebra in the case that the manifold is a compact Lie group G. Our answer ...
Koszul information geometry and Souriau Lie group thermodynamics
Energy Technology Data Exchange (ETDEWEB)
Barbaresco, Frédéric, E-mail: frederic.barbaresco@thalesgroup.com
2015-01-13
The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from 'Characteristic Functions', was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of 'Information Geometry' theory, defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, Souriau has extended the Characteristic Function in Statistical Physics looking for other kinds of invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean 'Moment map' by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. These elements has been developed by author in [10][11].
The theory of nilpotent groups
Clement, Anthony E; Zyman, Marcos
2017-01-01
This monograph presents both classical and recent results in the theory of nilpotent groups and provides a self-contained, comprehensive reference on the topic. While the theorems and proofs included can be found throughout the existing literature, this is the first book to collect them in a single volume. Details omitted from the original sources, along with additional computations and explanations, have been added to foster a stronger understanding of the theory of nilpotent groups and the techniques commonly used to study them. Topics discussed include collection processes, normal forms and embeddings, isolators, extraction of roots, P-localization, dimension subgroups and Lie algebras, decision problems, and nilpotent groups of automorphisms. Requiring only a strong undergraduate or beginning graduate background in algebra, graduate students and researchers in mathematics will find The Theory of Nilpotent Groups to be a valuable resource.
Lie symmetries and differential galois groups of linear equations
Oudshoorn, W.R.; Put, M. van der
2002-01-01
For a linear ordinary differential equation the Lie algebra of its infinitesimal Lie symmetries is compared with its differential Galois group. For this purpose an algebraic formulation of Lie symmetries is developed. It turns out that there is no direct relation between the two above objects. In
MINAMI, Haruo
2016-01-01
For a compact simple Lie group $G$, we show that the element $[G, \\mathcal{L}] \\in \\pi^S_*(S^0)$ represented by the pair $(G, \\mathcal{L})$ is zero, where $\\mathcal{L}$ denotes the left invariant framing of $G$. The proof relies on the method of E. Ossa [Topology, 21 (1982), 315–323].
On the Lie symmetry group for classical fields in noncommutative space
Energy Technology Data Exchange (ETDEWEB)
Pereira, Ricardo Martinho Lima Santiago [Universidade Federal da Bahia (UFBA), BA (Brazil); Instituto Federal da Bahia (IFBA), BA (Brazil); Ressureicao, Caio G. da [Universidade Federal da Bahia (UFBA), BA (Brazil). Inst. de Fisica; Vianna, Jose David M. [Universidade Federal da Bahia (UFBA), BA (Brazil); Universidade de Brasilia (UnB), DF (Brazil)
2011-07-01
Full text: An alternative way to include effects of noncommutative geometries in field theory is based on the concept of noncommutativity among degrees of freedom of the studied system. In this context it is reasonable to consider that, in the multiparticle noncommutative quantum mechanics (NCQM), the noncommutativity among degrees of freedom to discrete system with N particles is also verified. Further, an analysis of the classical limit of the single particle NCQM leads to a deformed Newtonian mechanics where the Newton's second law is modified in order to include the noncommutative parameter {theta}{sub {iota}j} and, for a one-dimensional discrete system with N particles, the dynamical evolution of each particle is given by this modified Newton's second law. Hence, applying the continuous limit to this multiparticle classical system it is possible to obtain a noncommutative extension of two -dimensional field theory in a noncommutative space. In the present communication we consider a noncommutative extension of the scalar field obtained from this approach and we analyze the Lie symmetries in order to compare the Lie group of this field with the usual scalar field in the commutative space. (author)
Motivation and Consequences of Lying. A Qualitative Analysis of Everyday Lying
Directory of Open Access Journals (Sweden)
Beata Arcimowicz
2015-09-01
Full Text Available This article presents findings of qualitative analysis of semi-structured interviews with a group of "frequent liars" and another of "rare liars" who provided their subjective perspectives on the phenomenon of lying. Participants in this study previously had maintained a diary of their social interactions and lies over the course of one week, which allowed to assign them to one of the two groups: frequent or rare liars. Thematic analysis of the material followed by elements of theory formulation resulted in an extended lying typology that includes not only the target of the lie (the liar vs. other but also the motivation (protection vs. bringing benefits. We offer an analysis of what prevents from telling the truth, i.e. penalties, relationship losses, distress of the lied-to, and anticipated lack of criticism for telling the truth. We also focus on understanding moderatorsof consequences of lying (significance of the area of life, the type of lie and capacity to understand the liar that can be useful in future studies. URN: http://nbn-resolving.de/urn:nbn:de:0114-fqs1503318
Controllability of linear vector fields on Lie groups
International Nuclear Information System (INIS)
Ayala, V.; Tirao, J.
1994-11-01
In this paper, we shall deal with a linear control system Σ defined on a Lie group G with Lie algebra g. The dynamic of Σ is determined by the drift vector field which is an element in the normalizer of g in the Lie algebra of all smooth vector field on G and by the control vectors which are elements in g considered as left-invariant vector fields. We characterize the normalizer of g identifying vector fields on G with C ∞ -functions defined on G into g. For this class of control systems we study algebraic conditions for the controllability problem. Indeed, we prove that if the drift vector field has a singularity then the Lie algebra rank condition is necessary for the controllability property, but in general this condition does not determine this property. On the other hand, we show that the rank (ad-rank) condition is sufficient for the controllability of Σ. In particular, we extend the fundamental Kalman's theorem when G is an Abelian connected Lie group. Our work is related with a paper of L. Markus and we also improve his results. (author). 7 refs
Transformation groups and Lie algebras
Ibragimov, Nail H
2013-01-01
This book is based on the extensive experience of teaching for mathematics, physics and engineering students in Russia, USA, South Africa and Sweden. The author provides students and teachers with an easy to follow textbook spanning a variety of topics. The methods of local Lie groups discussed in the book provide universal and effective method for solving nonlinear differential equations analytically. Introduction to approximate transformation groups also contained in the book helps to develop skills in constructing approximate solutions for differential equations with a small parameter.
Exponentiation and deformations of Lie-admissible algebras
International Nuclear Information System (INIS)
Myung, H.C.
1982-01-01
The exponential function is defined for a finite-dimensional real power-associative algebra with unit element. The application of the exponential function is focused on the power-associative (p,q)-mutation of a real or complex associative algebra. Explicit formulas are computed for the (p,q)-mutation of the real envelope of the spin 1 algebra and the Lie algebra so(3) of the rotation group, in light of earlier investigations of the spin 1/2. A slight variant of the mutated exponential is interpreted as a continuous function of the Lie algebra into some isotope of the corresponding linear Lie group. The second part of this paper is concerned with the representation and deformation of a Lie-admissible algebra. The second cohomology group of a Lie-admissible algebra is introduced as a generalization of those of associative and Lie algebras in the Hochschild and Chevalley-Eilenberg theory. Some elementary theory of algebraic deformation of Lie-admissible algebras is discussed in view of generalization of that of associative and Lie algebras. Lie-admissible deformations are also suggested by the representation of Lie-admissible algebras. Some explicit examples of Lie-admissible deformation are given in terms of the (p,q)-mutation of associative deformation of an associative algebra. Finally, we discuss Lie-admissible deformations of order one
Gradings on simple Lie algebras
Elduque, Alberto
2013-01-01
Gradings are ubiquitous in the theory of Lie algebras, from the root space decomposition of a complex semisimple Lie algebra relative to a Cartan subalgebra to the beautiful Dempwolff decomposition of E_8 as a direct sum of thirty-one Cartan subalgebras. This monograph is a self-contained exposition of the classification of gradings by arbitrary groups on classical simple Lie algebras over algebraically closed fields of characteristic not equal to 2 as well as on some nonclassical simple Lie algebras in positive characteristic. Other important algebras also enter the stage: matrix algebras, the octonions, and the Albert algebra. Most of the presented results are recent and have not yet appeared in book form. This work can be used as a textbook for graduate students or as a reference for researchers in Lie theory and neighboring areas.
The role of executive functions and theory of mind in children's prosocial lie-telling.
Williams, Shanna; Moore, Kelsey; Crossman, Angela M; Talwar, Victoria
2016-01-01
Children's prosocial lying was examined in relation to executive functioning skills and theory of mind development. Prosocial lying was observed using a disappointing gift paradigm. Of the 79 children (ages 6-12 years) who completed the disappointing gift paradigm, 47 (59.5%) told a prosocial lie to a research assistant about liking their prize. In addition, of those children who told prosocial lies, 25 (53.2%) maintained semantic leakage control during follow-up questioning, thereby demonstrating advanced lie-telling skills. When executive functioning was examined, children who told prosocial lies were found to have significantly higher performance on measures of working memory and inhibitory control. In addition, children who lied and maintained semantic leakage control also displayed more advanced theory of mind understanding. Although children's age was not a predictor of lie-telling behavior (i.e., truthful vs. lie-teller), age was a significant predictor of semantic leakage control, with older children being more likely to maintain their lies during follow-up questioning. Copyright © 2015 Elsevier Inc. All rights reserved.
Directory of Open Access Journals (Sweden)
Frédéric Barbaresco
2016-11-01
Full Text Available We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie group thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant–Kirillov–Souriau 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of information geometry for the action of an affine group for exponential families, and provide some illustrations of use cases for multivariate gaussian densities. Information geometry is presented in the context of the seminal work of Fréchet and his Clairaut-Legendre equation. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.
Renormalization group flows and continual Lie algebras
International Nuclear Information System (INIS)
Bakas, Ioannis
2003-01-01
We study the renormalization group flows of two-dimensional metrics in sigma models using the one-loop beta functions, and demonstrate that they provide a continual analogue of the Toda field equations in conformally flat coordinates. In this algebraic setting, the logarithm of the world-sheet length scale, t, is interpreted as Dynkin parameter on the root system of a novel continual Lie algebra, denoted by (d/dt;1), with anti-symmetric Cartan kernel K(t,t') = δ'(t-t'); as such, it coincides with the Cartan matrix of the superalgebra sl(N vertical bar N+1) in the large-N limit. The resulting Toda field equation is a non-linear generalization of the heat equation, which is integrable in target space and shares the same dissipative properties in time, t. We provide the general solution of the renormalization group flows in terms of free fields, via Baecklund transformations, and present some simple examples that illustrate the validity of their formal power series expansion in terms of algebraic data. We study in detail the sausage model that arises as geometric deformation of the O(3) sigma model, and give a new interpretation to its ultra-violet limit by gluing together two copies of Witten's two-dimensional black hole in the asymptotic region. We also provide some new solutions that describe the renormalization group flow of negatively curved spaces in different patches, which look like a cane in the infra-red region. Finally, we revisit the transition of a flat cone C/Z n to the plane, as another special solution, and note that tachyon condensation in closed string theory exhibits a hidden relation to the infinite dimensional algebra (d/dt;1) in the regime of gravity. Its exponential growth holds the key for the construction of conserved currents and their systematic interpretation in string theory, but they still remain unknown. (author)
An introduction to tensors and group theory for physicists
Jeevanjee, Nadir
2015-01-01
The second edition of this highly praised textbook provides an introduction to tensors, group theory, and their applications in classical and quantum physics. Both intuitive and rigorous, it aims to demystify tensors by giving the slightly more abstract but conceptually much clearer definition found in the math literature, and then connects this formulation to the component formalism of physics calculations. New pedagogical features, such as new illustrations, tables, and boxed sections, as well as additional “invitation” sections that provide accessible introductions to new material, offer increased visual engagement, clarity, and motivation for students. Part I begins with linear algebraic foundations, follows with the modern component-free definition of tensors, and concludes with applications to physics through the use of tensor products. Part II introduces group theory, including abstract groups and Lie groups and their associated Lie algebras, then intertwines this material with that of Part...
Non-commutative representation for quantum systems on Lie groups
Energy Technology Data Exchange (ETDEWEB)
Raasakka, Matti Tapio
2014-01-27
The topic of this thesis is a new representation for quantum systems on weakly exponential Lie groups in terms of a non-commutative algebra of functions, the associated non-commutative harmonic analysis, and some of its applications to specific physical systems. In the first part of the thesis, after a review of the necessary mathematical background, we introduce a {sup *}-algebra that is interpreted as the quantization of the canonical Poisson structure of the cotangent bundle over a Lie group. From the physics point of view, this represents the algebra of quantum observables of a physical system, whose configuration space is a Lie group. We then show that this quantum algebra can be represented either as operators acting on functions on the group, the usual group representation, or (under suitable conditions) as elements of a completion of the universal enveloping algebra of the Lie group, the algebra representation. We further apply the methods of deformation quantization to obtain a representation of the same algebra in terms of a non-commutative algebra of functions on a Euclidean space, which we call the non-commutative representation of the original quantum algebra. The non-commutative space that arises from the construction may be interpreted as the quantum momentum space of the physical system. We derive the transform between the group representation and the non-commutative representation that generalizes in a natural way the usual Fourier transform, and discuss key properties of this new non-commutative harmonic analysis. Finally, we exhibit the explicit forms of the non-commutative Fourier transform for three elementary Lie groups: R{sup d}, U(1) and SU(2). In the second part of the thesis, we consider application of the non-commutative representation and harmonic analysis to physics. First, we apply the formalism to quantum mechanics of a point particle on a Lie group. We define the dual non-commutative momentum representation, and derive the phase
Non-commutative representation for quantum systems on Lie groups
International Nuclear Information System (INIS)
Raasakka, Matti Tapio
2014-01-01
The topic of this thesis is a new representation for quantum systems on weakly exponential Lie groups in terms of a non-commutative algebra of functions, the associated non-commutative harmonic analysis, and some of its applications to specific physical systems. In the first part of the thesis, after a review of the necessary mathematical background, we introduce a * -algebra that is interpreted as the quantization of the canonical Poisson structure of the cotangent bundle over a Lie group. From the physics point of view, this represents the algebra of quantum observables of a physical system, whose configuration space is a Lie group. We then show that this quantum algebra can be represented either as operators acting on functions on the group, the usual group representation, or (under suitable conditions) as elements of a completion of the universal enveloping algebra of the Lie group, the algebra representation. We further apply the methods of deformation quantization to obtain a representation of the same algebra in terms of a non-commutative algebra of functions on a Euclidean space, which we call the non-commutative representation of the original quantum algebra. The non-commutative space that arises from the construction may be interpreted as the quantum momentum space of the physical system. We derive the transform between the group representation and the non-commutative representation that generalizes in a natural way the usual Fourier transform, and discuss key properties of this new non-commutative harmonic analysis. Finally, we exhibit the explicit forms of the non-commutative Fourier transform for three elementary Lie groups: R d , U(1) and SU(2). In the second part of the thesis, we consider application of the non-commutative representation and harmonic analysis to physics. First, we apply the formalism to quantum mechanics of a point particle on a Lie group. We define the dual non-commutative momentum representation, and derive the phase space path
Algebras of Complete Hörmander Vector Fields, and Lie-Group Construction
Directory of Open Access Journals (Sweden)
Andrea Bonfiglioli
2014-12-01
Full Text Available The aim of this note is to characterize the Lie algebras g of the analytic vector fields in RN which coincide with the Lie algebras of the (analytic Lie groups defined on RN (with its usual differentiable structure. We show that such a characterization amounts to asking that: (i g is N-dimensional; (ii g admits a set of Lie generators which are complete vector fields; (iii g satisfies Hörmander’s rank condition. These conditions are necessary, sufficient and mutually independent. Our approach is constructive, in that for any such g we show how to construct a Lie group G = (RN, * whose Lie algebra is g. We do not make use of Lie’s Third Theorem, but we only exploit the Campbell-Baker-Hausdorff-Dynkin Theorem for ODE’s.
Riesz transforms and Lie groups of polynomial growth
Elst, ter A.F.M.; Robinson, D.W.; Sikora, A.
1999-01-01
Let G be a Lie group of polynomial growth. We prove that the second-order Riesz transforms onL2(G; dg) are bounded if, and only if, the group is a direct product of a compact group and a nilpotent group, in which case the transforms of all orders are bounded.
Bounds on the number of possible Higgs particles using grand unification and exceptional Lie groups
International Nuclear Information System (INIS)
El Naschie, M.S.
2008-01-01
The total sum of dimensions of a magnum exceptional Lie symmetry groups hierarchy is 4α-bar o =(4)(137+k o )≅548. Dividing this value among the various quantum fields leads to the possibility of an eight degrees of freedom Higgs field. However analyzing the same situation using sub groups of the largest exceptional Lie group leads to the conclusion that we are likely to find three Higgs particles only at the energy scale of the standard model. Consequently five of the eight degrees of freedom are unlikely to materialize as particles at this particular energy scale. This conclusion is reinforced by an entirely different approach based on grand unification analysis which excludes any grand unification using 4HD, i.e. four Higgs doublets. This leaves us with one, two and three Higgs doublets. Noting that a super symmetric standard model with two Higgs doublets gives almost perfect grand unification and that the result agrees with our exceptional Lie symmetry groups analysis, we exclude everything else. The final result is that we expect to find at least three more Higgs particles leading to a total of 66 elementary particles while at a somewhat higher energy, the expected number of 69 particles found using E-infinity theory is obtained
Lie Group Classifications and Non-differentiable Solutions for Time-Fractional Burgers Equation
International Nuclear Information System (INIS)
Wu Guocheng
2011-01-01
Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)
Ma, Fengling; Evans, Angela D.; Liu, Ying; Luo, Xianming; Xu, Fen
2015-01-01
Prior studies have demonstrated that social-cognitive factors such as children's false-belief understanding and parenting style are related to children's lie-telling behaviors. The present study aimed to investigate how earlier forms of theory-of-mind understanding contribute to children's lie-telling as well as how parenting practices are related…
Statistics on Lie groups: A need to go beyond the pseudo-Riemannian framework
Miolane, Nina; Pennec, Xavier
2015-01-01
Lie groups appear in many fields from Medical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ's shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group G is a manifold that carries an additional group structure. Statistics on Riemannian manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall [1, 2, 3, 4] followed by others [5, 6, 7, 8, 9]. In order to use such a Riemannian structure for statistics on Lie groups, one needs to define a Riemannian metric that is compatible with the group structure, i.e a bi-invariant metric. However, it is well known that general Lie groups which cannot be decomposed into the direct product of compact and abelian groups do not admit a bi-invariant metric. One may wonder if removing the positivity of the metric, thus asking only for a bi-invariant pseudo-Riemannian metric, would be sufficient for most of the groups used in Computational Anatomy. In this paper, we provide an algorithmic procedure that constructs bi-invariant pseudo-metrics on a given Lie group G. The procedure relies on a classification theorem of Medina and Revoy. However in doing so, we prove that most Lie groups do not admit any bi-invariant (pseudo-) metric. We conclude that the (pseudo-) Riemannian setting is not the richest setting if one wants to perform statistics on Lie groups. One may have to rely on another framework, such as affine connection space.
Linear algebra and group theory for physicists
Rao, K N Srinivasa
2006-01-01
Professor Srinivasa Rao's text on Linear Algebra and Group Theory is directed to undergraduate and graduate students who wish to acquire a solid theoretical foundation in these mathematical topics which find extensive use in physics. Based on courses delivered during Professor Srinivasa Rao's long career at the University of Mysore, this text is remarkable for its clear exposition of the subject. Advanced students will find a range of topics such as the Representation theory of Linear Associative Algebras, a complete analysis of Dirac and Kemmer algebras, Representations of the Symmetric group via Young Tableaux, a systematic derivation of the Crystallographic point groups, a comprehensive and unified discussion of the Rotation and Lorentz groups and their representations, and an introduction to Dynkin diagrams in the classification of Lie groups. In addition, the first few chapters on Elementary Group Theory and Vector Spaces also provide useful instructional material even at an introductory level. An author...
International Nuclear Information System (INIS)
Bonora, Loriano; Bytsenko, Andrey; Elizalde, Emilio
2012-01-01
This review paper contains a concise introduction to highest weight representations of infinite-dimensional Lie algebras, vertex operator algebras and Hilbert schemes of points, together with their physical applications to elliptic genera of superconformal quantum mechanics and superstring models. The common link of all these concepts and of the many examples considered in this paper is to be found in a very important feature of the theory of infinite-dimensional Lie algebras: the modular properties of the characters (generating functions) of certain representations. The characters of the highest weight modules represent the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. We discuss the role of the unimodular (and modular) groups and the (Selberg-type) Ruelle spectral functions of hyperbolic geometry in the calculation of elliptic genera and associated q-series. For mathematicians, elliptic genera are commonly associated with new mathematical invariants for spaces, while for physicists elliptic genera are one-loop string partition function. (Therefore, they are applicable, for instance, to topological Casimir effect calculations.) We show that elliptic genera can be conveniently transformed into product expressions, which can then inherit the homology properties of appropriate polygraded Lie algebras. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker’s 75th birthday devoted to ‘Applications of zeta functions and other spectral functions in mathematics and physics’. (review)
Special functions and the theory of group representations
Vilenkin, N Ja
1968-01-01
A standard scheme for a relation between special functions and group representation theory is the following: certain classes of special functions are interpreted as matrix elements of irreducible representations of a certain Lie group, and then properties of special functions are related to (and derived from) simple well-known facts of representation theory. The book combines the majority of known results in this direction. In particular, the author describes connections between the exponential functions and the additive group of real numbers (Fourier analysis), Legendre and Jacobi polynomials and representations of the group SU(2), and the hypergeometric function and representations of the group SL(2,R), as well as many other classes of special functions.
Chern-Simons theory, 2d Yang-Mills, and Lie algebra wanderers
International Nuclear Information System (INIS)
Haro, Sebastian de
2005-01-01
We work out the relation between Chern-Simons, 2d Yang-Mills on the cylinder, and Brownian motion. We show that for the unitary, orthogonal and symplectic groups, various observables in Chern-Simons theory on S 3 and lens spaces are exactly given by counting the number of paths of a Brownian particle wandering in the fundamental Weyl chamber of the corresponding Lie algebra. We construct a fermionic formulation of Chern-Simons on S 3 which allows us to identify the Brownian particles as B-model branes moving on a noncommutative two-sphere, and construct 1- and 2-matrix models to compute Brownian motion ensemble averages
Application of Lie group analysis in geophysical fluid dynamics
Ibragimov, Ranis
2011-01-01
This is the first monograph dealing with the applications of the Lie group analysis to the modeling equations governing internal wave propagation in the deep ocean. A new approach to describe the nonlinear interactions of internal waves in the ocean is presented. While the central idea of the book is to investigate oceanic internal waves through the prism of Lie group analysis, it is also shown for the first time that internal wave beams, representing exact solutions to the equation of motion of stratified fluid, can be found by solving the given model as invariant solutions of nonlinear equat
Quantum theory, groups and representations an introduction
Woit, Peter
2017-01-01
This text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful to the reader, no specific ...
International Nuclear Information System (INIS)
Wu Ming-Zhong; Bai Cheng-Ming
2015-01-01
A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket. We construct a bialgebra theory of compatible Lie algebras as an analogue of a Lie bialgebra. They can also be regarded as a “compatible version” of Lie bialgebras, that is, a pair of Lie bialgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra. Many properties of compatible Lie bialgebras as the “compatible version” of the corresponding properties of Lie bialgebras are presented. In particular, there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang–Baxter equation in compatible Lie algebras as a combination of two classical Yang–Baxter equations in Lie algebras. Furthermore, a notion of compatible pre-Lie algebra is introduced with an interpretation of its close relation with the classical Yang–Baxter equation in compatible Lie algebras which leads to a construction of the solutions of the latter. As a byproduct, the compatible Lie bialgebras fit into the framework to construct non-constant solutions of the classical Yang–Baxter equation given by Golubchik and Sokolov. (paper)
Observability of linear control systems on Lie groups
International Nuclear Information System (INIS)
Ayala, V.; Hacibekiroglu, A.K.
1995-01-01
In this paper, we study the observability problem for a linear control system Σ on a Lie group G. The drift vector field of Σ is an infinitesimal automorphism of G and the control vectors are elements in the Lie algebra of G. We establish algebraic conditions to characterize locally and globally observability for Σ. As in the linear case on R n , these conditions are independent of the control vector. We give an algorithm on the co-tangent bundle of G to calculate the equivalence class of the neutral element. (author). 6 refs
Valued Graphs and the Representation Theory of Lie Algebras
Directory of Open Access Journals (Sweden)
Joel Lemay
2012-07-01
Full Text Available Quivers (directed graphs, species (a generalization of quivers and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this paper, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field. Namely, we show that the category of K -species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K -species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.
Symmetry breaking in fluid dynamics: Lie group reducible motions for real fluids
International Nuclear Information System (INIS)
Holm, D.D.
1976-07-01
The physics of fluids is based on certain kinematical invariance principles, which refer to coordinate systems, dimensions, and Galilean reference frames. Other, thermodynamic, symmetry principles are introduced by the material description. In the present work, the interplay between these two kinds of invariance principles is used to solve for classes of one-dimensional non-steady isentropic motions of a fluid whose equation of state is of Mie-Gruneisen type. Also, the change in profile and attenuation of weak shock waves in a dissipative medium is studied at the level of Burgers' approximation from the viewpoint of its underlying symmetry structure. The mathematical method of approach is based on the theory of infinitesimal Lie groups. Fluid motions are characterized according to inequivalent subgroups of the full invariance group of the flow description and exact group reducible solutions are presented
Symmetry breaking in fluid dynamics: Lie group reducible motions for real fluids
Energy Technology Data Exchange (ETDEWEB)
Holm, D.D.
1976-07-01
The physics of fluids is based on certain kinematical invariance principles, which refer to coordinate systems, dimensions, and Galilean reference frames. Other, thermodynamic, symmetry principles are introduced by the material description. In the present work, the interplay between these two kinds of invariance principles is used to solve for classes of one-dimensional non-steady isentropic motions of a fluid whose equation of state is of Mie-Gruneisen type. Also, the change in profile and attenuation of weak shock waves in a dissipative medium is studied at the level of Burgers' approximation from the viewpoint of its underlying symmetry structure. The mathematical method of approach is based on the theory of infinitesimal Lie groups. Fluid motions are characterized according to inequivalent subgroups of the full invariance group of the flow description and exact group reducible solutions are presented.
International Nuclear Information System (INIS)
El Naschie, M.S.
2008-01-01
The short note gives a derivation for a new E12 exceptional Lie group corresponding to affine KAC-Moody algebra. We derive the dimension of the group by intersectionally embedding the intrinsic dimension of E8 namely D(E8) = 57 into the 12 spacetime dimensions of F theory and finding that Dim E12 = D(E8) (DF) + 1 = (57)(12) + 1 = 685
Differential calculus on quantized simple Lie groups
International Nuclear Information System (INIS)
Jurco, B.
1991-01-01
Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q ε R are also discussed. (orig.)
International Nuclear Information System (INIS)
Dobrev, V.K.
1986-11-01
Let G be a real linear connected semisimple Lie group. We present a canonical construction of the differential operators intertwining elementary (≡ generalized principal series) representations of G. The results are easily extended to real linear reductive Lie groups. (author). 20 refs
6th Hilbert's problem and S.Lie's infinite groups
International Nuclear Information System (INIS)
Konopleva, N.P.
1999-01-01
The progress in Hilbert's sixth problem solving is demonstrated. That became possible thanks to the gauge field theory in physics and to the geometrical treatment of the gauge fields. It is shown that the fibre bundle spaces geometry is the best basis for solution of the problem being discussed. This talk has been reported at the International Seminar '100 Years after Sophus Lie' (Leipzig, Germany)
Unitary representations of some infinite-dimensional Lie algebras motivated by string theory on AdS3
International Nuclear Information System (INIS)
Andreev, Oleg
1999-01-01
We consider some unitary representations of infinite-dimensional Lie algebras motivated by string theory on AdS 3 . These include examples of two kinds: the A,D,E type affine Lie algebras and the N=4 superconformal algebra. The first presents a new construction for free field representations of affine Lie algebras. The second is of a particular physical interest because it provides some hints that a hybrid of the NSR and GS formulations for string theory on AdS 3 exists
Lie group classification and exact solutions of the generalized Kompaneets equations
Directory of Open Access Journals (Sweden)
Oleksii Patsiuk
2015-04-01
Full Text Available We study generalized Kompaneets equations (GKEs with one functional parameter, and using the Lie-Ovsiannikov algorithm, we carried out the group classification. It is shown that the kernel algebra of the full groups of the GKEs is the one-dimensional Lie algebra. Using the direct method, we find the equivalence group. We obtain six non-equivalent (up to transformations from the equivalence group GKEs that allow wider invariance algebras than the kernel one. We find a number of exact solutions of the non-linear GKE which has the maximal symmetry properties.
Masip, Jaume; Blandón-Gitlin, Iris; de la Riva, Clara; Herrero, Carmen
2016-09-01
Meta-analyses reveal that behavioral differences between liars and truth tellers are small. To facilitate lie detection, researchers are currently developing interviewing approaches to increase these differences. Some of these approaches assume that lying is cognitively more difficult than truth telling; however, they are not based on specific cognitive theories of lie production, which are rare. Here we examined one existing theory, Walczyk et al.'s (2014) Activation-Decision-Construction-Action Theory (ADCAT). We tested the Decision component. According to ADCAT, people decide whether to lie or tell the truth as if they were using a specific mathematical formula to calculate the motivation to lie from (a) the probability of a number of outcomes derived from lying vs. telling the truth, and (b) the costs/benefits associated with each outcome. In this study, participants read several hypothetical scenarios and indicated whether they would lie or tell the truth in each scenario (Questionnaire 1). Next, they answered several questions about the consequences of lying vs. telling the truth in each scenario, and rated the probability and valence of each consequence (Questionnaire 2). Significant associations were found between the participants' dichotomous decision to lie/tell the truth in Questionnaire 1 and their motivation to lie scores calculated from the Questionnaire 2 data. However, interestingly, whereas the expected consequences of truth telling were associated with the decision to lie vs. tell the truth, the expected consequences of lying were not. Suggestions are made to refine ADCAT, which can be a useful theoretical framework to guide deception research. Copyright © 2016 Elsevier B.V. All rights reserved.
Theory-of-Mind Training Causes Honest Young Children to Lie.
Ding, Xiao Pan; Wellman, Henry M; Wang, Yu; Fu, Genyue; Lee, Kang
2015-11-01
Theory of mind (ToM) has long been recognized to play a major role in children's social functioning. However, no direct evidence confirms the causal linkage between the two. In the current study, we addressed this significant gap by examining whether ToM causes the emergence of lying, an important social skill. We showed that after participating in ToM training to learn about mental-state concepts, 3-year-olds who originally had been unable to lie began to deceive consistently. This training effect lasted for more than a month. In contrast, 3-year-olds who participated in control training to learn about physical concepts were significantly less inclined to lie than the ToM-trained children. These findings provide the first experimental evidence supporting the causal role of ToM in the development of social competence in early childhood. © The Author(s) 2015.
Differential calculus on quantized simple Lie groups
Energy Technology Data Exchange (ETDEWEB)
Jurco, B. (Dept. of Optics, Palacky Univ., Olomouc (Czechoslovakia))
1991-07-01
Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU{sub q}(2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q {epsilon} R are also discussed. (orig.).
On approximation of Lie groups by discrete subgroups
Indian Academy of Sciences (India)
1Department of Mathematics, Faculty of Sciences at Sfax, University of Sfax,. Route Soukra ... Let S (G) denote the space of discrete co-compact subgroup of a Lie group G. We ..... For example, it suffices to apply the following fact: The mapping.
Olver, Peter J; the American Mathematical Society on Lie Algebras, Cohomology and New Applications to Quantum Mechanics
1994-01-01
This volume is devoted to a range of important new ideas arising in the applications of Lie groups and Lie algebras to Schrödinger operators and associated quantum mechanical systems. In these applications, the group does not appear as a standard symmetry group, but rather as a "hidden" symmetry group whose representation theory can still be employed to analyze at least part of the spectrum of the operator. In light of the rapid developments in this subject, a Special Session was organized at the AMS meeting at Southwest Missouri State University in March 1992 in order to bring together, perhaps for the first time, mathematicians and physicists working in closely related areas. The contributions to this volume cover Lie group methods, Lie algebras and Lie algebra cohomology, representation theory, orthogonal polynomials, q-series, conformal field theory, quantum groups, scattering theory, classical invariant theory, and other topics. This volume, which contains a good balance of research and survey papers, p...
Analytic factorization of Lie group representations
DEFF Research Database (Denmark)
Gimperlein, Heiko; Krötz, Bernhard; Lienau, Christoph
2012-01-01
For every moderate growth representation (p,E)(p,E) of a real Lie group G on a Fréchet space, we prove a factorization theorem of Dixmier–Malliavin type for the space of analytic vectors E¿E¿. There exists a natural algebra of superexponentially decreasing analytic functions A(G)A(G), such that E......¿=¿(A(G))E¿E¿=¿(A(G))E¿. As a corollary we obtain that E¿E¿ coincides with the space of analytic vectors for the Laplace–Beltrami operator on G....
Simple Lie groups without the approximation property
DEFF Research Database (Denmark)
Haagerup, Uffe; de Laat, Tim
2013-01-01
For a locally compact group G, let A(G) denote its Fourier algebra, and let M0A(G) denote the space of completely bounded Fourier multipliers on G. The group G is said to have the Approximation Property (AP) if the constant function 1 can be approximated by a net in A(G) in the weak-∗ topology...... on the space M0A(G). Recently, Lafforgue and de la Salle proved that SL(3,R) does not have the AP, implying the first example of an exact discrete group without it, namely, SL(3,Z). In this paper we prove that Sp(2,R) does not have the AP. It follows that all connected simple Lie groups with finite center...
Nonparametric Second-Order Theory of Error Propagation on Motion Groups.
Wang, Yunfeng; Chirikjian, Gregory S
2008-01-01
Error propagation on the Euclidean motion group arises in a number of areas such as in dead reckoning errors in mobile robot navigation and joint errors that accumulate from the base to the distal end of kinematic chains such as manipulators and biological macromolecules. We address error propagation in rigid-body poses in a coordinate-free way. In this paper we show how errors propagated by convolution on the Euclidean motion group, SE(3), can be approximated to second order using the theory of Lie algebras and Lie groups. We then show how errors that are small (but not so small that linearization is valid) can be propagated by a recursive formula derived here. This formula takes into account errors to second-order, whereas prior efforts only considered the first-order case. Our formulation is nonparametric in the sense that it will work for probability density functions of any form (not only Gaussians). Numerical tests demonstrate the accuracy of this second-order theory in the context of a manipulator arm and a flexible needle with bevel tip.
Mapping Spaces, Centralizers, and p-Local Finite Groups of Lie Type
DEFF Research Database (Denmark)
Laude, Isabelle
We study the space of maps from the classifying space of a finite p-group to theBorel construction of a finite group of Lie type G in characteristic p acting on itsbuilding. The first main result is a description of the homology with Fp-coefficients,showing that the mapping space, up to p...... between a finite p-group and theuncompleted classifying space of the p-local finite group coming from a finite groupof Lie type in characteristic p, providing some of the first results in this uncompletedsetting.......-completion, is a disjoint union indexedover the group homomorphism up to conjugation of classifying spaces of centralizersof p-subgroups in the underlying group G. We complement this description bydetermining the actual homotopy groups of the mapping space. These resultstranslate to descriptions of the space of maps...
Cluster X-varieties, amalgamation, and Poisson-Lie groups
DEFF Research Database (Denmark)
Fock, V. V.; Goncharov, A. B.
2006-01-01
In this paper, starting from a split semisimple real Lie group G with trivial center, we define a family of varieties with additional structures. We describe them as cluster χ-varieties, as defined in [FG2]. In particular they are Poisson varieties. We define canonical Poisson maps of these varie...
Diagram Techniques in Group Theory
Stedman, Geoffrey E.
2009-09-01
Preface; 1. Elementary examples; 2. Angular momentum coupling diagram techniques; 3. Extension to compact simple phase groups; 4. Symmetric and unitary groups; 5. Lie groups and Lie algebras; 6. Polarisation dependence of multiphoton processes; 7. Quantum field theoretic diagram techniques for atomic systems; 8. Applications; Appendix; References; Indexes.
Quantization and harmonic analysis on nilpotent Lie groups
International Nuclear Information System (INIS)
Wildberger, N.J.
1983-01-01
Weyl Quantization is a procedure for associating a function on which the canonical commutation relations are realized. If G is a simply-connected, connected nilpotent Lie group with Lie algebra g and dual g/sup */, it is shown how to inductively construct symplectic isomorphisms between every co-adjoint orbit O and the bundle in Hilbert Space for some m. Weyl Quantization can then be used to associate to each orbit O a unitary representation rho 0 of G, recovering the classification of the unitary dual by Kirillov. It is used to define a geometric Fourier transform, F : L 1 (G) → functions on g/sup */, and it is shown that the usual operator-valued Fourier transform can be recovered from F, characters are inverse Fourier transforms of invariant measures on orbits, and matrix coefficients are inverse Fourier transforms of non-invariant measures supported on orbits. Realizations of the representations rho 0 in subspaces of L 2 (O) are obtained.. Finally, the kernel function is computed for the upper triangular unipotent group and one other example
Unbounded representations of symmetry groups in gauge quantum field theory. II. Integration
International Nuclear Information System (INIS)
Voelkel, A.H.
1986-01-01
Within the gauge quantum field theory of the Wightman--Garding type, the integration of representations of Lie algebras is investigated. By means of the covariance condition (substitution rules) for the basic fields, it is shown that a form skew-symmetric representation of a Lie algebra can be integrated to a form isometric and in general unbounded representation of the universal covering group of a corresponding Lie group provided the conditions (Nelson, Sternheimer, etc.), which are well known for the case of Hilbert or Banach representations, hold. If a form isometric representation leaves the subspace from which the physical Hilbert space is obtained via factorization and completion invariant, then the same is proved to be true for its differential. Conversely, a necessary and sufficient condition is derived for the transmission of the invariance of this subspace under a form skew-symmetric representation of a Lie algebra to its integral
Davis, Joshua R.; Titus, Sarah J.; Horsman, Eric
2013-11-01
The dynamic theory of deformable ellipsoidal inclusions in slow viscous flows was worked out by J.D. Eshelby in the 1950s, and further developed and applied by various authors. We describe three approaches to computing Eshelby's ellipsoid dynamics and other homogeneous deformations. The most sophisticated of our methods uses differential-geometric techniques on Lie groups. This Lie group method is faster and more precise than earlier methods, and perfectly preserves certain geometric properties of the ellipsoids, including volume. We apply our method to the analysis of naturally deformed clasts from the Gem Lake shear zone in the Sierra Nevada mountains of California, USA. This application demonstrates how, given three-dimensional strain data, we can solve simultaneously for best-fit bulk kinematics of the shear zone, as well as relative viscosities of clasts and matrix rocks.
Hierarchy of kissing numbers for exceptional Lie symmetry groups in high energy physics
International Nuclear Information System (INIS)
El Naschie, M.S.
2008-01-01
We are constructing a hierarchy of kissing numbers representing singular contact points of hyper-spheres in exceptional Lie symmetry groups lattice arrangement embedded in the 26 dimensional bosonic strings spacetime. That way we find a total number of points and dimensions equal to 548. This is 52 more than the order of E 8 E 8 of heterotic string theory and leads to the prediction of 69 elementary particles at an energy scale under 1 T. In other words, our mathematical model predicts nine more particles than what is currently experimentally known to exist in the standard model of high energy physics namely only 60. The result is thus in full agreement with all our previous theoretical findings
Diffeomorphism Group Representations in Relativistic Quantum Field Theory
Energy Technology Data Exchange (ETDEWEB)
Goldin, Gerald A. [Rutgers Univ., Piscataway, NJ (United States); Sharp, David H. [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2017-12-20
We explore the role played by the di eomorphism group and its unitary representations in relativistic quantum eld theory. From the quantum kinematics of particles described by representations of the di eomorphism group of a space-like surface in an inertial reference frame, we reconstruct the local relativistic neutral scalar eld in the Fock representation. An explicit expression for the free Hamiltonian is obtained in terms of the Lie algebra generators (mass and momentum densities). We suggest that this approach can be generalized to elds whose quanta are spatially extended objects.
Knot wormholes and the dimensional invariant of exceptional Lie groups and Stein space hierarchies
International Nuclear Information System (INIS)
Elokaby, Ayman
2009-01-01
The present short note points out a most interesting and quite unexpected connection between the number of distinct knot as a function of their crossing number and exceptional Lie groups and Stein space hierarchies. It is found that the crossing number 7 plays the role of threshold similar to 4 and 5 in E-infinity theory and for the 11 crossing the number of distinct knots is very close to 4α-bar 0 +1=548+1=549, where α-bar 0 =137 is the inverse integer electromagnetic fine structure constant. This is particularly intriguing in view of a similar relation pertinent to the 17 two and three Stein spaces where the total dimension is Σ 1 17 Stein=5α-bar 0 +1=685+1=686, as well as the sum of the eight exceptional Lie symmetry groups Σ i=1 8 |E i |=4α-bar 0 =548. The slight discrepancy of one is explained in both cases by the inclusion of El Naschie's transfinite corrections leading to Σ i=1 8 |E i |=(4)(137+k 0 )=548.328157 and Σ i=1 17 Stein=(5)(137+k 0 )=685.41097, where k o = φ 5 (1 - φ 5 ) and φ=(√(5)-1)/2.
Sweet, Monica A; Heyman, Gail D; Fu, Genyue; Lee, Kang
2010-07-01
This study explored the effects of collectivism on lying to conceal a group transgression. Seven-, 9-, and 11-year-old US and Chinese children (N = 374) were asked to evaluate stories in which protagonists either lied or told the truth about their group's transgression and were then asked about either the protagonist's motivations or justification for their own evaluations. Previous research suggests that children in collectivist societies such as China find lying for one's group to be more acceptable than do children from individualistic societies such as the United States. The current study provides evidence that this is not always the case: Chinese children in this study viewed lies told to conceal a group's transgressions less favourably than did US children. An examination of children's reasoning about protagonists' motivations for lying indicated that children in both countries focused on an impact to self when discussing motivations for protagonists to lie for their group. Overall, results suggest that children living in collectivist societies do not always focus on the needs of the group.
Sugawara operators for classical Lie algebras
Molev, Alexander
2018-01-01
The celebrated Schur-Weyl duality gives rise to effective ways of constructing invariant polynomials on the classical Lie algebras. The emergence of the theory of quantum groups in the 1980s brought up special matrix techniques which allowed one to extend these constructions beyond polynomial invariants and produce new families of Casimir elements for finite-dimensional Lie algebras. Sugawara operators are analogs of Casimir elements for the affine Kac-Moody algebras. The goal of this book is to describe algebraic structures associated with the affine Lie algebras, including affine vertex algebras, Yangians, and classical \\mathcal{W}-algebras, which have numerous ties with many areas of mathematics and mathematical physics, including modular forms, conformal field theory, and soliton equations. An affine version of the matrix technique is developed and used to explain the elegant constructions of Sugawara operators, which appeared in the last decade. An affine analogue of the Harish-Chandra isomorphism connec...
International Nuclear Information System (INIS)
Zhi Hongyan
2009-01-01
In this paper, based on the symbolic computing system Maple, the direct method for Lie symmetry groups presented by Sen-Yue Lou [J. Phys. A: Math. Gen. 38 (2005) L129] is extended from the continuous differential equations to the differential-difference equations. With the extended method, we study the well-known differential-difference KP equation, KZ equation and (2+1)-dimensional ANNV system, and both the Lie point symmetry groups and the non-Lie symmetry groups are obtained.
Gauge theory of gravity and supergravity on a group manifold
International Nuclear Information System (INIS)
Ne'eman, Y.; Regge, T.
1977-12-01
The natural arena for the physics of gravity, supergravity and their enlargements appears to be the group manifold of the Poincare group P, the graded Poincare group GP of supersymmetry, and the corresponding enlargements. The dynamics of these theories correspond to geometrical algorithms in P and GP. Differential geometry on Lie groups is reviewed and results applied to P and GP. Curvature, gauge transformations and factorization are introduced. Also reviewed is the general coordinate transformation group and a hybrid gauge transformation, the anholonomized G.C.T. gauge. A study is made of the construction of an action, including the introduction of a set of special 2 forms, the ''pseudo curvatures.'' The possibilities of factorization in supersymmetry are analyzed. The version of supergravity is present which has now become a completely geometrical theory
An isomorphism for algebra of distributions with compact support on Lie groups
International Nuclear Information System (INIS)
El-Hussein, K.
1991-08-01
Let (H, H 0 ,...,H L L is an element of IN) be a finite sequence of abelian connected Lie Groups, G L = H, G 1 G i+1 χ ρi+1 H i+1 (0 ≤ i ≤ L - 1) and G = G 0 χ ρo H 0 the Lie groups which are the semi-direct product of G i by H-i (0 ≤ i ≤ L), where ρ i : H i → Aut(G i ) is a group homomorphism (0 ≤ i ≤ L). Let G-tilde = H x H L x...xH 0 be the Lie group of the direct product of H, H L ,..., and H 0 and let ε'(G-tilde) the Topological vector space of all distributions with compact support on G-tilde. In this paper, we prove that there is a structure of algebra on ε'(G-tilde) such that the algebra (convolution) of all distributions with compact support on G is isomorphic onto ε'(G-tilde). (author). 7 refs
Lying in business : Insights from Hanna Arendt's 'Lying in Politics'
Eenkhoorn, P.; Graafland, J.J.
2011-01-01
The political philosopher Hannah Arendt develops several arguments regarding why truthfulness cannot be counted among the political virtues. This article shows that similar arguments apply to lying in business. Based on Hannah Arendt's theory, we distinguish five reasons why lying is a structural
Nazarov, Anton
2012-11-01
In this paper we present Affine.m-a program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. The algorithms are based on the properties of weights and Weyl symmetry. Computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition are the most important problems for us. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in the popular computer algebra system Mathematica and works with finite-dimensional and affine Lie algebras. Catalogue identifier: AENA_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENB_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, UK Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 24 844 No. of bytes in distributed program, including test data, etc.: 1 045 908 Distribution format: tar.gz Programming language: Mathematica. Computer: i386-i686, x86_64. Operating system: Linux, Windows, Mac OS, Solaris. RAM: 5-500 Mb Classification: 4.2, 5. Nature of problem: Representation theory of finite-dimensional Lie algebras has many applications in different branches of physics, including elementary particle physics, molecular physics, nuclear physics. Representations of affine Lie algebras appear in string theories and two-dimensional conformal field theory used for the description of critical phenomena in two-dimensional systems. Also Lie symmetries play a major role in a study of quantum integrable systems. Solution method: We work with weights and roots of finite-dimensional and affine Lie algebras and use Weyl symmetry extensively. Central problems which are the computations of weight multiplicities, branching and fusion coefficients are solved using one general recurrent
Lie algebroids in derived differential topology
Nuiten, J.J.
2018-01-01
A classical principle in deformation theory asserts that any formal deformation problem is controlled by a differential graded Lie algebra. This thesis studies a generalization of this principle to Lie algebroids, and uses this to examine the interactions between the theory of Lie algebroids and the
Papi, Paolo; Advances in Lie Superalgebras
2014-01-01
The volume is the outcome of the conference "Lie superalgebras," which was held at the Istituto Nazionale di Alta Matematica, in 2012. The conference gathered many specialists in the subject, and the talks held provided comprehensive insights into the newest trends in research on Lie superalgebras (and related topics like vertex algebras, representation theory and supergeometry). The book contains contributions of many leading esperts in the field and provides a complete account of the newest trends in research on Lie Superalgebras.
On discretization of tori of compact simple Lie groups: II
International Nuclear Information System (INIS)
Hrivnák, Jiří; Motlochová, Lenka; Patera, Jiří
2012-01-01
The discrete orthogonality of special function families, called C- and S-functions, which are derived from the characters of compact simple Lie groups, is described in Hrivnák and Patera (2009 J. Phys. A: Math. Theor. 42 385208). Here, the results of Hrivnák and Patera are extended to two additional recently discovered families of special functions, called S s - and S l -functions. The main result is an explicit description of their pairwise discrete orthogonality within each family, when the functions are sampled on finite fragments F s M and F l M of a lattice in any dimension n ⩾ 2 and of any density controlled by M, and of the symmetry of the weight lattice of any compact simple Lie group with two different lengths of roots. (paper)
On the q-exponential of matrix q-Lie algebras
Directory of Open Access Journals (Sweden)
Ernst Thomas
2017-01-01
Full Text Available In this paper, we define several new concepts in the borderline between linear algebra, Lie groups and q-calculus.We first introduce the ring epimorphism r, the set of all inversions of the basis q, and then the important q-determinant and corresponding q-scalar products from an earlier paper. Then we discuss matrix q-Lie algebras with a modified q-addition, and compute the matrix q-exponential to form the corresponding n × n matrix, a so-called q-Lie group, or manifold, usually with q-determinant 1. The corresponding matrix multiplication is twisted under τ, which makes it possible to draw diagrams similar to Lie group theory for the q-exponential, or the so-called q-morphism. There is no definition of letter multiplication in a general alphabet, but in this article we introduce new q-number systems, the biring of q-integers, and the extended q-rational numbers. Furthermore, we provide examples of matrices in suq(4, and its corresponding q-Lie group. We conclude with an example of system of equations with Ward number coeficients.
Shany-Ur, Tal; Poorzand, Pardis; Grossman, Scott N; Growdon, Matthew E; Jang, Jung Y; Ketelle, Robin S; Miller, Bruce L; Rankin, Katherine P
2012-01-01
Comprehension of insincere communication is an important aspect of social cognition requiring visual perspective taking, emotion reading, and understanding others' thoughts, opinions, and intentions. Someone who is lying intends to hide their insincerity from the listener, while a sarcastic speaker wants the listener to recognize they are speaking insincerely. We investigated whether face-to-face testing of comprehending insincere communication would effectively discriminate among neurodegenerative disease patients with different patterns of real-life social deficits. We examined ability to comprehend lies and sarcasm from a third-person perspective, using contextual cues, in 102 patients with one of four neurodegenerative diseases (behavioral variant frontotemporal dementia [bvFTD], Alzheimer's disease [AD], progressive supranuclear palsy [PSP], and vascular cognitive impairment) and 77 healthy older adults (normal controls--NCs). Participants answered questions about videos depicting social interactions involving deceptive, sarcastic, or sincere speech using The Awareness of Social Inference Test. All subjects equally understood sincere remarks, but bvFTD patients displayed impaired comprehension of lies and sarcasm compared with NCs. In other groups, impairment was not disease-specific but was proportionate to general cognitive impairment. Analysis of the task components revealed that only bvFTD patients were impaired on perspective taking and emotion reading elements and that both bvFTD and PSP patients had impaired ability to represent others' opinions and intentions (i.e., theory of mind). Test performance correlated with informants' ratings of subjects' empathy, perspective taking and neuropsychiatric symptoms in everyday life. Comprehending insincere communication is complex and requires multiple cognitive and emotional processes vulnerable across neurodegenerative diseases. However, bvFTD patients show uniquely focal and severe impairments at every level
Lie-superalgebraical aspects of quantum statistics
International Nuclear Information System (INIS)
Palev, T.D.
1978-01-01
The Lie-superalgebraical properties of the ordinary quantum statistics are discussed with the aim of possible generalization in quantum theory and in theoretical physics. It is indicated that the algebra generated by n pairs of Fermi or paraFermi operators is isomorphic to the classical simple Lie algebra Bsub(n) of the SO(2n+1) orthogonal group, whereas n pairs of Bose or paraBose operators generate the simple orthosympletic superalgebra B(O,n). The transition to infinite number of creation and annihilation operators (n → infinity) does not change a superalgebraic structure. Hence, ordinary Bose and Fermi quantization can be considered as quantization over definite irreducible representations of two simple Lie superalgebras. The idea is given of how one can introduce creation and annihilation operators that satisfy the second quantization postulates and generate other simple Lie superalgebras
Renormalization group aspects of 3-dimensional Pure U(1) lattice gauge theory
International Nuclear Information System (INIS)
Gopfert, M.; Mack, G.
1983-01-01
A few surprises in a recent study of the 3-dimensional pure U(1) lattice gauge theory model, from the point of view of the renormalization group theory, are discussed. Since the gauge group U(1) of this model is abelian, the model is subject to KramersWannier duality transformation. One obtains a ferromagnet with a global symmetry group Z. The duality transformation shows that the surface tension alpha of the model equals the strong tension of the U(1) gauge model. A theorem to represent the true asymptotic behaviour of alpha is derived. A second theorem considers the correlation functions. Discrepiancies between the theorems result in a solution that ''is regarded as a catastrophe'' in renormalization group theory. A lesson is drawn: To choose a good block spin in a renormalization group procedure, know what the low lying excitations of the theory are, to avoid integrating some of them by mischief
A very strong difference property for semisimple compact connected lie groups
Shtern, A. I.
2011-06-01
Let G be a topological group. For a function f: G → ℝ and h ∈ G, the difference function Δ h f is defined by the rule Δ h f( x) = f( xh) - f( x) ( x ∈ G). A function H: G → ℝ is said to be additive if it satisfies the Cauchy functional equation H( x + y) = H( x) + H( y) for every x, y ∈ G. A class F of real-valued functions defined on G is said to have the difference property if, for every function f: G → ℝ satisfying Δ h f ∈ F for each h ∈ G, there is an additive function H such that f - H ∈ F. Erdős' conjecture claiming that the class of continuous functions on ℝ has the difference property was proved by N. G. de Bruijn; later on, F. W. Carroll and F. S. Koehl obtained a similar result for compact Abelian groups and, under the additional assumption that the other one-sided difference function ∇ h f defined by ∇ h f( x) = f( xh) - f( x) ( x ∈ G, h ∈ G) is measurable for any h ∈ G, also for noncommutative compact metric groups. In the present paper, we consider a narrower class of groups, namely, the family of semisimple compact connected Lie groups. It turns out that these groups admit a significantly stronger difference property. Namely, if a function f: G → ℝ on a semisimple compact connected Lie group has continuous difference functions Δ h f for any h ∈ G (without the additional assumption concerning the measurability of the functions of the form ∇ h f), then f is automatically continuous, and no nontrivial additive function of the form H is needed. Some applications are indicated, including difference theorems for homogeneous spaces of compact connected Lie groups.
Group theory for unified model building
International Nuclear Information System (INIS)
Slansky, R.
1981-01-01
The results gathered here on simple Lie algebras have been selected with attention to the needs of unified model builders who study Yang-Mills theories based on simple, local-symmetry groups that contain as a subgroup the SUsup(w) 2 x Usup(w) 1 x SUsup(c) 3 symmetry of the standard theory of electromagnetic, weak, and strong interactions. The major topics include, after a brief review of the standard model and its unification into a simple group, the use of Dynkin diagrams to analyze the structure of the group generators and to keep track of the weights (quantum numbers) of the representation vectors; an analysis of the subgroup structure of simple groups, including explicit coordinatizations of the projections in weight space; lists of representations, tensor products and branching rules for a number of simple groups; and other details about groups and their representations that are often helpful for surveying unified models, including vector-coupling coefficient calculations. Tabulations of representations, tensor products, and branching rules for E 6 , SO 10 , SU 6 , F 4 , SO 9 , SO 5 , SO 8 , SO 7 , SU 4 , E 7 , E 8 , SU 8 , SO 14 , SO 18 , SO 22 , and for completeness, SU 3 are included. (These tables may have other applications.) Group-theoretical techniques for analyzing symmetry breaking are described in detail and many examples are reviewed, including explicit parameterizations of mass matrices. (orig.)
Invariant differential operators for non-compact Lie groups: an introduction
International Nuclear Information System (INIS)
Dobrev, V.K.
2015-01-01
In the present paper we review the progress of the project of classification and construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we called earlier 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduced recently the new notion of parabolic relation between two non-compact semisimple Lie algebras G and G' that have the same complexification and possess maximal parabolic subalgebras with the same complexification. In the present paper we consider in detail the orthogonal algebras so(p,q) all of which are parabolically related to the conformal algebra so(n,2) with p+q=n+2, the parabolic subalgebras including the Lorentz subalgebra so(n-1,1) and its analogs so(p-1,q-1)
K-theory and representation theory
International Nuclear Information System (INIS)
Kuku, A.O.
2003-01-01
This contribution includes K-theory of orders, group-rings and modules over EI categories, equivariant higher algebraic K-theory for finite, profinite and compact Lie group actions together with their relative generalisations and applications
Particle-like structure of coaxial Lie algebras
Vinogradov, A. M.
2018-01-01
This paper is a natural continuation of Vinogradov [J. Math. Phys. 58, 071703 (2017)] where we proved that any Lie algebra over an algebraically closed field or over R can be assembled in a number of steps from two elementary constituents, called dyons and triadons. Here we consider the problems of the construction and classification of those Lie algebras which can be assembled in one step from base dyons and triadons, called coaxial Lie algebras. The base dyons and triadons are Lie algebra structures that have only one non-trivial structure constant in a given basis, while coaxial Lie algebras are linear combinations of pairwise compatible base dyons and triadons. We describe the maximal families of pairwise compatible base dyons and triadons called clusters, and, as a consequence, we give a complete description of the coaxial Lie algebras. The remarkable fact is that dyons and triadons in clusters are self-organised in structural groups which are surrounded by casings and linked by connectives. We discuss generalisations and applications to the theory of deformations of Lie algebras.
Group-geometric methods in supergravity and superstring theories
International Nuclear Information System (INIS)
Castellani, L.
1992-01-01
The purpose of this paper is to give a brief and pedagogical account of the group-geometric approach to (super)gravity and superstring theories. The authors summarize the main ideas and apply them to selected examples. Group geometry provides a natural and unified formulation of gravity and gauge theories. The invariance of both are interpreted as diffeomorphisms on a suitable group manifold. This geometrical framework has a fruitful output, in that it provides a systematic algorithm for the gauging of Lie algebras and the construction of (super)gravity or (super)string Lagrangians. The basic idea is to associate fundamental fields to the group generators. This is done by considering first a basis of tangent vectors on the group manifold. These vectors close on the same algebra as the abstract group generators. The dual basis, i.e. the vielbeins (cotangent basis of one-forms) is then identified with the set of fundamental fields. Thus, for example, the vielbein V a and the spin connection ω ab of ordinary Einstein-Cartan gravity are seen as the duals of the tangent vectors corresponding to translations and Lorentz rotations, respectively
Matrix groups for undergraduates
Tapp, Kristopher
2016-01-01
Matrix groups touch an enormous spectrum of the mathematical arena. This textbook brings them into the undergraduate curriculum. It makes an excellent one-semester course for students familiar with linear and abstract algebra and prepares them for a graduate course on Lie groups. Matrix Groups for Undergraduates is concrete and example-driven, with geometric motivation and rigorous proofs. The story begins and ends with the rotations of a globe. In between, the author combines rigor and intuition to describe the basic objects of Lie theory: Lie algebras, matrix exponentiation, Lie brackets, maximal tori, homogeneous spaces, and roots. This second edition includes two new chapters that allow for an easier transition to the general theory of Lie groups. From reviews of the First Edition: This book could be used as an excellent textbook for a one semester course at university and it will prepare students for a graduate course on Lie groups, Lie algebras, etc. … The book combines an intuitive style of writing w...
Filiform Lie algebras of order 3
Navarro, R. M.
2014-04-01
The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, "Cohomologie des algèbres de Lie nilpotentes. Application à l'étude de la variété des algèbres de Lie nilpotentes," Bull. Soc. Math. France 98, 81-116 (1970)]. Also we give the dimension, using an adaptation of the {sl}(2,{C})-module Method, and a basis of such infinitesimal deformations in some generic cases.
Biyogmam, Guy Roger
2011-01-01
In this paper, we introduce the category of Lie $n$-racks and generalize several results known on racks. In particular, we show that the tangent space of a Lie $n$-Rack at the neutral element has a Leibniz $n$-algebra structure. We also define a cohomology theory of $n$-racks..
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Larouche, M [Departement de Mathematiques et Statistique, Universite de Montreal, 2920 chemin de la Tour, Montreal, Quebec H3T 1J4 (Canada); Lemire, F W [Department of Mathematics, University of Windsor, Windsor, Ontario (Canada); Patera, J, E-mail: larouche@dms.umontreal.ca, E-mail: lemire@uwindsor.ca, E-mail: patera@crm.umontreal.ca [Centre de Recherches Mathematiques, Universite de Montreal, CP 6128-Centre ville, Montreal, Quebec H3C 3J7 (Canada)
2011-10-14
In this paper, we present a new, uniform and comprehensive description of centralizers of the maximal regular subgroups in compact simple Lie groups of all types and ranks. The centralizer is either a direct product of finite cyclic groups, a continuous group of rank 1, or a product, not necessarily direct, of a continuous group of rank 1 with a finite cyclic group. Explicit formulas for the action of such centralizers on irreducible representations of the simple Lie algebras are given. (paper)
Invariance Lie algebra and group of the non relativistic hydrogen atom
International Nuclear Information System (INIS)
Decoster, Alain
1970-01-01
The first part of this work contains a general survey of the use of Lie groups and algebras in quantum mechanics, followed by an extensive description of tbe invariance algebra and invariance group of the non-relativistic hydrogen atom; the realization of this group discovered by FOCK is specially examined. The second part is a two-hundred items bibliography on invariance groups and algebras of classical and quantum-mechanical simple systems. (author) [fr
Filiform Lie algebras of order 3
International Nuclear Information System (INIS)
Navarro, R. M.
2014-01-01
The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases
A density matrix renormalization group study of low-lying excitations ...
Indian Academy of Sciences (India)
Symmetrized density-matrix-renormalization-group calculations have been carried out, within Pariser-Parr-Pople Hamiltonian, to explore the nature of the ground and low-lying excited states of long polythiophene oligomers. We have exploited 2 symmetry and spin parity of the system to obtain excited states of ...
Koszul Information Geometry and Souriau Geometric Temperature/Capacity of Lie Group Thermodynamics
Directory of Open Access Journals (Sweden)
Frédéric Barbaresco
2014-08-01
Full Text Available The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from “Characteristic Functions”, was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF on convex cones will be presented as cornerstone of “Information Geometry” theory, defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, Souriau has extended the Characteristic Function in Statistical Physics looking for other kinds of invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric (Planck Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean “Moment map” by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. We will synthetize the analogies between both Koszul and Souriau models, and will reduce their definitions to the exclusive Cartan “Inner Product”. Interpreting Legendre transform as Fourier transform in (Min,+ algebra, we conclude with a definition of Entropy given by a relation mixing Fourier/Laplace transforms: Entropy = (minus Fourier(Min,+ o Log o Laplace(+,X.
Lying in Business : Insights from Hannah Arendt’s ‘Lying in Politics’
Eenkhoorn, P.; Graafland, J.J.
2010-01-01
The famous political philosopher Hannah Arendt develops several arguments why truthfulness cannot be counted among the political virtues. This article shows that similar arguments apply to lying in business. Based on Hannah Arendt’s theory, we distinguish five reasons why lying is a structural
International Workshop "Groups, Rings, Lie and Hopf Algebras"
2003-01-01
The volume is almost entirely composed of the research and expository papers by the participants of the International Workshop "Groups, Rings, Lie and Hopf Algebras", which was held at the Memorial University of Newfoundland, St. John's, NF, Canada. All four areas from the title of the workshop are covered. In addition, some chapters touch upon the topics, which belong to two or more areas at the same time. Audience: The readership targeted includes researchers, graduate and senior undergraduate students in mathematics and its applications.
International Nuclear Information System (INIS)
El-Hussein, K.
1991-08-01
Let V be a real finite dimensional vector space and let K be a connected compact Lie group, which acts on V by means of a continuous linear representation ρ. Let G=V x p K be the motion group which is the semi-direct product of V by K and let P be an invariant differential operator on G. In this paper we give a necessary and sufficient condition for the global solvability of P on G. Now let G be a connected semi-simple Lie group with finite centre and let P be an invariant differential operator on G. We give also a necessary and sufficient condition for the global solvability of P on G. (author). 8 refs
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Sati, Hisham [University of Pittsburgh,Pittsburgh, PA, 15260 (United States); Mathematics Program, Division of Science and Mathematics, New York University Abu Dhabi,Saadiyat Island, Abu Dhabi (United Arab Emirates); Schreiber, Urs [Mathematics Institute of the Academy,Žitna 25, Praha 1, 115 67 (Czech Republic)
2017-03-16
We uncover higher algebraic structures on Noether currents and BPS charges. It is known that equivalence classes of conserved currents form a Lie algebra. We show that at least for target space symmetries of higher parameterized WZW-type sigma-models this naturally lifts to a Lie (p+1)-algebra structure on the Noether currents themselves. Applied to the Green-Schwarz-type action functionals for super p-brane sigma-models this yields super Lie (p+1)-algebra refinements of the traditional BPS brane charge extensions of supersymmetry algebras. We discuss this in the generality of higher differential geometry, where it applies also to branes with (higher) gauge fields on their worldvolume. Applied to the M5-brane sigma-model we recover and properly globalize the M-theory super Lie algebra extension of 11-dimensional superisometries by 2-brane and 5-brane charges. Passing beyond the infinitesimal Lie theory we find cohomological corrections to these charges in higher analogy to the familiar corrections for D-brane charges as they are lifted from ordinary cohomology to twisted K-theory. This supports the proposal that M-brane charges live in a twisted cohomology theory.
Particle-like structure of Lie algebras
Vinogradov, A. M.
2017-07-01
If a Lie algebra structure 𝔤 on a vector space is the sum of a family of mutually compatible Lie algebra structures 𝔤i's, we say that 𝔤 is simply assembled from the 𝔤i's. Repeating this procedure with a number of Lie algebras, themselves simply assembled from the 𝔤i's, one obtains a Lie algebra assembled in two steps from 𝔤i's, and so on. We describe the process of modular disassembling of a Lie algebra into a unimodular and a non-unimodular part. We then study two inverse questions: which Lie algebras can be assembled from a given family of Lie algebras, and from which Lie algebras can a given Lie algebra be assembled. We develop some basic assembling and disassembling techniques that constitute the elements of a new approach to the general theory of Lie algebras. The main result of our theory is that any finite-dimensional Lie algebra over an algebraically closed field of characteristic zero or over R can be assembled in a finite number of steps from two elementary constituents, which we call dyons and triadons. Up to an abelian summand, a dyon is a Lie algebra structure isomorphic to the non-abelian 2-dimensional Lie algebra, while a triadon is isomorphic to the 3-dimensional Heisenberg Lie algebra. As an example, we describe constructions of classical Lie algebras from triadons.
Directory of Open Access Journals (Sweden)
Fedriani Martel, Eugenio M.
2006-06-01
Full Text Available En la presente comunicación explicamos algunas de las herramientas de la Geometría Diferencial y, en concreto, de la Teoría de Lie con las que se trabaja actualmente en Economía. Se indican las condiciones que se exigen a las funciones de producción y la definición de un tipo de progreso técnico denominado de tipo Lie, consistente en exigir las tres propiedades que han de verificar los grupos de Lie. También se expone el uso del operador de Lie en interpretaciones económicas y en la cuantificación del impacto del progreso técnico. Dicho operador permite dar una respuesta a la Controversia Solow-Stigler. Por último, se indican varias aplicaciones de la Teoría de Lie en los estudios económicos, que permiten abrir futuras líneas de investigación,de las que se apuntan algunas. De este modo, nuestro objetivo principal es mostrar el uso, actual y futuro, de la Teoría de Lie en el campo de la Economía.
Global dimensions for Lie groups at level k and their conformally exceptional quantum subgroups
Coquereaux, Robert
2010-01-01
We obtain formulae giving global dimensions for fusion categories defined by Lie groups G at level k and for the associated module-categories obtained via conformal embeddings. The results can be expressed in terms of Lie quantum superfactorials of type G. The later are related, for the type Ar, to the quantum Barnes function.
$C^1$ actions on manifolds by lattices in Lie groups with sufficiently high rank
Damjanovic, Danijela; Zhang, Zhiyuan
2018-01-01
In this paper we study Zimmer's conjecture for $C^1$ actions of higher-rank lattices of a connected, semisimple Lie group with finite center on compact manifolds. We show that if the Lie group has no compact factor, and all of whose non-compact factors are of ranks in some sense sufficiently large with respect to the dimension of the manifold, then every $C^1$ action of an irreducible, co-compact lattice has a finite image. As a corollary of our results, for every (uniform or non-uniform) lat...
Smooth vectors and Weyl-Pedersen calculus for representations of nilpotent Lie groups
Beltita, Ingrid; Beltita, Daniel
2009-01-01
We present some recent results on smooth vectors for unitary irreducible representations of nilpotent Lie groups. Applications to the Weyl-Pedersen calculus of pseudo-differential operators with symbols on the coadjoint orbits are also discussed.
Advances in geometry and Lie algebras from supergravity
Frè, Pietro Giuseppe
2018-01-01
This book aims to provide an overview of several topics in advanced Differential Geometry and Lie Group Theory, all of them stemming from mathematical problems in supersymmetric physical theories. It presents a mathematical illustration of the main development in geometry and symmetry theory that occurred under the fertilizing influence of supersymmetry/supergravity. The contents are mainly of mathematical nature, but each topic is introduced by historical information and enriched with motivations from high energy physics, which help the reader in getting a deeper comprehension of the subject. .
Lie Symmetries and Solitons in Nonlinear Systems with Spatially Inhomogeneous Nonlinearities
International Nuclear Information System (INIS)
Belmonte-Beitia, Juan; Perez-Garcia, Victor M.; Vekslerchik, Vadym; Torres, Pedro J.
2007-01-01
Using Lie group theory and canonical transformations, we construct explicit solutions of nonlinear Schroedinger equations with spatially inhomogeneous nonlinearities. We present the general theory, use it to show that localized nonlinearities can support bound states with an arbitrary number solitons, and discuss other applications of interest to the field of nonlinear matter waves
Closure of the gauge algebra, generalized Lie equations and Feynman rules
International Nuclear Information System (INIS)
Batalin, I.A.
1984-01-01
A method is given by which an open gauge algebra can always be closed and even made abelian. As a preliminary the generalized Lie equations for the open group are obtained. The Feynman rules for gauge theories with open algebras are derived by reducing the gauge theory to a non-gauge one. (orig.)
Lie-algebra expansions, Chern-Simons theories and the Einstein-Hilbert Lagrangian
International Nuclear Information System (INIS)
Edelstein, Jose D.; Hassaine, Mokhtar; Troncoso, Ricardo; Zanelli, Jorge
2006-01-01
Starting from gravity as a Chern-Simons action for the AdS algebra in five dimensions, it is possible to modify the theory through an expansion of the Lie algebra that leads to a system consisting of the Einstein-Hilbert action plus non-minimally coupled matter. The modified system is gauge invariant under the Poincare group enlarged by an Abelian ideal. Although the resulting action naively looks like general relativity plus corrections due to matter sources, it is shown that the non-minimal couplings produce a radical departure from GR. Indeed, the dynamics is not continuously connected to the one obtained from Einstein-Hilbert action. In a matter-free configuration and in the torsionless sector, the field equations are too strong a restriction on the geometry as the metric must satisfy both the Einstein and pure Gauss-Bonnet equations. In particular, the five-dimensional Schwarzschild geometry fails to be a solution; however, configurations corresponding to a brane-world with positive cosmological constant on the worldsheet are admissible when one of the matter fields is switched on. These results can be extended to higher odd dimensions
Automorphic Lie algebras with dihedral symmetry
International Nuclear Information System (INIS)
Knibbeler, V; Lombardo, S; A Sanders, J
2014-01-01
The concept of automorphic Lie algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems. automorphic Lie algebras are obtained by imposing a discrete group symmetry on a current algebra of Krichever–Novikov type. Past work shows remarkable uniformity between algebras associated to different reduction groups. For example, if the base Lie algebra is sl 2 (C) and the poles of the automorphic Lie algebra are restricted to an exceptional orbit of the symmetry group, changing the reduction group does not affect the Lie algebra structure. In this research we fix the reduction group to be the dihedral group and vary the orbit of poles as well as the group action on the base Lie algebra. We find a uniform description of automorphic Lie algebras with dihedral symmetry, valid for poles at exceptional and generic orbits. (paper)
Lectures on the theory of group properties of differential equations
Ovsyannikov, LV
2013-01-01
These lecturers provide a clear introduction to Lie group methods for determining and using symmetries of differential equations, a variety of their applications in gas dynamics and other nonlinear models as well as the author's remarkable contribution to this classical subject. It contains material that is useful for students and teachers but cannot be found in modern texts. For example, the theory of partially invariant solutions developed by Ovsyannikov provides a powerful tool for solving systems of nonlinear differential equations and investigating complicated mathematical models. Readers
Control Algorithms Along Relative Equilibria of Underactuated Lagrangian Systems on Lie Groups
DEFF Research Database (Denmark)
Nordkvist, Nikolaj; Bullo, F.
2008-01-01
We present novel algorithms to control underactuated mechanical systems. For a class of invariant systems on Lie groups, we design iterative small-amplitude control forces to accelerate along, decelerate along, and stabilize relative equilibria. The technical approach is based upon a perturbation...
Control algorithms along relative equilibria of underactuated Lagrangian systems on Lie groups
DEFF Research Database (Denmark)
Nordkvist, Nikolaj; Bullo, Francesco
2007-01-01
We present novel algorithms to control underactuated mechanical systems. For a class of invariant systems on Lie groups, we design iterative small-amplitude control forces to accelerate along, decelerate along, and stabilize relative equilibria. The technical approach is based upon a perturbation...
Real representations of Lie groups and a theorem of H. Pittie
International Nuclear Information System (INIS)
Freitas, R.
1992-01-01
In this paper, we prove a structure theorem of the real representation ring RO(T) as a module over the real representation ring RO(G), where G is a compact, connected and simply connected Lie group and T is a maximal torus of G. This provides a real version to a theorem of H. Pittie. (author). 24 refs
Low-dimensional filiform Lie algebras over finite fields
Falcón Ganfornina, Óscar Jesús; Núñez Valdés, Juan; Pacheco Martínez, Ana María; Villar Liñán, María Trinidad; Vasek, Vladimir (Coordinador); Shmaliy, Yuriy S. (Coordinador); Trcek, Denis (Coordinador); Kobayashi, Nobuhiko P. (Coordinador); Choras, Ryszard S. (Coordinador); Klos, Zbigniew (Coordinador)
2011-01-01
In this paper we use some objects of Graph Theory to classify low-dimensional filiform Lie algebras over finite fields. The idea lies in the representation of each Lie algebra by a certain type of graphs. Then, some properties on Graph Theory make easier to classify the algebras. As results, which can be applied in several branches of Physics or Engineering, for instance, we find out that there exist, up to isomorphism, six 6-dimensional filiform Lie algebras over Z/pZ, for p = 2, 3, 5. Pl...
The Exceptional Lie symmetry groups hierarchy and the expected number of Higgs bosons
International Nuclear Information System (INIS)
El Naschie, M.S.
2008-01-01
New insights into the structure of various exceptional Lie symmetry groups hierarchies are utilized to shed light on various problems pertinent to the standard model of high energy physics and the Higgs
International Nuclear Information System (INIS)
Baeuerle, G.G.A.; Kerf, E.A. de
1990-01-01
The structure of the laws in physics is largely based on symmetries. This book is on Lie algebras, the mathematics of symmetry. It gives a thorough mathematical treatment of finite dimensional Lie algebras and Kac-Moody algebras. Concepts such as Cartan matrix, root system, Serre's construction are carefully introduced. Although the book can be read by an undergraduate with only an elementary knowledge of linear algebra, the book will also be of use to the experienced researcher. Experience has shown that students who followed the lectures are well-prepared to take on research in the realms of string-theory, conformal field-theory and integrable systems. 48 refs.; 66 figs.; 3 tabs
Group formalism of Lie transformations to time-fractional partial ...
Indian Academy of Sciences (India)
Lie symmetry analysis; Fractional partial differential equation; Riemann–Liouville fractional derivative ... science and engineering. It is known that while ... differential equations occurring in different areas of applied science [11,14]. The Lie ...
Nonlinear wave evolution in VLASOV plasma: a lie-transform analysis
International Nuclear Information System (INIS)
Cary, J.R.
1979-08-01
Nonlinear wave evolution in Vlasov plasma is analyzed using the Lie transform, a powerful mathematical tool which is applicable to Hamiltonian systems. The first part of this thesis is an exposition of the Lie transform. Dewar's general Lie transform theory is explained and is used to construct Deprit's Lie transform perturbation technique. The basic theory is illustrated by simple examples
Matrix groups for undergraduates
Tapp, Kristopher
2005-01-01
Matrix groups touch an enormous spectrum of the mathematical arena. This textbook brings them into the undergraduate curriculum. It makes an excellent one-semester course for students familiar with linear and abstract algebra and prepares them for a graduate course on Lie groups. Matrix Groups for Undergraduates is concrete and example-driven, with geometric motivation and rigorous proofs. The story begins and ends with the rotations of a globe. In between, the author combines rigor and intuition to describe basic objects of Lie theory: Lie algebras, matrix exponentiation, Lie brackets, and maximal tori.
Possible identification of quarks with leptons in Lie-isotopic SU(3) theory
International Nuclear Information System (INIS)
Animalu, A.O.E.
1984-01-01
A possible identification of the six quarks (d,s,c;u,t,b) with the corresponding leptons (e - ,μ - ,tau - ;v/sub e/,v/sub μ/,v/sub tau/) is attempted via the corrspondence principle, dapprox.(uv-bar/sub e/)e - , sapprox.(tv-bar/sub μ/)μ - , c(bv-bar/sub t/)t - ,uapprox.(uv/sub e/) v/sub e/,..., and its inverse, which are formally represented by a non-unitary integral transformation (with kernel P) and its inverse or dual (with kernel Q), connecting the quark and lepton fields. It is shown that PQ and QP may be interpreted as hadronic and leptonic density matrix operators which obey the quantum mechanical analog of the Liouville equation of conservation from which a Lie-isotopic generalization of Heisenberg's equation of motion is abstracted. P and Q form iso-canonically conjugate dynamical veriables, i.e., Q is the isotpic element for the isoassociative product H*Q = HPQ in the equation of motion for Q. It is also shown that PQ and QP, being idempotent operators, have eigenvalues 0 or 1, which imply that both P and Q can be singular, leading to a further differentiation of ''hadronic mechanics'' into the conventional ''isotopic'' theory in which the isotopic element (g) in the isoassociative product A*B = AgB is non-singular and Hermitian, and a new ''homotopic'' theory in which g is singular and non-Hermitian A Lie-admissible generalization is also obained, and SU(2)-spin realizations are indicated
International Nuclear Information System (INIS)
Foroutan, A.
1992-05-01
The essential mathematical challenge in transport theory is based on the nonlinearity of the integro-differential equations governing classical thermodynamic systems on molecular kinetic level. It is the aim of this thesis to gain exact analytical solutions to the model Boltzmann equation suggested by Tjon and Wu. Such solutions afford a deeper insight into the dynamics of rarefied gases. Tjon and Wu have provided a stochastic model of a Boltzmann equation. Its transition probability depends only on the relative speed of the colliding particles. This assumption leads in the case of two translational degrees of freedom to an integro-differential equation of convolution type. According to this convolution structure the integro-differential equation is Laplace transformed. The result is a nonlinear partial differential equation. The investigation of the symmetries of this differential equation by means of Lie groups of transformation enables us to transform the originally nonlinear partial differential equation into ordinary differential equation into ordinary differential equations of Bernoulli type. (author)
System theory on group manifolds and coset spaces.
Brockett, R. W.
1972-01-01
The purpose of this paper is to study questions regarding controllability, observability, and realization theory for a particular class of systems for which the state space is a differentiable manifold which is simultaneously a group or, more generally, a coset space. We show that it is possible to give rather explicit expressions for the reachable set and the set of indistinguishable states in the case of autonomous systems. We also establish a type of state space isomorphism theorem. Our objective is to reduce all questions about the system to questions about Lie algebras generated from the coefficient matrices entering in the description of the system and in that way arrive at conditions which are easily visualized and tested.
Harmonic Analysis and Group Representation
Figa-Talamanca, Alessandro
2011-01-01
This title includes: Lectures - A. Auslander, R. Tolimeri - Nilpotent groups and abelian varieties, M Cowling - Unitary and uniformly bounded representations of some simple Lie groups, M. Duflo - Construction de representations unitaires d'un groupe de Lie, R. Howe - On a notion of rank for unitary representations of the classical groups, V.S. Varadarajan - Eigenfunction expansions of semisimple Lie groups, and R. Zimmer - Ergodic theory, group representations and rigidity; and, Seminars - A. Koranyi - Some applications of Gelfand pairs in classical analysis.
Global solvability of the differential operators non-invariants on semi-simple Lie groups
International Nuclear Information System (INIS)
El Hussein, K.
1991-09-01
Let G be a connected semi-simple Lie group with finite centre and let G=KAN be the Iwasawa decomposition of G. Let P be a differential operator on G, which is right invariant by the sub-group AN and left invariant by the sub-group K. In this paper, we give a necessary and sufficient condition for the global solvability of P on G. (author). 5 refs
Group theory for chemists fundamental theory and applications
Molloy, K C
2010-01-01
The basics of group theory and its applications to themes such as the analysis of vibrational spectra and molecular orbital theory are essential knowledge for the undergraduate student of inorganic chemistry. The second edition of Group Theory for Chemists uses diagrams and problem-solving to help students test and improve their understanding, including a new section on the application of group theory to electronic spectroscopy.Part one covers the essentials of symmetry and group theory, including symmetry, point groups and representations. Part two deals with the application of group theory t
Druţu, Cornelia
2018-01-01
The key idea in geometric group theory is to study infinite groups by endowing them with a metric and treating them as geometric spaces. This applies to many groups naturally appearing in topology, geometry, and algebra, such as fundamental groups of manifolds, groups of matrices with integer coefficients, etc. The primary focus of this book is to cover the foundations of geometric group theory, including coarse topology, ultralimits and asymptotic cones, hyperbolic groups, isoperimetric inequalities, growth of groups, amenability, Kazhdan's Property (T) and the Haagerup property, as well as their characterizations in terms of group actions on median spaces and spaces with walls. The book contains proofs of several fundamental results of geometric group theory, such as Gromov's theorem on groups of polynomial growth, Tits's alternative, Stallings's theorem on ends of groups, Dunwoody's accessibility theorem, the Mostow Rigidity Theorem, and quasiisometric rigidity theorems of Tukia and Schwartz. This is the f...
Analytic vectors and irreducible representations of nilpotent Lie groups and algebras
International Nuclear Information System (INIS)
Arnal, D.
1978-01-01
Let U be a unitary irreducible locally faithful representation of a nilpotent Lie group G, V the universal enveloping algebra of G, M a simple module on V with kernel ker dU, then there exists an automorphism of V keeping ker dU invariant such that, after transport of structure, M is isomorphic to a submodule of the space of analytic vectors for U. (Auth.)
Du, Xia-Xia; Tian, Bo; Chai, Jun; Sun, Yan; Yuan, Yu-Qiang
2017-11-01
In this paper, we investigate a (3+1)-dimensional modified Zakharov-Kuznetsov equation, which describes the nonlinear plasma-acoustic waves in a multicomponent magnetised plasma. With the aid of the Hirota method and symbolic computation, bilinear forms and one-, two- and three-soliton solutions are derived. The characteristics and interaction of the solitons are discussed graphically. We present the effects on the soliton's amplitude by the nonlinear coefficients which are related to the ratio of the positive-ion mass to negative-ion mass, number densities, initial densities of the lower- and higher-temperature electrons and ratio of the lower temperature to the higher temperature for electrons, as well as by the dispersion coefficient, which is related to the ratio of the positive-ion mass to the negative-ion mass and number densities. Moreover, using the Lie symmetry group theory, we derive the Lie point symmetry generators and the corresponding symmetry reductions, through which certain analytic solutions are obtained via the power series expansion method and the (G'/G) expansion method. We demonstrate that such an equation is strictly self-adjoint, and the conservation laws associated with the Lie point symmetry generators are derived.
Representation Theory of Algebraic Groups and Quantum Groups
Gyoja, A; Shinoda, K-I; Shoji, T; Tanisaki, Toshiyuki
2010-01-01
Invited articles by top notch expertsFocus is on topics in representation theory of algebraic groups and quantum groupsOf interest to graduate students and researchers in representation theory, group theory, algebraic geometry, quantum theory and math physics
Finding a nonlinear lattice with improved integrability using Lie transform perturbation theory
International Nuclear Information System (INIS)
Sonnad, Kiran G.; Cary, John R.
2004-01-01
A condition for improved dynamic aperture for nonlinear, alternating gradient transport systems is derived using Lie transform perturbation theory. The Lie transform perturbation method is used here to perform averaging over fast oscillations by canonically transforming to slowly oscillating variables. This is first demonstrated for a linear sinusoidal focusing system. This method is then employed to average the dynamics over a lattice period for a nonlinear focusing system, provided by the use of higher order poles such as sextupoles and octupoles along with alternate gradient quadrupoles. Unlike the traditional approach, the higher order focusing is not treated as a perturbation. The Lie transform method is particularly advantageous for such a system where the form of the Hamiltonian is complex. This is because the method exploits the property of canonical invariance of Poisson brackets so that the change of variables is accomplished by just replacing the old ones with the new. The analysis shows the existence of a condition in which the system is azimuthally symmetric in the transformed, slowly oscillating frame. Such a symmetry in the time averaged frame renders the system nearly integrable in the laboratory frame. This condition leads to reduced chaos and improved confinement when compared to a system that is not close to integrability. Numerical calculations of single-particle trajectories and phase space projections of the dynamic aperture performed for a lattice with quadrupoles and sextupoles confirm that this is indeed the case
Celse, Jérémy; Chang, Kirk
2017-11-30
This research analyzed whether political leaders make people lie via priming experiments. Priming is a non-conscious and implicit memory effect in which exposure to one stimulus affects the response to another. Following priming theories, we proposed an innovative concept that people who perceive leaders to be dishonest (such as liars) are likely to lie themselves. We designed three experiments to analyze and critically discussed the potential influence of prime effect on lying behavior, through the prime effect of French political leaders (including general politicians, presidents and parties). Experiment 1 discovered that participants with non-politician-prime were less likely to lie (compared to politician-prime). Experiment 2A discovered that, compared to Hollande-prime, Sarkozy-prime led to lying behavior both in gravity (i.e., bigger lies) and frequency (i.e., lying more frequently). Experiment 2B discovered that Republicans-prime yielded an impact on more lying behavior, and Sarkozy-prime made such impact even stronger. Overall, the research findings suggest that lying can be triggered by external influencers such as leaders, presidents and politicians in the organizations. Our findings have provided valuable insights into organizational leaders and managers in their personnel management practice, especially in the intervention of lying behavior. Our findings also have offered new insights to explain non-conscious lying behavior.
Renormalized Lie perturbation theory
International Nuclear Information System (INIS)
Rosengaus, E.; Dewar, R.L.
1981-07-01
A Lie operator method for constructing action-angle transformations continuously connected to the identity is developed for area preserving mappings. By a simple change of variable from action to angular frequency a perturbation expansion is obtained in which the small denominators have been renormalized. The method is shown to lead to the same series as the Lagrangian perturbation method of Greene and Percival, which converges on KAM surfaces. The method is not superconvergent, but yields simple recursion relations which allow automatic algebraic manipulation techniques to be used to develop the series to high order. It is argued that the operator method can be justified by analytically continuing from the complex angular frequency plane onto the real line. The resulting picture is one where preserved primary KAM surfaces are continuously connected to one another
Directory of Open Access Journals (Sweden)
Hernández Fernández, Isabel
2008-01-01
Full Text Available En este artículo, los autores pretenden mostrar y explicar cómo la Teoría de Lie se puede aplicar a la resolución de algunos problemas relativos a la Economía y a las Finanzas. Concretamente, se realiza un análisis de dos de esos problemas y se discuten tanto sus aspectos matemáticos como el acercamiento hecho desde la Teoría de Lie para su resolución. Igualmente, se indican los avances más recientes existentes en esta línea de investigación, mencionando también algunos problemas abiertos que pueden ser tratados en futuros trabajos. = This paper shows and explains two problems in Economics and Finance, both dealt with a Lie Theory approach. So, mathematical aspects for these approaches are put forward and discussed in several economic problems which have been previously considered in the literature. Besides, some advances on this topic are also shown, mentioning some open problems for future research.
Homotopy Lie superalgebra in Yang-Mills theory
International Nuclear Information System (INIS)
Zeitlin, Anton M.
2007-01-01
The Yang-Mills equations are formulated in the form of generalized Maurer-Cartan equations, such that the corresponding algebraic operations are shown to satisfy the defining relations of homotopy Lie superalgebra
Group-theoretical method in the many-beam theory of electron diffraction
International Nuclear Information System (INIS)
Kogiso, Motokazu; Takahashi, Hidewo.
1977-01-01
A group-theoretical method is developed for the many-beam dynamical theory of the symmetric Laue case. When the incident wave is directed so that the Laue point lies on a symmetric position in the reciprocal lattice, the dispersion matrix in the fundamental equation can be reduced to a block diagonal form. The transformation matrix is composed of column vectors belonging to irreducible representations of the group of the incident wave vector. Without performing reduction, the reduced form of the dispersion matrix is determined from characters of representations. Practical application is made to the case of symmorphic crystals, where general reduced forms and all solvable examples are given in terms of some geometrical factors of reciprocal lattice arrangements. (auth.)
Quantum control and representation theory
International Nuclear Information System (INIS)
Ibort, A; Perez-Pardo, J M
2009-01-01
A new notion of controllability for quantum systems that takes advantage of the linear superposition of quantum states is introduced. We call such a notion von Neumann controllability, and it is shown that it is strictly weaker than the usual notion of pure state and operator controllability. We provide a simple and effective characterization of it by using tools from the theory of unitary representations of Lie groups. In this sense, we are able to approach the problem of control of quantum states from a new perspective, that of the theory of unitary representations of Lie groups. A few examples of physical interest and the particular instances of compact and nilpotent dynamical Lie groups are discussed
From groups to geometry and back
Climenhaga, Vaughn
2017-01-01
Groups arise naturally as symmetries of geometric objects, and so groups can be used to understand geometry and topology. Conversely, one can study abstract groups by using geometric techniques and ultimately by treating groups themselves as geometric objects. This book explores these connections between group theory and geometry, introducing some of the main ideas of transformation groups, algebraic topology, and geometric group theory. The first half of the book introduces basic notions of group theory and studies symmetry groups in various geometries, including Euclidean, projective, and hyperbolic. The classification of Euclidean isometries leads to results on regular polyhedra and polytopes; the study of symmetry groups using matrices leads to Lie groups and Lie algebras. The second half of the book explores ideas from algebraic topology and geometric group theory. The fundamental group appears as yet another group associated to a geometric object and turns out to be a symmetry group using covering space...
Linear algebra meets Lie algebra: the Kostant-Wallach theory
Shomron, Noam; Parlett, Beresford N.
2008-01-01
In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.
Coproduct and star product in field theories on Lie-algebra noncommutative space-times
International Nuclear Information System (INIS)
Amelino-Camelia, Giovanni; Arzano, Michele
2002-01-01
We propose a new approach to field theory on κ-Minkowski noncommutative space-time, a popular example of Lie-algebra space-time. Our proposal is essentially based on the introduction of a star product, a technique which is proving to be very fruitful in analogous studies of canonical noncommutative space-times, such as the ones recently found to play a role in the description of certain string-theory backgrounds. We find to be incorrect the expectation, previously reported in the literature, that the lack of symmetry of the κ-Poincare coproduct should lead to interaction vertices that are not symmetric under exchanges of the momenta of identical particles entering the relevant processes. We show that in κ-Minkowski the coproduct and the star product must indeed treat momenta in a nonsymmetric way, but the overall structure of interaction vertices is symmetric under exchange of identical particles. We also show that in κ-Minkowski field theories it is convenient to introduce the concepts of 'planar' and 'nonplanar' Feynman loop diagrams, again in close analogy with the corresponding concepts previously introduced in the study of field theories in canonical noncommutative space-times
Hermann Weyl and Representation Theory
Indian Academy of Sciences (India)
His work on the theory ofLie groups was motivated by his life-long interest in quantummechanics and relativity. When Weyl entered Lie theory,it mostly focussed on the infinitesimal, and he strove to bringin a global perspective. Time and again, Weyl's ideas arisingin one context have been adapted and applied to wholly ...
Bestvina, Mladen; Vogtmann, Karen
2014-01-01
Geometric group theory refers to the study of discrete groups using tools from topology, geometry, dynamics and analysis. The field is evolving very rapidly and the present volume provides an introduction to and overview of various topics which have played critical roles in this evolution. The book contains lecture notes from courses given at the Park City Math Institute on Geometric Group Theory. The institute consists of a set of intensive short courses offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The courses begin at an introductory level suitable for graduate students and lead up to currently active topics of research. The articles in this volume include introductions to CAT(0) cube complexes and groups, to modern small cancellation theory, to isometry groups of general CAT(0) spaces, and a discussion of nilpotent genus in the context of mapping class groups and CAT(0) gro...
Lie bialgebras with triangular decomposition
International Nuclear Information System (INIS)
Andruskiewitsch, N.; Levstein, F.
1992-06-01
Lie bialgebras originated in a triangular decomposition of the underlying Lie algebra are discussed. The explicit formulas for the quantization of the Heisenberg Lie algebra and some motion Lie algebras are given, as well as the algebra of rational functions on the quantum Heisenberg group and the formula for the universal R-matrix. (author). 17 refs
On the exceptional generalised Lie derivative for d≥7
International Nuclear Information System (INIS)
Rosabal, J.A.
2015-01-01
In this work we revisit the E_8×ℝ"+ generalised Lie derivative encoding the algebra of diffeomorphisms and gauge transformations of compactifications of M-theory on eight-dimensional manifolds, by extending certain features of the E_7×ℝ"+ one. Compared to its E_d×ℝ"+, d≤7 counterparts, a new term is needed for consistency. However, we find that no compensating parameters need to be introduced, but rather that the new term can be written in terms of the ordinary generalised gauge parameters by means of a connection. This implies that no further degrees of freedom, beyond those of the field content of the E_8 group, are needed to have a well defined theory. We discuss the implications of the structure of the E_8×ℝ"+ generalised transformation on the construction of the d=8 generalised geometry. Finally, we suggest how to lift the generalised Lie derivative to eleven dimensions.
Unbounded representations of symmetry groups in gauge quantum field theory. Pt. 1
International Nuclear Information System (INIS)
Voelkel, A.H.
1983-01-01
Symmetry groups and especially the covariance (substitution rules) of the basic fields in a gauge quantum field theory of the Wightman-Garding type are investigated. By means of the continuity properties hidden in the substitution rules it is shown that every unbounded form-isometric representation U of a Lie group has a form-skew-symmetric differential deltaU with dense domain in the unphysical Hilbert space. Necessary and sufficient conditions for the existence of the closures of U and deltaU as well as for the isometry of U are derived. It is proved that a class of representations of the transition group enforces a relativistic confinement mechanism, by which some or all basic fields are confined but certain mixed products of them are not. (orig.)
Dual Solutions for Nonlinear Flow Using Lie Group Analysis.
Directory of Open Access Journals (Sweden)
Muhammad Awais
Full Text Available `The aim of this analysis is to investigate the existence of the dual solutions for magnetohydrodynamic (MHD flow of an upper-convected Maxwell (UCM fluid over a porous shrinking wall. We have employed the Lie group analysis for the simplification of the nonlinear differential system and computed the absolute invariants explicitly. An efficient numerical technique namely the shooting method has been employed for the constructions of solutions. Dual solutions are computed for velocity profile of an upper-convected Maxwell (UCM fluid flow. Plots reflecting the impact of dual solutions for the variations of Deborah number, Hartman number, wall mass transfer are presented and analyzed. Streamlines are also plotted for the wall mass transfer effects when suction and blowing situations are considered.
New Pathways between Group Theory and Model Theory
Fuchs, László; Goldsmith, Brendan; Strüngmann, Lutz
2017-01-01
This volume focuses on group theory and model theory with a particular emphasis on the interplay of the two areas. The survey papers provide an overview of the developments across group, module, and model theory while the research papers present the most recent study in those same areas. With introductory sections that make the topics easily accessible to students, the papers in this volume will appeal to beginning graduate students and experienced researchers alike. As a whole, this book offers a cross-section view of the areas in group, module, and model theory, covering topics such as DP-minimal groups, Abelian groups, countable 1-transitive trees, and module approximations. The papers in this book are the proceedings of the conference “New Pathways between Group Theory and Model Theory,” which took place February 1-4, 2016, in Mülheim an der Ruhr, Germany, in honor of the editors’ colleague Rüdiger Göbel. This publication is dedicated to Professor Göbel, who passed away in 2014. He was one of th...
Loop groups and Yang-Mills theory in dimension two
DEFF Research Database (Denmark)
Gravesen, Jens
1990-01-01
Given a connection ω in a G-bundle over S2, then a process called radial trivialization from the poles gives a unique clutching function, i.e., an element γ of the loop group ΩG. Up to gauge equivalence, ω is completely determined by γ and a map f:S2 →g into the Lie algebra. Moreover, the Yang......-Mills function of ω is the sum of the energy of γ and the square of a certain norm of f. In particular, the Yang-Mills functional has the same Morse theory as the energy functional on ΩG. There is a similar description of connections in a G-bundle over an arbitrary Riemann surface, but so far not of the Yang...
On the group theoretical meaning of conformal field theories in the framework of coadjoint orbits
International Nuclear Information System (INIS)
Aratyn, H.; Nissimov, E.; Pacheva, S.
1990-01-01
We present a unifying approach to conformal field theories and other geometric models within the formalism of coadjoint orbits of infinite dimensional Lie groups with central extensions. Starting from the previously obtained general formula for the symplectic action in terms of two fundamental group one-cocycles, we derive the most general form of the Polyakov-Wiegmann composition laws for any geometric model. These composition laws are succinct expressions of all pertinent Noether symmetries. As a basic consequence we obtain Ward identities allowing for the exact quantum solvability of any geometric model. (orig.)
Bismut's way of the Malliavin calculus for large order generators on a Lie group
Léandre, Rémi
2018-01-01
We adapt Bismut's mechanism of the Malliavin Calculus to right invariant big order generator on a Lie group. We use deeply the symmetry in order to avoid the use of the Malliavin matrix. As an application, we deduce logarithmic estimates in small time of the heat kernel.
Cartan determinants, LIE algebra extensions, and the exceptional group series
International Nuclear Information System (INIS)
Capps, R.H.
1986-01-01
In this note the author utilizes the determinant of the generalized Cartan matrix for candidate Dynkin systems for two purposes. The first is to provide an uncomplicated criterion for classifying candidate one-root extensions of diagrams for semisimple Lie algebras. The second is to help determine some important properties of related Lie algebras and their representations
Directory of Open Access Journals (Sweden)
Jen-Cheng Wang
Full Text Available Lie group analysis of the photo-induced fluorescence of Drosophila oogenesis with the asymmetrically localized Gurken protein has been performed systematically to assess the roles of ligand-receptor complexes in follicle cells. The (2×2 matrix representations resulting from the polarized tissue spectra were employed to characterize the asymmetrical Gurken distributions. It was found that the fluorescence of the wild-type egg shows the Lie point symmetry X 23 at early stages of oogenesis. However, due to the morphogen regulation by intracellular proteins and extracellular proteins, the fluorescence of the embryogenesis with asymmetrically localized Gurken expansions exhibits specific symmetry features: Lie point symmetry Z 1 and Lie point symmetry X 1. The novel approach developed herein was successfully used to validate that the invariant-theoretical characterizations are consonant with the observed asymmetric fluctuations during early embryological development.
Lie algebras and linear differential equations.
Brockett, R. W.; Rahimi, A.
1972-01-01
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
Geometric group theory an introduction
Löh, Clara
2017-01-01
Inspired by classical geometry, geometric group theory has in turn provided a variety of applications to geometry, topology, group theory, number theory and graph theory. This carefully written textbook provides a rigorous introduction to this rapidly evolving field whose methods have proven to be powerful tools in neighbouring fields such as geometric topology. Geometric group theory is the study of finitely generated groups via the geometry of their associated Cayley graphs. It turns out that the essence of the geometry of such groups is captured in the key notion of quasi-isometry, a large-scale version of isometry whose invariants include growth types, curvature conditions, boundary constructions, and amenability. This book covers the foundations of quasi-geometry of groups at an advanced undergraduate level. The subject is illustrated by many elementary examples, outlooks on applications, as well as an extensive collection of exercises.
International Nuclear Information System (INIS)
Geng, L. S.; Camalich, J. Martin; Vacas, M. J. Vicente
2009-01-01
We present a calculation of the leading SU(3)-breaking O(p 3 ) corrections to the electromagnetic moments and charge radius of the lowest-lying decuplet resonances in covariant chiral perturbation theory. In particular, the magnetic dipole moment of the members of the decuplet is predicted fixing the only low-energy constant (LEC) present up to this order with the well-measured magnetic dipole moment of the Ω - . We predict μ Δ ++ =6.04(13) and μ Δ + =2.84(2), which agree well with the current experimental information. For the electric quadrupole moment and the charge radius, we use state-of-the-art lattice QCD results to determine the corresponding LECs, whereas for the magnetic octupole moment there is no unknown LEC up to the order considered here, and we obtain a pure prediction. We compare our results with those reported in large N c , lattice QCD, heavy-baryon chiral perturbation theory, and other models.
Hermann Weyl and Representation Theory
Indian Academy of Sciences (India)
told us to hear the volume of a drum but, about the shape, ... resentation theory of Lie groups which solved fundamental problems, and ..... Cartan's classification of simple Lie algebras depended ..... age of 27 due to general sepsis. Weyl was ...
Jurco, Branislav
2011-01-01
Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, which is simply connected in each simplicial level. We use the 1-jet of the classifying space of G to construct, starting from g, a Lie k-algebra L. The so constructed Lie k-algebra L is actually a differential graded Lie algebra. The differential and the brackets are explicitly described in terms (of a part) of the corresponding k-hypercrossed complex structure of Ng. The res...
Sequences, groups, and number theory
Rigo, Michel
2018-01-01
This collaborative book presents recent trends on the study of sequences, including combinatorics on words and symbolic dynamics, and new interdisciplinary links to group theory and number theory. Other chapters branch out from those areas into subfields of theoretical computer science, such as complexity theory and theory of automata. The book is built around four general themes: number theory and sequences, word combinatorics, normal numbers, and group theory. Those topics are rounded out by investigations into automatic and regular sequences, tilings and theory of computation, discrete dynamical systems, ergodic theory, numeration systems, automaton semigroups, and amenable groups. This volume is intended for use by graduate students or research mathematicians, as well as computer scientists who are working in automata theory and formal language theory. With its organization around unified themes, it would also be appropriate as a supplemental text for graduate level courses.
What is new in the study of differential equations by group theoretical methods
International Nuclear Information System (INIS)
Winternitz, P.
1986-11-01
Several recent developments have made the application of group theory to the solving of differential equations more powerful than it used to be. The ones discussed here are: 1. The advent of symbol manipulating computer languages that greatly simplify the construction of the symmetry group of an equation 2. Methods of finding all subgroups of a given Lie symmetry group 3. The theory of infinite dimensional Lie algebras 4. The combination of group theory and singularity analysis
A quantum group structure in integrable conformal field theories
International Nuclear Information System (INIS)
Smit, D.J.
1990-01-01
We discuss a quantization prescription of the conformal algebras of a class of d=2 conformal field theories which are integrable. We first give a geometrical construction of certain extensions of the classical Virasoro algebra, known as classical W algebras, in which these algebras arise as the Lie algebra of the second Hamiltonian structure of a generalized Korteweg-de Vries hierarchy. This fact implies that the W algebras, obtained as a reduction with respect to the nilpotent subalgebras of the Kac-Moody algebra, describe the intergrability of a Toda field theory. Subsequently we determine the coadjoint operators of the W algebras, and relate these to classical Yang-Baxter matrices. The quantization of these algebras can be carried out using the concept of a so-called quantum group. We derive the condition under which the representations of these quantum groups admit a Hilbert space completion by exploring the relation with the braid group. Then we consider a modification of the Miura transformation which we use to define a quantum W algebra. This leads to an alternative interpretation of the coset construction for Kac-Moody algebras in terms of nonlinear integrable hierarchies. Subsequently we use the connection between the induced braid group representations and the representations of the mapping class group of Riemann surfaces to identify an action of the W algebras on the moduli space of stable curves, and we give the invariants of this action. This provides a generalization of the situation for the Virasoro algebra, where such an invariant is given by the so-called Mumford form which describes the partition function of the bosonic string. (orig.)
Lie Algebroids in Classical Mechanics and Optimal Control
Directory of Open Access Journals (Sweden)
Eduardo Martínez
2007-03-01
Full Text Available We review some recent results on the theory of Lagrangian systems on Lie algebroids. In particular we consider the symplectic and variational formalism and we study reduction. Finally we also consider optimal control systems on Lie algebroids and we show how to reduce Pontryagin maximum principle.
International Nuclear Information System (INIS)
Thierry-Mieg, Jean
2006-01-01
In Yang-Mills theory, the charges of the left and right massless Fermions are independent of each other. We propose a new paradigm where we remove this freedom and densify the algebraic structure of Yang-Mills theory by integrating the scalar Higgs field into a new gauge-chiral 1-form which connects Fermions of opposite chiralities. Using the Bianchi identity, we prove that the corresponding covariant differential is associative if and only if we gauge a Lie-Kac super-algebra. In this model, spontaneous symmetry breakdown naturally occurs along an odd generator of the super-algebra and induces a representation of the Connes-Lott non commutative differential geometry of the 2-point finite space
International Nuclear Information System (INIS)
Fradkin, E.S.; Linetsky, V.Ya.
1990-06-01
With any semisimple Lie algebra g we associate an infinite-dimensional Lie algebra AC(g) which is an analytic continuation of g from its root system to its root lattice. The manifest expressions for the structure constants of analytic continuations of the symplectic Lie algebras sp2 n are obtained by Poisson-bracket realizations method and AC(g) for g=sl n and so n are discussed. The representations, central extension, supersymmetric and higher spin generalizations are considered. The Virasoro theory is a particular case when g=sp 2 . (author). 9 refs
A Third-Order p-Laplacian Boundary Value Problem Solved by an SL(3,ℝ Lie-Group Shooting Method
Directory of Open Access Journals (Sweden)
Chein-Shan Liu
2013-01-01
Full Text Available The boundary layer problem for power-law fluid can be recast to a third-order p-Laplacian boundary value problem (BVP. In this paper, we transform the third-order p-Laplacian into a new system which exhibits a Lie-symmetry SL(3,ℝ. Then, the closure property of the Lie-group is used to derive a linear transformation between the boundary values at two ends of a spatial interval. Hence, we can iteratively solve the missing left boundary conditions, which are determined by matching the right boundary conditions through a finer tuning of r∈[0,1]. The present SL(3,ℝ Lie-group shooting method is easily implemented and is efficient to tackle the multiple solutions of the third-order p-Laplacian. When the missing left boundary values can be determined accurately, we can apply the fourth-order Runge-Kutta (RK4 method to obtain a quite accurate numerical solution of the p-Laplacian.
Lie Group Classification of a Generalized Lane-Emden Type System in Two Dimensions
Directory of Open Access Journals (Sweden)
Motlatsi Molati
2012-01-01
Full Text Available The aim of this work is to perform a complete Lie symmetry classification of a generalized Lane-Emden type system in two dimensions which models many physical phenomena in biological and physical sciences. The classical approach of group classification is employed for classification. We show that several cases arise in classifying the arbitrary parameters, the forms of which include amongst others the power law nonlinearity, and exponential and quadratic forms.
Morozov, Oleg I.
2018-06-01
The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given PDE. The exotic cohomology of the symmetry algebras opens a way to formulate such conditions in internal terms of the PDE s under the study. In this paper we consider certain examples of infinite-dimensional Lie algebras with nontrivial second exotic cohomology groups and show that the Maurer-Cartan forms of the associated extensions of these Lie algebras generate Lax representations for integrable systems, both known and new ones.
Smooth homogeneous structures in operator theory
Beltita, Daniel
2005-01-01
Geometric ideas and techniques play an important role in operator theory and the theory of operator algebras. Smooth Homogeneous Structures in Operator Theory builds the background needed to understand this circle of ideas and reports on recent developments in this fruitful field of research. Requiring only a moderate familiarity with functional analysis and general topology, the author begins with an introduction to infinite dimensional Lie theory with emphasis on the relationship between Lie groups and Lie algebras. A detailed examination of smooth homogeneous spaces follows. This study is illustrated by familiar examples from operator theory and develops methods that allow endowing such spaces with structures of complex manifolds. The final section of the book explores equivariant monotone operators and Kähler structures. It examines certain symmetry properties of abstract reproducing kernels and arrives at a very general version of the construction of restricted Grassmann manifolds from the theory of loo...
Dynamics on the group manifolds and path integral
International Nuclear Information System (INIS)
Marinov, M.S.; Terentyev, M.V.
1979-01-01
Classical and quantum dynamics onn the compact simple Lie group and on the sphere of arbitrary dimensionality are considered. The accuracy of the semiclassical approximation for Green functions is discussed. Various path integral representations of the Green functions are presented. The special features of these representations due to the compactness and curvature are analysed. Basic results of the theory of Lie algebras and Lie groups used in the main text are presented
Group theory and its applications
Patra, Prasanta Kumar
2018-01-01
Every molecule possesses symmetry and hence has symmetry operations and symmetry elements. From symmetry properties of a system we can deduce its significant physical results. Consequently it is essential to operations of a system forms a group. Group theory is an abstract mathematical tool that underlies the study of symmetry and invariance. By using the concepts of symmetry and group theory, it is possible to obtain the members of complete set of known basis functions of the various irreducible representations of the group. I practice this is achieved by applying the projection operators to linear combinations of atomic orbital (LCAO) when the valence electrons are tightly bound to the ions, to orthogonalized plane waves (OPW) when valence electrons are nearly free and to the other given functions that are judged to the particular system under consideration. In solid state physics the group theory is indispensable in the context of finding the energy bands of electrons in solids. Group theory can be applied...
Recoupling Lie algebra and universal ω-algebra
International Nuclear Information System (INIS)
Joyce, William P.
2004-01-01
We formulate the algebraic version of recoupling theory suitable for commutation quantization over any gradation. This gives a generalization of graded Lie algebra. Underlying this is the new notion of an ω-algebra defined in this paper. ω-algebra is a generalization of algebra that goes beyond nonassociativity. We construct the universal enveloping ω-algebra of recoupling Lie algebras and prove a generalized Poincare-Birkhoff-Witt theorem. As an example we consider the algebras over an arbitrary recoupling of Z n graded Heisenberg Lie algebra. Finally we uncover the usual coalgebra structure of a universal envelope and substantiate its Hopf structure
Walczyk, Jeffrey J.; Igou, Frank P.; Dixon, Alexa P.; Tcholakian, Talar
2013-01-01
This article critically reviews techniques and theories relevant to the emerging field of “lie detection by inducing cognitive load selectively on liars.” To help these techniques benefit from past mistakes, we start with a summary of the polygraph-based Controlled Question Technique (CQT) and the major criticisms of it made by the National Research Council (2003), including that it not based on a validated theory and administration procedures have not been standardized. Lessons from the more successful Guilty Knowledge Test are also considered. The critical review that follows starts with the presentation of models and theories offering insights for cognitive lie detection that can undergird theoretically load-inducing approaches. This is followed by evaluation of specific research-based, load-inducing proposals, especially for their susceptibility to rehearsal and other countermeasures. To help organize these proposals and suggest new direction for innovation and refinement, a theoretical taxonomy is presented based on the type of cognitive load induced in examinees (intrinsic or extraneous) and how open-ended the responses to test items are. Finally, four recommendations are proffered that can help researchers and practitioners to avert the corresponding mistakes with the CQT and yield new, valid cognitive lie detection technologies. PMID:23378840
Introduction to quantum groups
Chaichian, Masud
1996-01-01
In the past decade there has been an extemely rapid growth in the interest and development of quantum group theory.This book provides students and researchers with a practical introduction to the principal ideas of quantum groups theory and its applications to quantum mechanical and modern field theory problems. It begins with a review of, and introduction to, the mathematical aspects of quantum deformation of classical groups, Lie algebras and related objects (algebras of functions on spaces, differential and integral calculi). In the subsequent chapters the richness of mathematical structure
Mixed global anomalies and boundary conformal field theories
Numasawa, Tokiro; Yamaguchi, Satoshi
2017-01-01
We consider the relation of mixed global gauge gravitational anomalies and boundary conformal field theory in WZW models for simple Lie groups. The discrete symmetries of consideration are the centers of the simple Lie groups. These mixed anomalies prevent to gauge them i.e, take the orbifold by the center. The absence of anomalies impose conditions on the levels of WZW models. Next, we study the conformal boundary conditions for the original theories. We consider the existence of a conformal...
Vertex operator algebras and conformal field theory
International Nuclear Information System (INIS)
Huang, Y.Z.
1992-01-01
This paper discusses conformal field theory, an important physical theory, describing both two-dimensional critical phenomena in condensed matter physics and classical motions of strings in string theory. The study of conformal field theory will deepen the understanding of these theories and will help to understand string theory conceptually. Besides its importance in physics, the beautiful and rich mathematical structure of conformal field theory has interested many mathematicians. New relations between different branches of mathematics, such as representations of infinite-dimensional Lie algebras and Lie groups, Riemann surfaces and algebraic curves, the Monster sporadic group, modular functions and modular forms, elliptic genera and elliptic cohomology, Calabi-Yau manifolds, tensor categories, and knot theory, are revealed in the study of conformal field theory. It is therefore believed that the study of the mathematics involved in conformal field theory will ultimately lead to new mathematical structures which would be important to both mathematics and physics
Reductive Lie-admissible algebras applied to H-spaces and connections
International Nuclear Information System (INIS)
Sagle, A.A.
1982-01-01
An algebra A with multiplication xy is Lie-admissible if the vector space A with new multiplication [x,y] = xy-yx is a Lie algebra; we denote this Lie algebra by A - . Thus, an associative algebra is Lie-admissible but a Cayley algebra is not Lie-admissible. In this paper we show how Lie-admissible algebras arise from Lie groups and their application to differential geometry on Lie groups via the following theorem. Let A be an n-dimensional Lie-admissible algebra over the reals. Let G be a Lie group with multiplication function μ and with Lie algebra g which is isomorphic to A - . Then there exiss a corrdinate system at the identify e in G which represents μ by a function F:gxg→g defined locally at the origin, such that the second derivative, F 2 , at the origin defines on the vector space g the structure of a nonassociative algebra (g, F 2 ). Furthermore this algebra is isomorphic to A and (g, F 2 ) - is isomorphic to A - . Thus roughly, any Lie-admissible algebra is isomorphic to an algebra obtained from a Lie algebra via a change of coordinates in the Lie group. Lie algebras arise by using canonical coordinates and the Campbell-Hausdorff formula. Applications of this show that any G-invariant psuedo-Riemannian connection on G is completely determined by a suitable Lie-admissible algebra. These results extend to H-spaces, reductive Lie-admissible algebras and connections on homogeneous H-spaces. Thus, alternative and other non-Lie-admissible algebras can be utilized
Exceptional gauge groups and quantum theory
International Nuclear Information System (INIS)
Horwitz, L.P.; Biedenharn, L.C.
1979-01-01
It is shown that a Hilbert space over the real Clifford algebra C 7 provides a mathematical framework, consistent with the structure of the usual quantum mechanical formalism, for models for the unification of weak, electromagnetic and strong interactions utilizing the exceptional Lie groups. In particular, in case no further structure is assumed beyond that of C 7 , the group of automorphisms leaving invariant a minimal subspace acts, in the ideal generated by that subspace, as G 2 , and the subgroup of this group leaving one generating element (e 7 ) fixed acts, in this ideal, as the color gauge group SU(3). A generalized phase algebra AcontainsC 7 is defined by the requirement that quantum mechanical states can be consistently constructed for a theory in which the smallest linear manifolds are closed over the subalgebra C(1,e 7 ) (isomorphic to the complex field) of C 7 . Eight solutions are found for the generalized phase algebra, corresponding (up to an overall sign), in effect, to the use of +- e 7 as imaginary unit in each of four superselection sectors. Operators linear over these alternative forms of imanary unit provide distinct types of ''lepton--quark'' and ''quark--quark'' transitions. The subgroup in A which leaves expectation values of operators linear over A invariant is its unitary subgroup U(4), and is a realization (explicitly constructed) of the U(4) invariance of the complex scalar product. An embedding of the algebraic Hilbert space into the complex space defined over C(1,e 7 ) is shown to lead to a decomposition into ''lepton and ''quark'' superselection subspaces. The color SU(3) subgroup of G 2 coincides with the SU(3) subgroup of the generalized phase U(4) which leaves the ''lepton'' space invariant. The problem of constructing tensor products is studied, and some remarks are made on observability and the role of nonassociativity
Twisted equivariant K-theory, groupoids and proper actions
Cantarero, Jose
2009-01-01
In this paper we define twisted equivariant K-theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid, this defines a periodic cohomology theory on the category of finite CW-complexes with equivariant stable projective bundles. A classification of these bundles is shown. We also obtain a completion theorem and apply these results to proper actions of groups.
Symmetry and group theory in chemistry
Ladd, M
1998-01-01
A comprehensive discussion of group theory in the context of molecular and crystal symmetry, this book covers both point-group and space-group symmetries.Provides a comprehensive discussion of group theory in the context of molecular and crystal symmetryCovers both point-group and space-group symmetriesIncludes tutorial solutions
The Relative Lie Algebra Cohomology of the Weil Representation
Ralston, Jacob
We study the relative Lie algebra cohomology of so(p,q) with values in the Weil representation piof the dual pair Sp(2k, R) x O(p,q ). Using the Fock model defined in Chapter 2, we filter this complex and construct the associated spectral sequence. We then prove that the resulting spectral sequence converges to the relative Lie algebra cohomology and has E0 term, the associated graded complex, isomorphic to a Koszul complex, see Section 3.4. It is immediate that the construction of the spectral sequence of Chapter 3 can be applied to any reductive subalgebra g ⊂ sp(2k(p + q), R). By the Weil representation of O( p,|q), we mean the twist of the Weil representation of the two-fold cover O(pq)[special character omitted] by a suitable character. We do this to make the center of O(pq)[special character omitted] act trivially. Otherwise, all relative Lie algebra cohomology groups would vanish, see Proposition 4.10.2. In case the symplectic group is large relative to the orthogonal group (k ≥ pq), the E 0 term is isomorphic to a Koszul complex defined by a regular sequence, see 3.4. Thus, the cohomology vanishes except in top degree. This result is obtained without calculating the space of cochains and hence without using any representation theory. On the other hand, in case k BMR], this author wrote with his advisor John Millson and Nicolas Bergeron of the University of Paris.
Description of a class of superstring compactifications related to semi-simple Lie algebras
International Nuclear Information System (INIS)
Markushevich, D.I.; Ol'shanetskij, M.A.; Perelomov, A.M.
1986-01-01
A class of vacuum configurations in the superstring theory obtained by compactification of physical dimensions from ten to four is constructed. The compactification scheme involves taking quotients of tori of semisimple Lie algebras by finite symmetry group actions. The complete list of such configurations arising from actions by a Coxeter transformation is given. Some topological invariants having physical interpretations are calculated
Extending Sociocultural Theory to Group Creativity
Sawyer, Keith
2012-01-01
Sociocultural theory focuses on group processes through time, and argues that group phenomena cannot be reduced to explanation in terms of the mental states or actions of the participating individuals. This makes sociocultural theory particularly useful in the analysis of group creativity and group learning, because both group creativity and group…
Noncommutative Geometry in M-Theory and Conformal Field Theory
International Nuclear Information System (INIS)
Morariu, Bogdan
1999-01-01
In the first part of the thesis I will investigate in the Matrix theory framework, the subgroup of dualities of the Discrete Light Cone Quantization of M-theory compactified on tori, which corresponds to T-duality in the auxiliary Type II string theory. After a review of matrix theory compactification leading to noncommutative supersymmetric Yang-Mills gauge theory, I will present solutions for the fundamental and adjoint sections on a two-dimensional twisted quantum torus and generalize to three-dimensional twisted quantum tori. After showing how M-theory T-duality is realized in supersymmetric Yang-Mills gauge theories on dual noncommutative tori I will relate this to the mathematical concept of Morita equivalence of C*-algebras. As a further generalization, I consider arbitrary Ramond-Ramond backgrounds. I will also discuss the spectrum of the toroidally compactified Matrix theory corresponding to quantized electric fluxes on two and three tori. In the second part of the thesis I will present an application to conformal field theory involving quantum groups, another important example of a noncommutative space. First, I will give an introduction to Poisson-Lie groups and arrive at quantum groups using the Feynman path integral. I will quantize the symplectic leaves of the Poisson-Lie group SU(2)*. In this way we obtain the unitary representations of U q (SU(2)). I discuss the X-structure of SU(2)* and give a detailed description of its leaves using various parametrizations. Then, I will introduce a new reality structure on the Heisenberg double of Fun q (SL(N,C)) for q phase, which can be interpreted as the quantum phase space of a particle on the q-deformed mass-hyperboloid. I also present evidence that the above real form describes zero modes of certain non-compact WZNW-models
Noncommutative Geometry in M-Theory and Conformal Field Theory
Energy Technology Data Exchange (ETDEWEB)
Morariu, Bogdan [Univ. of California, Berkeley, CA (United States)
1999-05-01
In the first part of the thesis I will investigate in the Matrix theory framework, the subgroup of dualities of the Discrete Light Cone Quantization of M-theory compactified on tori, which corresponds to T-duality in the auxiliary Type II string theory. After a review of matrix theory compactification leading to noncommutative supersymmetric Yang-Mills gauge theory, I will present solutions for the fundamental and adjoint sections on a two-dimensional twisted quantum torus and generalize to three-dimensional twisted quantum tori. After showing how M-theory T-duality is realized in supersymmetric Yang-Mills gauge theories on dual noncommutative tori I will relate this to the mathematical concept of Morita equivalence of C*-algebras. As a further generalization, I consider arbitrary Ramond-Ramond backgrounds. I will also discuss the spectrum of the toroidally compactified Matrix theory corresponding to quantized electric fluxes on two and three tori. In the second part of the thesis I will present an application to conformal field theory involving quantum groups, another important example of a noncommutative space. First, I will give an introduction to Poisson-Lie groups and arrive at quantum groups using the Feynman path integral. I will quantize the symplectic leaves of the Poisson-Lie group SU(2)*. In this way we obtain the unitary representations of U_{q}(SU(2)). I discuss the X-structure of SU(2)* and give a detailed description of its leaves using various parametrizations. Then, I will introduce a new reality structure on the Heisenberg double of Fun_{q} (SL(N,C)) for q phase, which can be interpreted as the quantum phase space of a particle on the q-deformed mass-hyperboloid. I also present evidence that the above real form describes zero modes of certain non-compact WZNW-models.
Classification and identification of Lie algebras
Snobl, Libor
2014-01-01
The purpose of this book is to serve as a tool for researchers and practitioners who apply Lie algebras and Lie groups to solve problems arising in science and engineering. The authors address the problem of expressing a Lie algebra obtained in some arbitrary basis in a more suitable basis in which all essential features of the Lie algebra are directly visible. This includes algorithms accomplishing decomposition into a direct sum, identification of the radical and the Levi decomposition, and the computation of the nilradical and of the Casimir invariants. Examples are given for each algorithm. For low-dimensional Lie algebras this makes it possible to identify the given Lie algebra completely. The authors provide a representative list of all Lie algebras of dimension less or equal to 6 together with their important properties, including their Casimir invariants. The list is ordered in a way to make identification easy, using only basis independent properties of the Lie algebras. They also describe certain cl...
The Higgs mass derived from the U(3) Lie group
DEFF Research Database (Denmark)
Trinhammer, Ole; Bohr, Henrik; Jensen, Mogens O Stibius
2015-01-01
The Higgs mass value is derived from a Hamiltonian on the Lie group U(3) where we relate strong and electroweak energy scales. The baryon states of nucleon and delta resonances originate in specific Bloch wave degrees of freedom coupled to a Higgs mechanism which also gives rise to the usual gauge...... boson masses. The derived Higgs mass is around 125 GeV. From the same Hamiltonian, we derive the relative neutron to proton mass ratio and the N and Delta mass spectra. All compare rather well with the experimental values. We predict scarce neutral flavor baryon singlets that should be visible...... in scattering cross-sections for negative pions on protons, in photoproduction on neutrons, in neutron diffraction dissociation experiments and in invariant mass spectra of protons and negative pions in B-decays. The fundamental predictions are based on just one length scale and the fine structure constant...
A Corresponding Lie Algebra of a Reductive homogeneous Group and Its Applications
International Nuclear Information System (INIS)
Zhang Yu-Feng; Rui Wen-Juan; Wu Li-Xin
2015-01-01
With the help of a Lie algebra of a reductive homogeneous space G/K, where G is a Lie group and K is a resulting isotropy group, we introduce a Lax pair for which an expanding (2+1)-dimensional integrable hierarchy is obtained by applying the binormial-residue representation (BRR) method, whose Hamiltonian structure is derived from the trace identity for deducing (2+1)-dimensional integrable hierarchies, which was proposed by Tu, et al. We further consider some reductions of the expanding integrable hierarchy obtained in the paper. The first reduction is just right the (2+1)-dimensional AKNS hierarchy, the second-type reduction reveals an integrable coupling of the (2+1)-dimensional AKNS equation (also called the Davey-Stewartson hierarchy), a kind of (2+1)-dimensional Schrödinger equation, which was once reobtained by Tu, Feng and Zhang. It is interesting that a new (2+1)-dimensional integrable nonlinear coupled equation is generated from the reduction of the part of the (2+1)-dimensional integrable coupling, which is further reduced to the standard (2+1)-dimensional diffusion equation along with a parameter. In addition, the well-known (1+1)-dimensional AKNS hierarchy, the (1+1)-dimensional nonlinear Schrödinger equation are all special cases of the (2+1)-dimensional expanding integrable hierarchy. Finally, we discuss a few discrete difference equations of the diffusion equation whose stabilities are analyzed by making use of the von Neumann condition and the Fourier method. Some numerical solutions of a special stationary initial value problem of the (2+1)-dimensional diffusion equation are obtained and the resulting convergence and estimation formula are investigated. (paper)
Symmetries and groups in particle physics
International Nuclear Information System (INIS)
Scherer, Stefan
2016-01-01
The aim of this book consists of a didactic introduction to the group-theoretical considerations and methods, which have led to an ever deeper understanding of the interactions of the elementary particles. The first three chapters deal primarily with the foundations of the representation theory of primarily finite groups, whereby many results are also transferable to compact Lie groups. In the third chapter we discuss the concept of Lie groups and their connection with Lie algebras. In the remaining chapter it is mainly about the application of group theory in physics. Chapter 4 deals with the groups SO(3) and SU(2), which occur in connection with the description of the angular momentum in quantum mechanics. We discuss the Wigner-Eckar theorem together with some applications. In chapter 5 we are employed to the composition properties of strongly interacting systems, so called hadrons, and discuss extensively the transformation properties of quarks with relation to the special unitary groups. The Noether theorem is generally treated in connection to the conservation laws belonging to the Galilei group and the Poincare group. We confine us in chapter 6 to internal symmetries, but explain for that extensively the application to quantum field theory. Especially an outlook on the effect of symmetries in form of so called Ward identities is granted. In chapter 7 we turn towards the gauge principle and discuss first the construction of quantum electrodynamics. In the following we generalize the gauge principle to non-Abelian groups (Yang-Mills theories) and formulate the quantum chromodynamics (QCD). Especially we take a view of ''random'' global symmetries of QCD, especially the chiral symmetry. In chapter 8 we illuminate the phenomenon of spontaneous symmetry breaking both for global and for local symmetries. In the final chapter we work out the group-theoretical structure of the Standard Model. Finally by means of the group SU(5) we take a view to
Testosterone administration reduces lying in men.
Directory of Open Access Journals (Sweden)
Matthias Wibral
Full Text Available Lying is a pervasive phenomenon with important social and economic implications. However, despite substantial interest in the prevalence and determinants of lying, little is known about its biological foundations. Here we study a potential hormonal influence, focusing on the steroid hormone testosterone, which has been shown to play an important role in social behavior. In a double-blind placebo-controlled study, 91 healthy men (24.32±2.73 years received a transdermal administration of 50 mg of testosterone (n=46 or a placebo (n=45. Subsequently, subjects participated in a simple task, in which their payoff depended on the self-reported outcome of a die-roll. Subjects could increase their payoff by lying without fear of being caught. Our results show that testosterone administration substantially decreases lying in men. Self-serving lying occurred in both groups, however, reported payoffs were significantly lower in the testosterone group (p<0.01. Our results contribute to the recent debate on the effect of testosterone on prosocial behavior and its underlying channels.
Grid-group cultural theory: an introduction
Mamadouh, V.
1999-01-01
This article offers an introduction to grid-group cultural theory (also known as grid-group analysis, Cultural Theory or theory of socio-cultural viability), an approach that has been developed over the past thirty years in the work of the British anthropologists Mary Douglas and Michael Thompson,
Gauge theory of the post-Galilean groups
International Nuclear Information System (INIS)
Dimakis, A.
1985-01-01
By means of an extension of the field of real numbers we construct post-Galilean groups, which in a sense lay between the Galilean group and the Lorentz group. By gauging these groups we obtain a frame theory of gravitation, which comprises Newton--Cartan theory, general relativity, and an infinite number of intermediate theories. This leads to a better understanding of how the structural differences of the two main theories of gravitation arise
Group theoretical methods in physics. [Tuebingen, July 18-22, 1977
Energy Technology Data Exchange (ETDEWEB)
Kramer, P; Rieckers, A
1978-01-01
This volume comprises the proceedings of the 6th International Colloquium on Group Theoretical Methods in Physics, held at Tuebingen in July 1977. Invited papers were presented on the following topics: supersymmetry and graded Lie algebras; concepts of order and disorder arising from molecular physics; symplectic structures and many-body physics; symmetry breaking in statistical mechanics and field theory; automata and systems as examples of applied (semi-) group theory; renormalization group; and gauge theories. Summaries are given of the contributed papers, which can be grouped as follows: supersymmetry, symmetry in particles and relativistic physics; symmetry in molecular and solid state physics; broken symmetry and phase transitions; structure of groups and dynamical systems; representations of groups and Lie algebras; and general symmetries, quantization. Those individual papers in scope for the TIC data base are being entered from ATOMINDEX tapes. (RWR)
Path integral quantization of the Symplectic Leaves of the SU(2)*Poisson-Lie Group
International Nuclear Information System (INIS)
Morariu, B.
1997-01-01
The Feynman path integral is used to quantize the symplectic leaves of the Poisson-Lie group SU(2)*. In this way we obtain the unitary representations of Uq(su(2)). This is achieved by finding explicit Darboux coordinates and then using a phase space path integral. I discuss the *-structure of SU(2)* and give a detailed description of its leaves using various parameterizations and also compare the results with the path integral quantization of spin
International Nuclear Information System (INIS)
Bauer, M.; Itzykson, C.
1990-01-01
Recent investigations on the classification of rational conformal theories have suggested relations with finite groups. It is not known at present if this is more than a happy coincidence in simple cases or possibly some more profound link exploiting the analogy between fusion rules and decompositions of tensor products of group representations or even in a more abstract context q-deformations of Lie algebras for roots of unity. Although finite group theory is a very elaborate subject the authors review on a slightly non-trivial example some of its numerous aspects, in particular those related to rings of invariants. The hope was to grasp, if possible, some properties which stand a chance of being related to conformal theories. Subgroups of SU(2) were found to be related to the A-D-E classification of Wess-Zumino-Witten models based on the corresponding affine Lie algebra. Extending the investigations to SU(3) the authors have picked one of its classical subgroups as a candidate of interest
On the use of the autonomous Birkhoff equations in Lie series perturbation theory
Boronenko, T. S.
2017-02-01
In this article, we present the Lie transformation algorithm for autonomous Birkhoff systems. Here, we are referring to Hamiltonian systems that obey a symplectic structure of the general form. The Birkhoff equations are derived from the linear first-order Pfaff-Birkhoff variational principle, which is more general than the Hamilton principle. The use of 1-form in formulating the equations of motion in dynamics makes the Birkhoff method more universal and flexible. Birkhoff's equations have a tensorial character, so their form is independent of the coordinate system used. Two examples of normalization in the restricted three-body problem are given to illustrate the application of the algorithm in perturbation theory. The efficiency of this algorithm for problems of asymptotic integration in dynamics is discussed for the case where there is a need to use non-canonical variables in phase space.
Bidirectional composition on lie groups for gradient-based image alignment.
Mégret, Rémi; Authesserre, Jean-Baptiste; Berthoumieu, Yannick
2010-09-01
In this paper, a new formulation based on bidirectional composition on Lie groups (BCL) for parametric gradient-based image alignment is presented. Contrary to the conventional approaches, the BCL method takes advantage of the gradients of both template and current image without combining them a priori. Based on this bidirectional formulation, two methods are proposed and their relationship with state-of-the-art gradient based approaches is fully discussed. The first one, i.e., the BCL method, relies on the compositional framework to provide the minimization of the compensated error with respect to an augmented parameter vector. The second one, the projected BCL (PBCL), corresponds to a close approximation of the BCL approach. A comparative study is carried out dealing with computational complexity, convergence rate and frequence of convergence. Numerical experiments using a conventional benchmark show the performance improvement especially for asymmetric levels of noise, which is also discussed from a theoretical point of view.
Normalization in Lie algebras via mould calculus and applications
Paul, Thierry; Sauzin, David
2017-11-01
We establish Écalle's mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to reduce the Lie-theoretic problem to a mould equation, the solutions of which are remarkably explicit and can be fully described by means of a gauge transformation group. The dynamical applications include the construction of Poincaré-Dulac formal normal forms for a vector field around an equilibrium point, a formal infinite-order multiphase averaging procedure for vector fields with fast angular variables (Hamiltonian or not), or the construction of Birkhoff normal forms both in classical and quantum situations. As a by-product we obtain, in the case of harmonic oscillators, the convergence of the quantum Birkhoff form to the classical one, without any Diophantine hypothesis on the frequencies of the unperturbed Hamiltonians.
Abelian Chern-Simons theory and contact torsion
DEFF Research Database (Denmark)
McLellan, Brendan Donald Kenneth
2013-01-01
Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A shift reduced abelian Chern-Simons partition function is introduced using an alternative formulation of the partition function using formal ideas in ...... in quantum field theory. We compare the shift reduced partition function with other formulations of the abelian Chern-Simons partition function. This study naturally motivates an Atiyah-Patodi-Singer type index problem in contact geometry.......Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A shift reduced abelian Chern-Simons partition function is introduced using an alternative formulation of the partition function using formal ideas...
Two Mentalizing Capacities and the Understanding of Two Types of Lie Telling in Children
Hsu, Yik Kwan; Cheung, Him
2013-01-01
This study examined the interrelationships among second-order belief, interpretive theory of mind, inhibitory control, and the understanding of strategic versus white lies in 54 children approximately 5 years 7 months old. Results showed that second-order belief was associated with strategic-lie understanding, whereas interpretive theory of mind…
International Nuclear Information System (INIS)
Alvarez, O.; Liu Chienhao
1996-01-01
It has been suggested that a possible classical remnant of the phenomenon of target-space duality (T-duality) would be the equivalence of the classical string Hamiltonian systems. Given a simple compact Lie group G with a bi-invariant metric and a generating function Γ suggested in the physics literature, we follow the above line of thought and work out the canonical transformation Φ generated by Γ together with an Ad-invariant metric and a B-field on the associated Lie algebra g of G so that G and g form a string target-space dual pair at the classical level under the Hamiltonian formalism. In this article, some general features of this Hamiltonian setting are discussed. We study properties of the canonical transformation Φ including a careful analysis of its domain and image. The geometry of the T-dual structure on g is lightly touched. We leave the task of tracing back the Hamiltonian formalism at the quantum level to the sequel of this paper. (orig.). With 4 figs
International Nuclear Information System (INIS)
Steinberg, S.; Wolf, K.B.
1979-01-01
The authors study the construction and action of certain Lie algebras of second- and higher-order differential operators on spaces of solutions of well-known parabolic, hyperbolic and elliptic linear differential equations. The latter include the N-dimensional quadratic quantum Hamiltonian Schroedinger equations, the one-dimensional heat and wave equations and the two-dimensional Helmholtz equation. In one approach, the usual similarity first-order differential operator algebra of the equation is embedded in the larger one, which appears as a quantum-mechanical dynamic algebra. In a second approach, the new algebra is built as the time evolution of a finite-transformation algebra on the initial conditions. In a third approach, the algebra to inhomogeneous similarity algebra is deformed to a noncompact classical one. In every case, we can integrate the algebra to a Lie group of integral transforms acting effectively on the solution space of the differential equation. (author)
Some quantum Lie algebras of type Dn positive
International Nuclear Information System (INIS)
Bautista, Cesar; Juarez-Ramirez, Maria Araceli
2003-01-01
A quantum Lie algebra is constructed within the positive part of the Drinfeld-Jimbo quantum group of type D n . Our quantum Lie algebra structure includes a generalized antisymmetry property and a generalized Jacobi identity closely related to the braid equation. A generalized universal enveloping algebra of our quantum Lie algebra of type D n positive is proved to be the Drinfeld-Jimbo quantum group of the same type. The existence of such a generalized Lie algebra is reduced to an integer programming problem. Moreover, when the integer programming problem is feasible we show, by means of the generalized Jacobi identity, that the Poincare-Birkhoff-Witt theorem (basis) is still true
On numerical characteristics of subvarieties for three varieties of Lie algebras
International Nuclear Information System (INIS)
Petrogradskii, V M
1999-01-01
Let V be a variety of Lie algebras. For each n we consider the dimension of the space of multilinear elements in n distinct letters of a free algebra of this variety. This gives rise to the codimension sequence c n (V). To study the exponential growth one defines the exponent of the variety. The variety of Lie algebras with nilpotent derived subalgebra N s A is known to have Exp(N s A)=s. Over a field of characteristic zero the exponent of every subvariety V subset of N s A is known to be an integer. We shall prove that this is true over any field. Unlike associative algebras, for varieties of Lie algebras it is typical to have superexponential growth for the codimension sequence. Earlier the author introduced a scale for measuring this growth. The following extreme property is established for two varieties AN 2 and A 3 . Any subvariety in each of them cannot be 'just slightly smaller' in terms of this scale. That is, either a subvariety lies at the same point of the scale as the variety itself, or it is situated substantially lower on the scale. These results are also established over an arbitrary field and without using the representation theory of symmetric groups
International Nuclear Information System (INIS)
Stephens, C. R.
2006-01-01
In this article I give a brief account of the development of research in the Renormalization Group in Mexico, paying particular attention to novel conceptual and technical developments associated with the tool itself, rather than applications of standard Renormalization Group techniques. Some highlights include the development of new methods for understanding and analysing two extreme regimes of great interest in quantum field theory -- the ''high temperature'' regime and the Regge regime
Modular groups in quantum field theory
International Nuclear Information System (INIS)
Borchers, H.-J.
2000-01-01
The author discusses the connection of Lagrangean quantum field theory, perturbation theory, the Lehmann-Symanzik-Zimmermann theory, Wightman's quantum field theory, the Euclidean quantum field theory, and the Araki-Haag-Kastler theory of local observables with modular groups. In this connection he considers the PCT-theorem, and the tensor product decomposition. (HSI)
Combination of activity and lying/standing data for detection of oestrus in cows
DEFF Research Database (Denmark)
Jónsson, Ragnar Ingi; Blanke, Mogens; Poulsen, Niels Kjølstad
2009-01-01
is measured by a sensor attached to the hind leg of the cow. Activity and lying/standing behaviour are modelled as a discrete event system, constructed using automata theory. In an attempt to estimate a biologically relevant lying balance, a lying balance indicator is constructed and is influencing transition...
BTZ black hole from Poisson–Lie T-dualizable sigma models with spectators
Directory of Open Access Journals (Sweden)
A. Eghbali
2017-09-01
Full Text Available The non-Abelian T-dualization of the BTZ black hole is discussed in detail by using the Poisson–Lie T-duality in the presence of spectators. We explicitly construct a dual pair of sigma models related by Poisson–Lie symmetry. The original model is built on a 2+1-dimensional manifold M≈O×G, where G as a two-dimensional real non-Abelian Lie group acts freely on M, while O is the orbit of G in M. The findings of our study show that the original model indeed is canonically equivalent to the SL(2,R Wess–Zumino–Witten (WZW model for a given value of the background parameters. Moreover, by a convenient coordinate transformation we show that this model describes a string propagating in a spacetime with the BTZ black hole metric in such a way that a new family of the solutions to low energy string theory with the BTZ black hole vacuum metric, constant dilaton field and a new torsion potential is found. The dual model is built on a 2+1-dimensional target manifold M˜ with two-dimensional real Abelian Lie group G˜ acting freely on it. We further show that the dual model yields a three-dimensional charged black string for which the mass M and axion charge Q per unit length are calculated. After that, the structure and asymptotic nature of the dual space–time including the horizon and singularity are determined.
Renormalization group theory of critical phenomena
International Nuclear Information System (INIS)
Menon, S.V.G.
1995-01-01
Renormalization group theory is a framework for describing those phenomena that involve a multitude of scales of variations of microscopic quantities. Systems in the vicinity of continuous phase transitions have spatial correlations at all length scales. The renormalization group theory and the pertinent background material are introduced and applied to some important problems in this monograph. The monograph begins with a historical survey of thermal phase transitions. The background material leading to the renormalization group theory is covered in the first three chapters. Then, the basic techniques of the theory are introduced and applied to magnetic critical phenomena in the next four chapters. The momentum space approach as well as the real space techniques are, thus, discussed in detail. Finally, brief outlines of applications of the theory to some of the related areas are presented in the last chapter. (author)
Linear algebra and group theory
Smirnov, VI
2011-01-01
This accessible text by a Soviet mathematician features material not otherwise available to English-language readers. Its three-part treatment covers determinants and systems of equations, matrix theory, and group theory. 1961 edition.
Lie Quasi-Bialgebras and Cohomology of Lie algebra
International Nuclear Information System (INIS)
Bangoura, Momo
2010-05-01
Lie quasi-bialgebras are natural generalisations of Lie bialgebras introduced by Drinfeld. To any Lie quasi-bialgebra structure of finite-dimensional (G, μ, γ, φ), corresponds one Lie algebra structure on D = G + G*, called the double of the given Lie quasi-bialgebra. We show that there exist on ΛG, the exterior algebra of G, a D-module structure and we establish an isomorphism of D-modules between ΛD and End(ΛG), D acting on ΛD by the adjoint action. (author) [fr
Tsao, Thomas R.; Tsao, Doris
1997-04-01
In the 1980's, neurobiologist suggested a simple mechanism in primate visual cortex for maintaining a stable and invariant representation of a moving object. The receptive field of visual neurons has real-time transforms in response to motion, to maintain a stable representation. When the visual stimulus is changed due to motion, the geometric transform of the stimulus triggers a dual transform of the receptive field. This dual transform in the receptive fields compensates geometric variation in the stimulus. This process can be modelled using a Lie group method. The massive array of affine parameter sensing circuits will function as a smart sensor tightly coupled to the passive imaging sensor (retina). Neural geometric engine is a neuromorphic computing device simulating our Lie group model of spatial perception of primate's primal visual cortex. We have developed the computer simulation and experimented on realistic and synthetic image data, and performed a preliminary research of using analog VLSI technology for implementation of the neural geometric engine. We have benchmark tested on DMA's terrain data with their result and have built an analog integrated circuit to verify the computational structure of the engine. When fully implemented on ANALOG VLSI chip, we will be able to accurately reconstruct a 3D terrain surface in real-time from stereoscopic imagery.
International Nuclear Information System (INIS)
Chodos, A.
1978-01-01
A version of lattice gauge theory is presented in which the shape of the lattice is not assumed at the outset but is a consequence of the dynamics. Other related features which are not specified a priori include the internal and space-time symmetry groups and the dimensionality of space-time. The theory possesses a much larger invariance group than the usual gauge group on a lattice, and has associated with it an integer k 0 analogous to the topological quantum numer of quantum chromodynamics. Families of semiclassical solutions are found which are labeled by k 0 and a second integer x, but the analysis is not carried far enough to determine which space-time and internal symmetry groups characterize the lowest-lying states of the theory
Lie groups and symmetric spaces in memory of F. I. Karpelevich
Gindikin, S G
2003-01-01
The book contains survey and research articles devoted mainly to geometry and harmonic analysis of symmetric spaces and to corresponding aspects of group representation theory. The volume is dedicated to the memory of Russian mathematician F. I. Karpelevich (1927-2000).
An introduction to algebraic geometry and algebraic groups
Geck, Meinolf
2003-01-01
An accessible text introducing algebraic geometries and algebraic groups at advanced undergraduate and early graduate level, this book develops the language of algebraic geometry from scratch and uses it to set up the theory of affine algebraic groups from first principles.Building on the background material from algebraic geometry and algebraic groups, the text provides an introduction to more advanced and specialised material. An example is the representation theory of finite groups of Lie type.The text covers the conjugacy of Borel subgroups and maximal tori, the theory of algebraic groups
N = 8 superconformal gauge theories and M2 branes
International Nuclear Information System (INIS)
Benvenuti, Sergio; Rodriguez-Gomez, Diego; Verlinde, Herman; Tonni, Erik
2009-01-01
Based on recent developments, in this letter we find 2+1 dimensional gauge theories with scale invariance and N = 8 supersymmetry. The gauge theories are defined by a Lagrangian and are based on an infinite set of 3-algebras, constructed as an extension of ordinary Lie algebras. Recent no-go theorems on the existence of 3-algebras are circumvented by relaxing the assumption that the invariant metric is positive definite. The gauge group is non compact, and its maximally compact subgroup can be chosen to be any ordinary Lie group, under which the matter fields are adjoints or singlets. Interestingly, the theories are parity invariant and do not admit any tunable coupling constant.
Directory of Open Access Journals (Sweden)
Kalidas Das
2018-03-01
Full Text Available The temperament of stream characteristic, heat and mass transfer of MHD forced convective flow over a linearly expanding porous medium has been scrutinized in the progress exploration. The germane possessions of the liquid like viscosity along with thermal conductivity are believed to be variable in nature, directly influenced by the temperature of flow. As soon as gaining the system of leading equations of the stream, Lie symmetric group transformations have been employed to come across the fitting parallel conversions to alter the central PDEs into a suit of ODEs. The renovated system of ODE with appropriate boundary conditions is numerically solved with the assistance of illustrative software MAPLE 17. The consequences of the relevant factors of the system have been exemplified through charts and graphs. An analogous qualified survey has been prepared among present inquiry and subsisting reads and achieved an admirable accord between them. The variable viscosity parameter has more significant effect on nanofluid velocity than regular fluid and temporal profile as well as nanoparticle concentration is also influenced with variable viscosity. Keywords: Nanofluid, Stretching sheet, Variable viscosity, Variable thermal conductivity, Lie symmetry group
Group theory and lattice gauge fields
International Nuclear Information System (INIS)
Creutz, M.
1988-09-01
Lattice gauge theory, formulated in terms of invariant integrals over group elements on lattice bonds, benefits from many group theoretical notions. Gauge invariance provides an enormous symmetry and powerful constraints on expectation values. Strong coupling expansions require invariant integrals over polynomials in group elements, all of which can be evaluated by symmetry considerations. Numerical simulations involve random walks over the group. These walks automatically generate the invariant group measure, avoiding explicit parameterization. A recently proposed overrelaxation algorithm is particularly efficient at exploring the group manifold. These and other applications of group theory to lattice gauge fields are reviewed in this talk. 17 refs
Group field theory with noncommutative metric variables.
Baratin, Aristide; Oriti, Daniele
2010-11-26
We introduce a dual formulation of group field theories as a type of noncommutative field theories, making their simplicial geometry manifest. For Ooguri-type models, the Feynman amplitudes are simplicial path integrals for BF theories. We give a new definition of the Barrett-Crane model for gravity by imposing the simplicity constraints directly at the level of the group field theory action.
Introduction to vertex algebras, Borcherds algebras and the Monster Lie algebras
International Nuclear Information System (INIS)
Gebert, R.W.
1993-09-01
The theory of vertex algebras constitutes a mathematically rigorous axiomatic formulation of the algebraic origins of conformal field theory. In this context Borcherds algebras arise as certain ''physical'' subspaces of vertex algebras. The aim of this review is to give a pedagogical introduction into this rapidly-developing area of mathematics. Based on the machinery of formal calculus we present the axiomatic definition of vertex algebras. We discuss the connection with conformal field theory by deriving important implications of these axioms. In particular, many explicit calculations are presented to stress the eminent role of the Jacobi identity axiom for vertex algebras. As a class of concrete examples the vertex algebras associated with even lattices are constructed and it is shown in detail how affine Lie algebras and the fake Monster Lie algebra naturally appear. This leads us to the abstract definition of Borcherds algebras as generalized Kac-Moody algebras and their basic properties. Finally, the results about the simplest generic Borcherds algebras are analysed from the point of view of symmetry in quantum theory and the construction of the Monster Lie algebra is sketched. (orig.)
Introduction to geometric nonlinear control; Controllability and lie bracket
Energy Technology Data Exchange (ETDEWEB)
Jakubczyk, B [Institute of Mathematics, Polish Academy of Sciences, Warsaw (Poland)
2002-07-15
We present an introduction to the qualitative theory of nonlinear control systems, with the main emphasis on controllability properties of such systems. We introduce the differential geometric language of vector fields, Lie bracket, distributions, foliations etc. One of the basic tools is the orbit theorem of Stefan and Sussmann. We analyse the basic controllability problems and give criteria for complete controllability, accessibility and related properties, using certain Lie algebras of ve fields defined by the system. A problem of path approximation is considered as an application of the developed theory. We illustrate our considerations with examples of simple systems or systems appearing in applications. The notes start from an elementary level and are self-contained. (author)
Gravitation as Gauge theory of Poincare Group
International Nuclear Information System (INIS)
Stedile, E.
1982-08-01
The geometrical approach to gauge theories, based on fiber-bundles, is shown in detail. Several gauge formalisms for gravitation are examined. In particular, it is shown how to build gauge theories for non-semisimple groups. A gravitational theory for the Poincare group, with all the essential characteristics of a Yang-Mills theory is proposed. Inonu-Wigner contractions of gauge theories are introduced, which provide a Lagrangian formalism, equivalent to a Lagrangian de Sitter theory supplemented by weak constraints. Yang and Einstein theories for gravitation become particular cases of a Yang-Mills theory. The classical limit of the proposed formalism leads to the Poisson equation, for the static case. (Author) [pt
Multi-group neutron transport theory
International Nuclear Information System (INIS)
Zelazny, R.; Kuszell, A.
1962-01-01
Multi-group neutron transport theory. In the paper the general theory of the application of the K. M. Case method to N-group neutron transport theory in plane geometry is given. The eigenfunctions (distributions) for the system of Boltzmann equations have been derived and the completeness theorem has been proved. By means of general solution two examples important for reactor and shielding calculations are given: the solution of a critical and albedo problem for a slab. In both cases the system of singular integral equations for expansion coefficients into a full set of eigenfunction distributions has been reduced to the system of Fredholm-type integral equations. Some results can be applied also to some spherical problems. (author) [fr
Naive Theories of Social Groups
Rhodes, Marjorie
2012-01-01
Four studies examined children's (ages 3-10, Total N = 235) naive theories of social groups, in particular, their expectations about how group memberships constrain social interactions. After introduction to novel groups of people, preschoolers (ages 3-5) reliably expected agents from one group to harm members of the other group (rather than…
Milewski, Emil G
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Group Theory I includes sets and mapping, groupoids and semi-groups, groups, isomorphisms and homomorphisms, cyclic groups, the Sylow theorems, and finite p-groups.
Lying to patients with dementia: Attitudes versus behaviours in nurses.
Cantone, Daniela; Attena, Francesco; Cerrone, Sabrina; Fabozzi, Antonio; Rossiello, Riccardo; Spagnoli, Laura; Pelullo, Concetta Paola
2017-01-01
Using lies, in dementia care, reveals a common practice far beyond the diagnosis and prognosis, extending to the entire care process. In this article, we report results about the attitude and the behaviour of nurses towards the use of lies to patients with dementia. An epidemiological cross-sectional study was conducted between September 2016 and February 2017 in 12 elderly residential facilities and in the geriatric, psychiatric and neurological wards of six specialised hospitals of Italy's Campania Region. In all, 106 nurses compiled an attitude questionnaire (A) where the main question was 'Do you think it is ethically acceptable to use lies to patients with dementia?', instead 106 nurses compiled a behaviour questionnaire (B), where the main question was 'Have you ever used lies to patients with dementia?' Ethical considerations: Using lies in dementia care, although topic ethically still controversial, reveals a common practice far beyond the diagnosis and prognosis, extending to the entire care process. Only a small percentage of the interviewed nurses stated that they never used lies/that it is never acceptable to use lies (behaviour 10.4% and attitude 12.3%; p = 0.66). The situation in which nurses were more oriented to use lies was 'to prevent or reduce aggressive behaviors'. Indeed, only the 6.7% in the attitude group and 3.8% in the behaviour group were against using lies. On the contrary, the case in which the nurses were less oriented to use lies was 'to avoid wasting time giving explanations', in this situation were against using lies the 51.0% of the behaviour group and the 44.6% of the attitude group. Our results, according to other studies, support the hypothesis of a low propensity of nurses to ethical reflection about use of lies. In our country, the implementation of guidelines about a correct use of lie in the relationship between health operators and patients would be desirable.
Classification of filiform Lie algebras up to dimension 7 over finite fields
Falcón Ganfornina, Óscar Jesús; Falcón Ganfornina, Raúl Manuel; Núñez Valdés, Juan; Pacheco Martínez, Ana María; Villar Liñán, María Trinidad
2016-01-01
This paper tries to develop a recent research which consists in using Discrete Mathematics as a tool in the study of the problem of the classification of Lie algebras in general, dealing in this case with filiform Lie algebras up to dimension 7 over finite fields. The idea lies in the representation of each Lie algebra by a certain type of graphs. Then, some properties on Graph Theory make easier to classify the algebras. As main results, we find out that there exist, up to isomor...
Gauge Theories of Vector Particles
Glashow, S. L.; Gell-Mann, M.
1961-04-24
The possibility of generalizing the Yang-Mills trick is examined. Thus we seek theories of vector bosons invariant under continuous groups of coordinate-dependent linear transformations. All such theories may be expressed as superpositions of certain "simple" theories; we show that each "simple theory is associated with a simple Lie algebra. We may introduce mass terms for the vector bosons at the price of destroying the gauge-invariance for coordinate-dependent gauge functions. The theories corresponding to three particular simple Lie algebras - those which admit precisely two commuting quantum numbers - are examined in some detail as examples. One of them might play a role in the physics of the strong interactions if there is an underlying super-symmetry, transcending charge independence, that is badly broken. The intermediate vector boson theory of weak interactions is discussed also. The so-called "schizon" model cannot be made to conform to the requirements of partial gauge-invariance.
Lagrangian submanifolds and dynamics on Lie algebroids
International Nuclear Information System (INIS)
Leon, Manuel de; Marrero, Juan C; MartInez, Eduardo
2005-01-01
In some previous papers, a geometric description of Lagrangian mechanics on Lie algebroids has been developed. In this topical review, we give a Hamiltonian description of mechanics on Lie algebroids. In addition, we introduce the notion of a Lagrangian submanifold of a symplectic Lie algebroid and we prove that the Lagrangian (Hamiltonian) dynamics on Lie algebroids may be described in terms of Lagrangian submanifolds of symplectic Lie algebroids. The Lagrangian (Hamiltonian) formalism on Lie algebroids permits us to deal with Lagrangian (Hamiltonian) functions not defined necessarily on tangent (cotangent) bundles. Thus, we may apply our results to the projection of Lagrangian (Hamiltonian) functions which are invariant under the action of a symmetry Lie group. As a consequence, we obtain that Lagrange-Poincare (Hamilton-Poincare) equations are the Euler-Lagrange (Hamilton) equations associated with the corresponding Atiyah algebroid. Moreover, we prove that Lagrange-Poincare (Hamilton-Poincare) equations are the local equations defining certain Lagrangian submanifolds of symplectic Atiyah algebroids. (topical review)
Turok, Neil
2018-01-01
Professor David Olive was a renowned British theoretical physicist who made seminal contributions to superstrings, quantum gauge theories and mathematical physics. He was awarded the Dirac Medal by the International Centre for Theoretical Physics in Trieste in 1997, with his long-standing collaborator Peter Goddard. David Olive was a Fellow of the Royal Society and a Founding Fellow of the Learned Society of Wales. David Olive was known for his visionary conjectures, including electromagnetic duality in spontaneously broken gauge theories, as well as his exceptionally clear and insightful style of exposition. These lectures, delivered by David Olive in 1982 at the University of Virginia, provide a pedagogical, self-contained introduction to gauge theory, Lie algebras, electromagnetic duality and integrable models. Despite enormous subsequent developments, they still provide a valuable entry point to some of the deepest topics in quantum gauge theory.
't Hooft's solution for arbitrary semisimple Lie group
International Nuclear Information System (INIS)
Leznov, A.N.; Mukhtarov, M.A.
1990-07-01
The generalization of the 't Hooft's A 1 solution for every semisimple Lie algebra is found. The solution depends on r-independent chains of linear self-dual systems (Δ s α ) z = (Δ s+1 α ) y -bar, (Δ s α ) y -bar = -(Δ s+1 α ) z (1 ≤ α ≤ r); the length of α chain is equal to 2ω α + 1, where ω α are the indexes of the semisimple algebra and r is its rank. In the special case the O(4)-invariant solutions with instanton number equal to one arises. (author). 6 refs
Symmetry and group theory throughout physics
Directory of Open Access Journals (Sweden)
Villain J.
2012-03-01
Full Text Available As noticed in 1884 by Pierre Curie [1], physical properties of matter are tightly related to the kind of symmetry of the medium. Group theory is a systematic tool, though not always easy to handle, to exploit symmetry properties, for instance to find the eigenvectors and eigenvalues of an operator. Certain properties (optical activity, piezoelectricity are forbidden in molecules or crystals of high symmetry. A few theorems (Noether, Goldstone establish general relations between physical properties and symmetry. Applications of group theory to condensed matter physics, elementary particle physics, quantum mechanics, electromagnetism are reviewed. Group theory is not only a tool, but also a beautiful construction which casts insight into natural phenomena.
The Lie algebra of the N=2-string
International Nuclear Information System (INIS)
Kugel, K.
2006-01-01
The theory of generalized Kac-Moody algebras is a generalization of the theory of finite dimensional simple Lie algebras. The physical states of some compactified strings give realizations of generalized Kac-Moody algebras. For example the physical states of a bosonic string moving on a 26 dimensional torus form a generalized Kac-Moody algebra and the physical states of a N=1 string moving on a 10 dimensional torus form a generalized Kac-Moody superalgebra. A natural question is whether the physical states of the compactified N=2-string also realize such an algebra. In this thesis we construct the Lie algebra of the compactified N=2-string, study its properties and show that it is not a generalized Kac-Moody algebra. The Fock space of a N=2-string moving on a 4 dimensional torus can be described by a vertex algebra constructed from a rational lattice of signature (8,4). Here 6 coordinates with signature (4,2) come from the matter part and 6 coordinates with signature (4,2) come from the ghost part. The physical states are represented by the cohomology of the BRST-operator. The vertex algebra induces a product on the vector space of physical states that defines the structure of a Lie algebra on this space. This Lie algebra shares many properties with generalized Kac-Moody algebra but we will show that it is not a generalized Kac-Moody algebra. (orig.)
The Lie algebra of the N=2-string
Energy Technology Data Exchange (ETDEWEB)
Kugel, K
2006-07-01
The theory of generalized Kac-Moody algebras is a generalization of the theory of finite dimensional simple Lie algebras. The physical states of some compactified strings give realizations of generalized Kac-Moody algebras. For example the physical states of a bosonic string moving on a 26 dimensional torus form a generalized Kac-Moody algebra and the physical states of a N=1 string moving on a 10 dimensional torus form a generalized Kac-Moody superalgebra. A natural question is whether the physical states of the compactified N=2-string also realize such an algebra. In this thesis we construct the Lie algebra of the compactified N=2-string, study its properties and show that it is not a generalized Kac-Moody algebra. The Fock space of a N=2-string moving on a 4 dimensional torus can be described by a vertex algebra constructed from a rational lattice of signature (8,4). Here 6 coordinates with signature (4,2) come from the matter part and 6 coordinates with signature (4,2) come from the ghost part. The physical states are represented by the cohomology of the BRST-operator. The vertex algebra induces a product on the vector space of physical states that defines the structure of a Lie algebra on this space. This Lie algebra shares many properties with generalized Kac-Moody algebra but we will show that it is not a generalized Kac-Moody algebra. (orig.)
Lie algebra lattices and strings on T-folds
Energy Technology Data Exchange (ETDEWEB)
Satoh, Yuji [Institute of Physics, University of Tsukuba,Ibaraki 305-8571 (Japan); Sugawara, Yuji [Department of Physical Sciences, College of Science and Engineering, Ritsumeikan University,Shiga 525-8577 (Japan)
2017-02-06
We study the world-sheet conformal field theories for T-folds systematically based on the Lie algebra lattices representing the momenta of strings. The fixed point condition required for the T-duality twist restricts the possible Lie algebras. When the T-duality acts as a simple chiral reflection, one is left with the four cases, A{sub 1},D{sub 2r},E{sub 7},E{sub 8}, among the simple simply-laced algebras. From the corresponding Englert-Neveu lattices, we construct the modular invariant partition functions for the T-fold CFTs in bosonic string theory. Similar construction is possible also by using Euclidean even self-dual lattices. We then apply our formulation to the T-folds in the E{sub 8}×E{sub 8} heterotic string theory. Incorporating non-trivial phases for the T-duality twist, we obtain, as simple examples, a class of modular invariant partition functions parametrized by three integers. Our construction includes the cases which are not reduced to the free fermion construction.
International Nuclear Information System (INIS)
Kanakoglou, K.; Daskaloyannis, C.; Herrera-Aguilar, A.
2010-01-01
The mathematical structure of a mixed paraparticle system (combining both parabosonic and parafermionic degrees of freedom) commonly known as the Relative Parabose Set, will be investigated and a braided group structure will be described for it. A new family of realizations of an arbitrary Lie superalgebra will be presented and it will be shown that these realizations possess the valuable representation-theoretic property of transferring invariably the super-Hopf structure. Finally two classes of virtual applications will be outlined: The first is of interest for both mathematics and mathematical physics and deals with the representation theory of infinite dimensional Lie superalgebras, while the second is of interest in theoretical physics and has to do with attempts to determine specific classes of solutions of the Skyrme model.
Nonabelian Jacobian of projective surfaces geometry and representation theory
Reider, Igor
2013-01-01
The Jacobian of a smooth projective curve is undoubtedly one of the most remarkable and beautiful objects in algebraic geometry. This work is an attempt to develop an analogous theory for smooth projective surfaces - a theory of the nonabelian Jacobian of smooth projective surfaces. Just like its classical counterpart, our nonabelian Jacobian relates to vector bundles (of rank 2) on a surface as well as its Hilbert scheme of points. But it also comes equipped with the variation of Hodge-like structures, which produces a sheaf of reductive Lie algebras naturally attached to our Jacobian. This constitutes a nonabelian analogue of the (abelian) Lie algebra structure of the classical Jacobian. This feature naturally relates geometry of surfaces with the representation theory of reductive Lie algebras/groups. This work’s main focus is on providing an in-depth study of various aspects of this relation. It presents a substantial body of evidence that the sheaf of Lie algebras on the nonabelian Jacobian is an effic...
Groups, graphs and random walks
Salvatori, Maura; Sava-Huss, Ecaterina
2017-01-01
An accessible and panoramic account of the theory of random walks on groups and graphs, stressing the strong connections of the theory with other branches of mathematics, including geometric and combinatorial group theory, potential analysis, and theoretical computer science. This volume brings together original surveys and research-expository papers from renowned and leading experts, many of whom spoke at the workshop 'Groups, Graphs and Random Walks' celebrating the sixtieth birthday of Wolfgang Woess in Cortona, Italy. Topics include: growth and amenability of groups; Schrödinger operators and symbolic dynamics; ergodic theorems; Thompson's group F; Poisson boundaries; probability theory on buildings and groups of Lie type; structure trees for edge cuts in networks; and mathematical crystallography. In what is currently a fast-growing area of mathematics, this book provides an up-to-date and valuable reference for both researchers and graduate students, from which future research activities will undoubted...
The metric-affine gravitational theory as the gauge theory of the affine group
International Nuclear Information System (INIS)
Lord, E.A.
1978-01-01
The metric-affine gravitational theory is shown to be the gauge theory of the affine group, or equivalently, the gauge theory of the group GL(4,R) of tetrad deformations in a space-time with a locally Minkowskian metric. The identities of the metric-affine theory, and the relationship between them and those of general relativity and Sciama-Kibble theory, are derived. (Auth.)
International Nuclear Information System (INIS)
Anon.
1993-01-01
Full text: In his review 'Genesis of Unified Gauge Theories' at the symposium in Honour of Abdus Salam (June, page 23), Tom Kibble of Imperial College, London, looked back to the physics events around Salam from 1959-67. He described how, in the early 1960s, people were pushing to enlarge the symmetry of strong interactions beyond the SU(2) of isospin and incorporate the additional strangeness quantum number. Kibble wrote - 'Salam had students working on every conceivable symmetry group. One of these was Yuval Ne'eman, who had the good fortune and/or prescience to work on SU(3). From that work, and of course from the independent work of Murray Gell- Mann, stemmed the Eightfold Way, with its triumphant vindication in the discovery of the omega-minus in 1964.' Yuval Ne'eman writes - 'I was the Defence Attaché at the Israeli Embassy in London and was admitted by Salam as a part-time graduate student when I arrived in 1958. I started research after resigning from the Embassy in May 1960. Salam suggested a problem: provide vector mesons with mass - the problem which was eventually solved by Higgs, Guralnik, Kibble,.... (as described by Kibble in his article). I explained to Salam that I had become interested in symmetry. Nobody at Imperial College at the time, other than Salam himself, was doing anything in groups, and attention further afield was focused on the rotation - SO(N) - groups. Reacting to my own half-baked schemes, Salam told me to forget about the rotation groups he taught us, and study group theory in depth, directing me to Eugene Dynkin's classification of Lie subalgebras, about which he had heard from Morton Hamermesh. I found Dynkin incomprehensible without first learning about Lie algebras from Henri Cartan's thesis, which luckily had been reproduced by Dynkin in his 1946 thesis, using his diagram method. From a copy of a translation of Dynkin's thesis which I found in the British Museum Library, I
International Nuclear Information System (INIS)
Srihirun, B; Meleshko, S V; Schulz, E
2006-01-01
The definition of an admitted Lie group of transformations for stochastic differential equations has been already presented for equations with one-dimensional Brownian motion. The transformation of the dependent variables involves time as well, and it has been proven that Brownian motion is transformed to Brownian motion. In this paper, we will discuss this concept for stochastic differential equations involving multi-dimensional Brownian motion and present applications to a variety of stochastic differential equations
Topological charge in non-abelian lattice gauge theory
International Nuclear Information System (INIS)
Lisboa, P.
1983-01-01
We report on a numerical calculation of topological charge densities in non-abelian gauge theory with gauge groups SU(2) and SU(3). The group manifold is represented by a discrete subset thereof which lies outside its finite subgroups. The results shed light on the usefulness of these representations in Monte Carlo evaluations of non-abelian lattice gauge theory. (orig.)
Hall, Marshall
2018-01-01
This 1959 text offers an unsurpassed resource for learning and reviewing the basics of a fundamental and ever-expanding area. "This remarkable book undoubtedly will become a standard text on group theory." - American Scientist.
Elements of theory of abelian groups
International Nuclear Information System (INIS)
Lebedenko, V.M.
1977-01-01
Some methods and results of studies on the abelian group theory being an important branch of modern algebra are presented. Some examples of the application of the abelian groups in physics are given. A primary information on commutative groups is presented. The concepts of a group, a subgroup, homomorphism, an order of element are given; those of torsion, torsion-free and mixed groups are considered, as well as the concepts of direct and full direct sums. The concepts of a free group and defining relations, of linear dependence and a rank are given. The main classes of abelian groups and subgroup types are described. Some classical results on the abelian group theory are presented, its modern state is described, the links with other regions of algebra are presented
International Nuclear Information System (INIS)
Kashaev, R.M.; Savel'ev, M.V.; Savel'eva, S.A.
1990-01-01
Nonlinear equations associated through a zero curvature type representation with Lie algebras S 0 Diff T 2 and of infinitesimal diffeomorphisms of (S 1 ) 2 , and also with a new infinite-dimensional Lie algebras. In particular, the general solution (in the sense of the Goursat problem) of the heavently equation which describes self-dual Einstein spaces with one rotational Killing symmetry is discussed, as well as the solutions to a generalized equation. The paper is supplied with Appendix containing the definition of the continuum graded Lie algebras and the general construction of the nonlinear equations associated with them. 11 refs
Rubio Martí, Vicente
2016-01-01
En el presente proyecto definimos lo que es un grupo de Lie, así como su respectiva álgebra de Lie canónica como aproximación lineal a dicho grupo de Lie. El proceso de linealización, que es hallar el algebra de Lie de un grupo de Lie dado, tiene su
Facilitating Group Decision-Making: Facilitator's Subjective Theories on Group Coordination
Directory of Open Access Journals (Sweden)
Michaela Kolbe
2008-10-01
Full Text Available A key feature of group facilitation is motivating and coordinating people to perform their joint work. This paper focuses on group coordination which is a prerequisite to group effectiveness, especially in complex tasks. Decision-making in groups is a complex task that consequently needs to be coordinated by explicit rather than implicit coordination mechanisms. Based on the embedded definition that explicit coordination does not just happen but is purposely executed by individuals, we argue that individual coordination intentions and mechanisms should be taken into account. Thus far, the subjective perspective of coordination has been neglected in coordination theory, which is understandable given the difficulties in defining and measuring subjective aspects of group facilitation. We therefore conducted focused interviews with eight experts who either worked as senior managers or as experienced group facilitators and analysed their approaches to group coordination using methods of content analysis. Results show that these experts possess sophisticated mental representations of their coordination behaviour. These subjective coordination theories can be organised in terms of coordination schemes in which coordination-releasing situations are facilitated by special coordination mechanisms that, in turn, lead to the perception of specific consequences. We discuss the importance of these subjective coordination theories for effectively facilitating group decision-making and minimising process losses. URN: urn:nbn:de:0114-fqs0901287
Exact renormalization group for gauge theories
International Nuclear Information System (INIS)
Balaban, T.; Imbrie, J.; Jaffe, A.
1984-01-01
Renormalization group ideas have been extremely important to progress in our understanding of gauge field theory. Particularly the idea of asymptotic freedom leads us to hope that nonabelian gauge theories exist in four dimensions and yet are capable of producing the physics we observe-quarks confined in meson and baryon states. For a thorough understanding of the ultraviolet behavior of gauge theories, we need to go beyond the approximation of the theory at some momentum scale by theories with one or a small number of coupling constants. In other words, we need a method of performing exact renormalization group transformations, keeping control of higher order effects, nonlocal effects, and large field effects that are usually ignored. Rigorous renormalization group methods have been described or proposed in the lectures of Gawedzki, Kupiainen, Mack, and Mitter. Earlier work of Glimm and Jaffe and Gallavotti et al. on the /phi/ model in three dimensions were quite important to later developments in this area. We present here a block spin procedure which works for gauge theories, at least in the superrenormalizable case. It should be enlightening for the reader to compare the various methods described in these proceedings-especially from the point of view of how each method is suited to the physics of the problem it is used to study
Small Group Learning: Do Group Members' Implicit Theories of Ability Make a Difference?
Beckmann, Nadin; Wood, Robert E.; Minbashian, Amirali; Tabernero, Carmen
2012-01-01
We examined the impact of members' implicit theories of ability on group learning and the mediating role of several group process variables, such as goal-setting, effort attributions, and efficacy beliefs. Comparisons were between 15 groups with a strong incremental view on ability (high incremental theory groups), and 15 groups with a weak…
Group manifold approach to gravity and supergravity theories
International Nuclear Information System (INIS)
d'Auria, R.; Fre, P.; Regge, T.
1981-05-01
Gravity theories are presented from the point of view of group manifold formulation. The differential geometry of groups and supergroups is discussed first; the notion of connection and related Yang-Mills potentials is introduced. Then ordinary Einstein gravity is discussed in the Cartan formulation. This discussion provides a first example which will then be generalized to more complicated theories, in particular supergravity. The distinction between ''pure'' and ''impure' theories is also set forth. Next, the authors develop an axiomatic approach to rheonomic theories related to the concept of Chevalley cohomology on group manifolds, and apply these principles to N = 1 supergravity. Then the panorama of so far constructed pure and impure group manifold supergravities is presented. The pure d = 5 N = 2 case is discussed in some detail, and N = 2 and N = 3 in d = 4 are considered as examples of the impure theories. The way a pure theory becomes impure after dimensional reduction is illustrated. Next, the role of kinematical superspace constraints as a subset of the group-manifold equations of motion is discussed, and the use of this approach to obtain the auxiliary fields is demonstrated. Finally, the application of the group manifold method to supersymmetric Super Yang-Mills theories is addressed
Directory of Open Access Journals (Sweden)
Canals B.
2012-03-01
Full Text Available This chapter is a concise mathematical introduction into the algebra of groups. It is build up in the way that definitions are followed by propositions and proofs. The concepts and the terminology introduced here will serve as a basis for the following chapters that deal with group theory in the stricter sense and its application to problems in physics. The mathematical prerequisites are at the bachelor level.1
International Nuclear Information System (INIS)
El Naschie, M.S.
2008-01-01
The maximal number of elementary particles which could be expected to be found within a modestly extended energy scale of the standard model was found using various methods to be N = 69. In particular using E-infinity theory the present Author found the exact transfinite expectation value to be =α-bar o /2≅69 where α-bar o =137.082039325 is the exact inverse fine structure constant. In the present work we show among other things how to derive the exact integer value 69 from the exceptional Lie symmetry groups hierarchy. It is found that the relevant number is given by dim H = 69 where H is the maximal compact subspace of E 7(-5) so that N = dim H = 69 while dim E 7 = 133
The group manifold approach to unified gravity
International Nuclear Information System (INIS)
Regge, T.
1984-01-01
These lectures start with a synopsis of historical results in the construction of unified theories of gravity. The author keeps some mathematical rigour throughout the lectures. He gives a provisional description of supermanifolds and a set of formal rules intended to manipulate superforms or supermanifolds. Super Lie groups are discussed as well as the dimensional reduction of gravity theories, the Kaluza-Klein theory. A formal introduction of supersymmetry is given. (Auth.)
Group field theories for all loop quantum gravity
Oriti, Daniele; Ryan, James P.; Thürigen, Johannes
2015-02-01
Group field theories represent a second quantized reformulation of the loop quantum gravity state space and a completion of the spin foam formalism. States of the canonical theory, in the traditional continuum setting, have support on graphs of arbitrary valence. On the other hand, group field theories have usually been defined in a simplicial context, thus dealing with a restricted set of graphs. In this paper, we generalize the combinatorics of group field theories to cover all the loop quantum gravity state space. As an explicit example, we describe the group field theory formulation of the KKL spin foam model, as well as a particular modified version. We show that the use of tensor model tools allows for the most effective construction. In order to clarify the mathematical basis of our construction and of the formalisms with which we deal, we also give an exhaustive description of the combinatorial structures entering spin foam models and group field theories, both at the level of the boundary states and of the quantum amplitudes.
The quest of a unified theory of interactions
International Nuclear Information System (INIS)
Weingerg, St.; Hawking, St.; Mlodinow, L.; Lisi, G.; Weatherall, J.
2011-01-01
The unification of the 4 basic interactions is far from being achieved despite all the efforts made during decades. One theory states that unification is not possible unless to have the point of view of an observer outside the universe...This document is composed of 3 articles. In the first article, stakes, difficulties and the existing research axis of unification are presented. The second article is dedicated to the string theory that is the most promising according to scientists. In fact there are 5 string theories, each one explaining a limited range of phenomena. Nevertheless, string theories share common concepts called dualities, which made physicists think of a unique theory: the M theory that might lie behind the string theories. The third article presents a recent attempt of unification based on the E8 Lie group. Even if this E8 theory appears to be wrong, it will have shed light on deep geometrical relationships between particles that the real theory will have to explain. (A.C.)
Computational invariant theory
Derksen, Harm
2015-01-01
This book is about the computational aspects of invariant theory. Of central interest is the question how the invariant ring of a given group action can be calculated. Algorithms for this purpose form the main pillars around which the book is built. There are two introductory chapters, one on Gröbner basis methods and one on the basic concepts of invariant theory, which prepare the ground for the algorithms. Then algorithms for computing invariants of finite and reductive groups are discussed. Particular emphasis lies on interrelations between structural properties of invariant rings and computational methods. Finally, the book contains a chapter on applications of invariant theory, covering fields as disparate as graph theory, coding theory, dynamical systems, and computer vision. The book is intended for postgraduate students as well as researchers in geometry, computer algebra, and, of course, invariant theory. The text is enriched with numerous explicit examples which illustrate the theory and should be ...
International Nuclear Information System (INIS)
Cadavid, A.C.
1989-01-01
The author constructs a non-Abelian field theory by gauging a Kac-Moody algebra, obtaining an infinite tower of interacting vector fields and associated ghosts, that obey slightly modified Feynman rules. She discusses the spontaneous symmetry breaking of such theory via the Higgs mechanism. If the Higgs particle lies in the Cartan subalgebra of the Kac-Moody algebra, the previously massless vectors acquire a mass spectrum that is linear in the Kac-Moody index and has additional fine structure depending on the associated Lie algebra. She proceeds to show that there is no obstacle in implementing the affine extension of supersymmetric Yang-Mills theories. The result is valid in four, six and ten space-time dimensions. Then the affine extension of supergravity is investigated. She discusses only the loop algebra since the affine extension of the super-Poincare algebra appears inconsistent. The construction of the affine supergravity theory is carried out by the group manifold method and leads to an action describing infinite towers of spin 2 and spin 3/2 fields that interact subject to the symmetries of the loop algebra. The equations of motion satisfy the usual consistency check. Finally, she postulates a theory in which both the vector and scalar fields lie in the loop algebra of SO(3). This theory has an expanded soliton sector, and corresponding to the original 't Hooft-Polyakov solitonic solutions she now finds an infinite family of exact, special solutions of the new equations. She also proposes a perturbation method for obtaining an arbitrary solution of those equations for each level of the affine index
A simple proof of orientability in colored group field theory.
Caravelli, Francesco
2012-01-01
Group field theory is an emerging field at the boundary between Quantum Gravity, Statistical Mechanics and Quantum Field Theory and provides a path integral for the gluing of n-simplices. Colored group field theory has been introduced in order to improve the renormalizability of the theory and associates colors to the faces of the simplices. The theory of crystallizations is instead a field at the boundary between graph theory and combinatorial topology and deals with n-simplices as colored graphs. Several techniques have been introduced in order to study the topology of the pseudo-manifold associated to the colored graph. Although of the similarity between colored group field theory and the theory of crystallizations, the connection between the two fields has never been made explicit. In this short note we use results from the theory of crystallizations to prove that color in group field theories guarantees orientability of the piecewise linear pseudo-manifolds associated to each graph generated perturbatively. Colored group field theories generate orientable pseudo-manifolds. The origin of orientability is the presence of two interaction vertices in the action of colored group field theories. In order to obtain the result, we made the connection between the theory of crystallizations and colored group field theory.
Transitive Lie algebras of vector fields: an overview
Draisma, J.
2011-01-01
This overview paper is intended as a quick introduction to Lie algebras of vector fields. Originally introduced in the late 19th century by Sophus Lie to capture symmetries of ordinary differential equations, these algebras, or infinitesimal groups, are a recurring theme in 20th-century research on
Civil-Military Relations and Strategy: Theory and Evidence
National Research Council Canada - National Science Library
Kimminau, Jon
2001-01-01
... between civilian and military strategy. There are a number of propositions about such differences that lie at the heart of theories of state and group behavior at international and domestic levels...
Some quantum Lie algebras of type D{sub n} positive
Energy Technology Data Exchange (ETDEWEB)
Bautista, Cesar [Facultad de Ciencias de la Computacion, Benemerita Universidad Autonoma de Puebla, Edif 135, 14 sur y Av San Claudio, Ciudad Universitaria, Puebla Pue. CP 72570 (Mexico); Juarez-Ramirez, Maria Araceli [Facultad de Ciencias Fisico-Matematicas, Benemerita Universidad Autonoma de Puebla, Edif 158 Av San Claudio y Rio Verde sn Ciudad Universitaria, Puebla Pue. CP 72570 (Mexico)
2003-03-07
A quantum Lie algebra is constructed within the positive part of the Drinfeld-Jimbo quantum group of type D{sub n}. Our quantum Lie algebra structure includes a generalized antisymmetry property and a generalized Jacobi identity closely related to the braid equation. A generalized universal enveloping algebra of our quantum Lie algebra of type D{sub n} positive is proved to be the Drinfeld-Jimbo quantum group of the same type. The existence of such a generalized Lie algebra is reduced to an integer programming problem. Moreover, when the integer programming problem is feasible we show, by means of the generalized Jacobi identity, that the Poincare-Birkhoff-Witt theorem (basis) is still true.
Young Children's Self-Benefiting Lies and Their Relation to Executive Functioning and Theory of Mind
Fu, Genyue; Sai, Liyang; Yuan, Fang; Lee, Kang
2018-01-01
It is well established that children lie in different social contexts for various purposes from the age of 2 years. Surprisingly, little is known about whether very young children will spontaneously lie for personal gain, how self-benefiting lies emerge, and what cognitive factors affect the emergence of self-benefiting lies. To bridge this gap in…
Group Theory, Computational Thinking, and Young Mathematicians
Gadanidis, George; Clements, Erin; Yiu, Chris
2018-01-01
In this article, we investigate the artistic puzzle of designing mathematics experiences (MEs) to engage young children with ideas of group theory, using a combination of hands-on and computational thinking (CT) tools. We elaborate on: (1) group theory and why we chose it as a context for young mathematicians' experiences with symmetry and…
Lie symmetries in differential equations
International Nuclear Information System (INIS)
Pleitez, V.
1979-01-01
A study of ordinary and Partial Differential equations using the symmetries of Lie groups is made. Following such a study, an application to the Helmholtz, Line-Gordon, Korleweg-de Vries, Burguer, Benjamin-Bona-Mahony and wave equations is carried out [pt
Transverse lie in labor: A study from Kaduna, Northern Nigeria ...
African Journals Online (AJOL)
Results: During the period there were 16633 deliveries and 30 women with transversely lying fetuses, giving an incidence of 1 in 554 deliveries. Forty percent of the cases were neglected transverse lies. The para 4 and above group had the highest incidence of 2.69/1000. Northern minorities ethnic group had the highest ...
From the geometric quantization to conformal field theory
International Nuclear Information System (INIS)
Alekseev, A.; Shatashvili, S.
1990-01-01
Investigation of 2d conformal field theory in terms of geometric quantization is given. We quantize the so-called model space of the compact Lie group, Virasoro group and Kac-Moody group. In particular, we give a geometrical interpretation of the Virasoro discrete series and explain that this type of geometric quantization reproduces the chiral part of CFT (minimal models, 2d-gravity, WZNW theory). In the appendix we discuss the relation between classical (constant) r-matrices and this geometrical approach. (orig.)
Sophus Lie une pensée audacieuse
Stubhaug, Arild
2006-01-01
Sophus Lie (1842-1899) compte parmi les plus grandes figures norvgiennes de la science. La notorit que lui valent ses travaux n'a rien envier celle de son illustre compatriote Niels Henrik Abel. Groupes et alg bres de Lie ont acquis droit de cit dans maints domaines. Dans cette biographie dtaille, l'crivain Arild Stubhaug, puisant dans la volumineuse correspondance de Lie, dcrit l'homme et la socit norvgienne dans la seconde moiti du XIXe si cle. Le lecteur peut ainsi suivre son enfance dans un presbyt re nich au fond d'un fjord, dcouvrir les rformes de l'enseignement, voyager en Europe, frque
"Lie to me"-Oxytocin impairs lie detection between sexes.
Pfundmair, Michaela; Erk, Wiebke; Reinelt, Annika
2017-10-01
The hormone oxytocin modulates various aspects of social behaviors and even seems to lead to a tendency for gullibility. The aim of the current study was to investigate the effect of oxytocin on lie detection. We hypothesized that people under oxytocin would be particularly susceptible to lies told by people of the opposite sex. After administration of oxytocin or a placebo, male and female participants were asked to judge the veracity of statements from same- vs. other-sex actors who either lied or told the truth. Results showed that oxytocin decreased the ability of both male and female participants to correctly classify other-sex statements as truths or lies compared to placebo. This effect was based on a lower ability to detect lies and not a stronger bias to regard truth statements as false. Revealing a new effect of oxytocin, the findings may support assumptions about the hormone working as a catalyst for social adaption. Copyright © 2017. Published by Elsevier Ltd.
Application of adult attachment theory to group member transference and the group therapy process.
Markin, Rayna D; Marmarosh, Cheri
2010-03-01
Although clinical researchers have applied attachment theory to client conceptualization and treatment in individual therapy, few researchers have applied this theory to group therapy. The purpose of this article is to begin to apply theory and research on adult dyadic and group attachment styles to our understanding of group dynamics and processes in adult therapy groups. In particular, we set forth theoretical propositions on how group members' attachment styles affect relationships within the group. Specifically, this article offers some predictions on how identifying group member dyadic and group attachment styles could help leaders predict member transference within the therapy group. Implications of group member attachment for the selection and composition of a group and the different group stages are discussed. Recommendations for group clinicians and researchers are offered. PsycINFO Database Record (c) 2010 APA, all rights reserved
Group Theory with Applications in Chemical Physics
Jacobs, Patrick
2005-10-01
Group Theory is an indispensable mathematical tool in many branches of chemistry and physics. This book provides a self-contained and rigorous account on the fundamentals and applications of the subject to chemical physics, assuming no prior knowledge of group theory. The first half of the book focuses on elementary topics, such as molecular and crystal symmetry, whilst the latter half is more advanced in nature. Discussions on more complex material such as space groups, projective representations, magnetic crystals and spinor bases, often omitted from introductory texts, are expertly dealt with. With the inclusion of numerous exercises and worked examples, this book will appeal to advanced undergraduates and beginning graduate students studying physical sciences and is an ideal text for use on a two-semester course. An introductory and advanced text that comprehensively covers fundamentals and applications of group theory in detail Suitable for a two-semester course with numerous worked examples and problems Includes several topics often omitted from introductory texts, such as rotation group, space groups and spinor bases
On low rank classical groups in string theory, gauge theory and matrix models
International Nuclear Information System (INIS)
Intriligator, Ken; Kraus, Per; Ryzhov, Anton V.; Shigemori, Masaki; Vafa, Cumrun
2004-01-01
We consider N=1 supersymmetric U(N), SO(N), and Sp(N) gauge theories, with two-index tensor matter and added tree-level superpotential, for general breaking patterns of the gauge group. By considering the string theory realization and geometric transitions, we clarify when glueball superfields should be included and extremized, or rather set to zero; this issue arises for unbroken group factors of low rank. The string theory results, which are equivalent to those of the matrix model, refer to a particular UV completion of the gauge theory, which could differ from conventional gauge theory results by residual instanton effects. Often, however, these effects exhibit miraculous cancellations, and the string theory or matrix model results end up agreeing with standard gauge theory. In particular, these string theory considerations explain and remove some apparent discrepancies between gauge theories and matrix models in the literature
International Nuclear Information System (INIS)
Park, Jeong-Hyuck; Sochichiu, Corneliu
2009-01-01
We propose a novel prescription to take off the square root of the Nambu-Goto action for a p-brane, which generalizes the Brink-Di Vecchia-Howe-Tucker, also known as the Polyakov method. With an arbitrary decomposition, d+n=p+1, our resulting action is a modified d-dimensional Polyakov action, which is gauged and possesses a Nambu n-bracket squared potential. We first spell out how the (p+1)-dimensional diffeomorphism is realized in the lower dimensional action. Then we discuss a possible gauge fixing of it to a direct product of d-dimensional diffeomorphism and n-dimensional volume preserving diffeomorphism. We show that the latter naturally leads to a novel Filippov-Lie n-algebra based gauge theory action in d dimensions. (orig.)
Lie algebras under constraints and nonbijective canonical transformations
International Nuclear Information System (INIS)
Kibler, M.; Winternitz, P.
1987-10-01
The concept of a Lie algebra under constraints is developed in connection with the theory of nonbijective canonical transformations. A finite dimensional vector space M, carrying a faithful linear representation of a Lie algebra L, is mapped into a lower dimensional space antiM in such a maner that a subalgebra L 0 of L is mapped into D(L 0 ) = 0. The Lie algebra L under the constraint D(L 0 ) = 0 is the largest subalgebra L 1 of L that can be represented faithfully on antiM. If L 0 is semi-simple, then L 1 is shown to be the centraliser cent L L 0 . If L is semi-simple and L 0 is an one-dimensional diagonal subalgebra of a Cartan subalgebra of L, then L 1 is shown to be the factor algebra cent L L 0 /L 0 . The latter two results are applied to nonbijective canonical transformations generalizing the Kustaanheimo-Stiefel transformation
Symmetries and groups in particle physics; Symmetrien und Gruppen in der Teilchenphysik
Energy Technology Data Exchange (ETDEWEB)
Scherer, Stefan [Mainz Univ. (Germany)
2016-07-01
The aim of this book consists of a didactic introduction to the group-theoretical considerations and methods, which have led to an ever deeper understanding of the interactions of the elementary particles. The first three chapters deal primarily with the foundations of the representation theory of primarily finite groups, whereby many results are also transferable to compact Lie groups. In the third chapter we discuss the concept of Lie groups and their connection with Lie algebras. In the remaining chapter it is mainly about the application of group theory in physics. Chapter 4 deals with the groups SO(3) and SU(2), which occur in connection with the description of the angular momentum in quantum mechanics. We discuss the Wigner-Eckar theorem together with some applications. In chapter 5 we are employed to the composition properties of strongly interacting systems, so called hadrons, and discuss extensively the transformation properties of quarks with relation to the special unitary groups. The Noether theorem is generally treated in connection to the conservation laws belonging to the Galilei group and the Poincare group. We confine us in chapter 6 to internal symmetries, but explain for that extensively the application to quantum field theory. Especially an outlook on the effect of symmetries in form of so called Ward identities is granted. In chapter 7 we turn towards the gauge principle and discuss first the construction of quantum electrodynamics. In the following we generalize the gauge principle to non-Abelian groups (Yang-Mills theories) and formulate the quantum chromodynamics (QCD). Especially we take a view of ''random'' global symmetries of QCD, especially the chiral symmetry. In chapter 8 we illuminate the phenomenon of spontaneous symmetry breaking both for global and for local symmetries. In the final chapter we work out the group-theoretical structure of the Standard Model. Finally by means of the group SU(5) we take a view to
Localization in abelian Chern-Simons theory
DEFF Research Database (Denmark)
McLellan, Brendan Donald Kenneth
2013-01-01
Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected, and abelian. The abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. The partition function is then formally computed...
Gruppi, anelli di Lie e teoria della coomologia
Zappa, G
2011-01-01
This book includes: R. Baer: Complementation in finite gropus; M. Lazard: Groupes, anneaux de Lie et probleme de Burnside; J. Tits: Sur les groupes algebriques afffines; Theoremes fondamentaux de structure; and, Classification des groupes semisimples et geometries associees.
Wess-Zumino-Novikov-Witten models based on Lie superalgebras
International Nuclear Information System (INIS)
Mohammedi, N.
1994-04-01
The affine current algebra for Lie superalgebras is examined. The bilinear invariant forms of the Lie superalgebra can be either degenerate or non-degenerate. We give the conditions for a Virasoro construction, in which the currents are primary fields of weight one, to exist. In certain cases, the Virasoro central charge is an integer equal to the super dimension of the group supermanifold. A Wess-Zumino-Novikov-Witten action based on these Lie superalgebras is also found. (orig.)
Hilbert schemes of points and infinite dimensional Lie algebras
Qin, Zhenbo
2018-01-01
Hilbert schemes, which parametrize subschemes in algebraic varieties, have been extensively studied in algebraic geometry for the last 50 years. The most interesting class of Hilbert schemes are schemes X^{[n]} of collections of n points (zero-dimensional subschemes) in a smooth algebraic surface X. Schemes X^{[n]} turn out to be closely related to many areas of mathematics, such as algebraic combinatorics, integrable systems, representation theory, and mathematical physics, among others. This book surveys recent developments of the theory of Hilbert schemes of points on complex surfaces and its interplay with infinite dimensional Lie algebras. It starts with the basics of Hilbert schemes of points and presents in detail an example of Hilbert schemes of points on the projective plane. Then the author turns to the study of cohomology of X^{[n]}, including the construction of the action of infinite dimensional Lie algebras on this cohomology, the ring structure of cohomology, equivariant cohomology of X^{[n]} a...
Cornwell, J F
1989-01-01
Recent devopments, particularly in high-energy physics, have projected group theory and symmetry consideration into a central position in theoretical physics. These developments have taken physicists increasingly deeper into the fascinating world of pure mathematics. This work presents important mathematical developments of the last fifteen years in a form that is easy to comprehend and appreciate.
Supersymmetric gauge theories with classical groups via M theory fivebrane
International Nuclear Information System (INIS)
Terashima, S.
1998-01-01
We study the moduli space of vacua of four-dimensional N=1 and N=2 supersymmetric gauge theories with the gauge groups Sp(2N c ), SO(2N c ) and SO(2N c +1) using the M theory fivebrane. Higgs branches of the N=2 supersymmetric gauge theories are interpreted in terms of the M theory fivebrane and the type IIA s-rule is realized in it. In particular, we construct the fivebrane configuration which corresponds to a special Higgs branch root. This root is analogous to the baryonic branch root in the SU(N c ) theory which remains as a vacuum after the adjoint mass perturbation to break N=2 to N=1. Furthermore, we obtain the monopole condensations and the meson vacuum expectation values in the confining phase of N=1 supersymmetric gauge theories using the fivebrane technique. These are in complete agreement with the field theory results for the vacua in the phase with a single confined photon. (orig.)
Basic gerbe over non simply connected compact groups
Gawedzki, Krzysztof; Reis, Nuno
2003-01-01
We present an explicit construction of the basic bundle gerbes with connection over all connected compact simple Lie groups. These are geometric objects that appear naturally in the Lagrangian approach to the WZW conformal field theories. Our work extends the recent construction of E. Meinrenken \\cite{Meinr} restricted to the case of simply connected groups.
Directory of Open Access Journals (Sweden)
Seiya Nishiyama
2009-01-01
Full Text Available The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective submanifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the conventional time-dependent (TD self-consistent field (SCF theory. Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system. This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective variable differentials of a canonical transformation on a group. The integrability condition of the system read the curvature C = 0. Our method is constructed manifesting itself the structure of the group under consideration. To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e., υ (external parameter-dependent Hartree-Fock (HF theory. Toward such an ultimate goal, the υ-HF theory has been reconstructed on an affine Kac-Moody algebra along the soliton theory, using infinite-dimensional fermion. An infinite-dimensional fermion operator is introduced through a Laurent expansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a υ-dependent potential with a Υ-periodicity. A bilinear equation for the υ-HF theory has been transcribed onto the corresponding τ-function using the regular representation for the group and the Schur-polynomials. The υ-HF SCF theory on an infinite-dimensional Fock space F∞ leads to a dynamics on an infinite-dimensional Grassmannian Gr∞ and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr
A Grounded Theory of Western-Trained Asian Group Leaders Leading Groups in Asia
Taephant, Nattasuda; Rubel, Deborah; Champe, Julia
2015-01-01
This grounded theory research explored the experiences of Western-trained Asian group leaders leading groups in Asia. A total of 6 participants from Japan, Taiwan, and Thailand were interviewed 3 times over 9 months. The recursive process of data collection and analysis yielded substantive theory describing the participants' process of reconciling…
Deformations of classical Lie algebras with homogeneous root system in characteristic two. I
International Nuclear Information System (INIS)
Chebochko, N G
2005-01-01
Spaces of local deformations of classical Lie algebras with a homogeneous root system over a field K of characteristic 2 are studied. By a classical Lie algebra over a field K we mean the Lie algebra of a simple algebraic Lie group or its quotient algebra by the centre. The description of deformations of Lie algebras is interesting in connection with the classification of the simple Lie algebras.
Lie-Nambu and Lie-Poisson structures in linear and nonlinear quantum mechanics
International Nuclear Information System (INIS)
Czachor, M.
1996-01-01
Space of density matrices in quantum mechanics can be regarded as a Poisson manifold with the dynamics given by certain Lie-Poisson bracket corresponding to an infinite dimensional Lie algebra. The metric structure associated with this Lie algebra is given by a metric tensor which is not equivalent to the Cartan-Killing metric. The Lie-Poisson bracket can be written in a form involving a generalized (Lie-)Nambu bracket. This bracket can be used to generate a generalized, nonlinear and completely integrable dynamics of density matrices. (author)
The group theory of oxidation II: cosets of non-split groups
International Nuclear Information System (INIS)
Keurentjes, Arjan
2003-01-01
The oxidation program given in the first article of this series (see preceding article in this issue) is extended to cover oxidation of 3d sigma model theories on a coset G/H, with G non-compact (but not necessarily split), and H the maximal compact subgroup. We recover the matter content, the equations of motion and Bianchi identities from group lattice and Cartan involution. Satake diagrams provide an elegant tool for the computations, the maximal oxidation dimension, and group disintegration chains can be directly read off. We give a complete list of theories that can be recovered from oxidation of a 3-dimensional coset sigma model on G/H, where G is a simple non-compact group
Multiplication: From Thales to Lie1
Indian Academy of Sciences (India)
Addition. To describe the geometric constructions of addition, as ..... general, we could apply the implicit function theorem of calculus to solve locally the defining ... and whose multiplication and inverse are analytic maps, is called a Lie group.
Renormalization group and fixed points in quantum field theory
International Nuclear Information System (INIS)
Hollowood, Timothy J.
2013-01-01
This Brief presents an introduction to the theory of the renormalization group in the context of quantum field theories of relevance to particle physics. Emphasis is placed on gaining a physical understanding of the running of the couplings. The Wilsonian version of the renormalization group is related to conventional perturbative calculations with dimensional regularization and minimal subtraction. An introduction is given to some of the remarkable renormalization group properties of supersymmetric theories.
Remainder Wheels and Group Theory
Brenton, Lawrence
2008-01-01
Why should prospective elementary and high school teachers study group theory in college? This paper examines applications of abstract algebra to the familiar algorithm for converting fractions to repeating decimals, revealing ideas of surprising substance beneath an innocent facade.
The Jordan structure of lie and Kac-Moody algebras
International Nuclear Information System (INIS)
Ferreira, L.A.; Gomes, J.F.; Teotonio Sobrinho, P.; Zimerman, A.H.
1989-01-01
A precise relation between the structures of Lie and Jordan algebras by presenting a method of constructing one type of algebra from the other is established. The method differs in some aspects of the Tits construction and Jordan pairs. The examples of the Lie algebras associated to simple Jordan algebras M m (n ) and Clifford algebras are discussed in detail. This approach will shed light on the role of the realizations of Jordan algebras through some types of Fermi fields used in the construction of Kac-Moodey and Virasoro algebras as well as its relevance in the study of some aspects of conformal fields theories. (author)
Practical impact of group communication theory
Schiper, A.
2003-01-01
Practical impact of group communication theory Andre Schiper Group communication is an important topic in fault-tolerant distributed applications. The paper summarizes the main contributions of practical importance that contributed to our current understanding of group communication. These contributions are classified into ''abstractions'' and ''specifications'', ''paradigms'', ''system models'', ''algorithms'', and ''theoretical results''. Some open issues are discussed at the end of the ...
Clifford theory for group representations
Karpilovsky, G
1989-01-01
Let N be a normal subgroup of a finite group G and let F be a field. An important method for constructing irreducible FG-modules consists of the application (perhaps repeated) of three basic operations: (i) restriction to FN. (ii) extension from FN. (iii) induction from FN. This is the `Clifford Theory' developed by Clifford in 1937. In the past twenty years, the theory has enjoyed a period of vigorous development. The foundations have been strengthened and reorganized from new points of view, especially from the viewpoint of graded rings and crossed products.The purpos
Lying in the Name of the Collective Good: A Developmental Study
Fu, Genyue; Evans, Angela D.; Wang, Lingfeng; Lee, Kang
2008-01-01
The present study examined the developmental origin of "blue lies", a pervasive form of lying in the adult world that is told purportedly to benefit a collective. Seven, 9-, and 11-year-old Chinese children were surreptitiously placed in a real-life situation where they decided whether to lie to conceal their group's cheating behavior. Children…
An elementary introduction to the Gauge theory approach to gravity. 23
International Nuclear Information System (INIS)
Mukunda, N.
1989-01-01
Can all the forces be unified by a gauge group? Can we get a clue by studying gravity itself which is also a gauge theory by gauging the Poincare group?. The main problems have been in the understanding of the role of invariants of the Lie algebra of the group if one has general covariance. One is led to theories more general than general relativity in that, in addition to curvature, one also has torsion. These and other aspects of gravitation as a gauge theory are treated. (author). 11 refs.; 1 fig
International Nuclear Information System (INIS)
Burde, G.I.
2002-01-01
A new approach to the use of the Lie group technique for partial and ordinary differential equations dependent on a small parameter is developed. In addition to determining approximate solutions to the perturbed equation, the approach allows constructing integrable equations that have solutions with (partially) prescribed features. Examples of application of the approach to partial differential equations are given
International Nuclear Information System (INIS)
Ghikas, D.K.P.; Ktorides, C.N.; Papaloukas, L.
1980-01-01
This mathematical note is motivated by an assessment concerning our current understanding of the role of Lie-admissible symmetries in connection with quantum structures. We identify the problem of representations of the universal enveloping (lambda, μ)-mutation algebra of a given Lie algebra on a suitable algebra of operators, as constituting a fundamental first step for improving the situation. We acknowledge a number of difficulties which are peculiar to the adopted nonassociative product for the operator algebra. In view of these difficulties, we are presently content in establishing the generalization, to the Lie-admissible case, of a certain theorem by Nelson. This theorem has been very instrumental in Nelson's treatment concerning the Lie symmetry content of quantum structures. It is hoped that a similar situation will eventually prevail for the Lie-admissible case. We offer a number of relevant suggestions
Fock model and Segal-Bargmann transform for minimal representations of Hermitian Lie groups
DEFF Research Database (Denmark)
Hilgert, Joachim; Kobayashi, Toshiyuki; Möllers, Jan
2012-01-01
For any Hermitian Lie group G of tube type we construct a Fock model of its minimal representation. The Fock space is defined on the minimal nilpotent K_C-orbit X in p_C and the L^2-inner product involves a K-Bessel function as density. Here K is a maximal compact subgroup of G, and g......_C=k_C+p_C is a complexified Cartan decomposition. In this realization the space of k-finite vectors consists of holomorphic polynomials on X. The reproducing kernel of the Fock space is calculated explicitly in terms of an I-Bessel function. We further find an explicit formula of a generalized Segal-Bargmann transform which...... intertwines the Schroedinger and Fock model. Its kernel involves the same I-Bessel function. Using the Segal--Bargmann transform we also determine the integral kernel of the unitary inversion operator in the Schroedinger model which is given by a J-Bessel function....
Nonlinear analysis of flexible plates lying on elastic foundation
Directory of Open Access Journals (Sweden)
Trushin Sergey
2017-01-01
Full Text Available This article describes numerical procedures for analysis of flexible rectangular plates lying on elastic foundation. Computing models are based on the theory of plates with account of transverse shear deformations. The finite difference energy method of discretization is used for reducing the initial continuum problem to finite dimensional problem. Solution procedures for nonlinear problem are based on Newton-Raphson method. This theory of plates and numerical methods have been used for investigation of nonlinear behavior of flexible plates on elastic foundation with different properties.
Solitonic Integrable Perturbations of Parafermionic Theories
Fernández-Pousa, C R; Hollowood, Timothy J; Miramontes, J L
1997-01-01
The quantum integrability of a class of massive perturbations of the parafermionic conformal field theories associated to compact Lie groups is established by showing that they have quantum conserved densities of scale dimension 2 and 3. These theories are integrable for any value of a continuous vector coupling constant, and they generalize the perturbation of the minimal parafermionic models by their first thermal operator. The classical equations-of-motion of these perturbed theories are the non-abelian affine Toda equations which admit (charged) soliton solutions whose semi-classical quantization is expected to permit the identification of the exact S-matrix of the theory.
Staircase Models from Affine Toda Field Theory
Dorey, P; Dorey, Patrick; Ravanini, Francesco
1993-01-01
We propose a class of purely elastic scattering theories generalising the staircase model of Al. B. Zamolodchikov, based on the affine Toda field theories for simply-laced Lie algebras g=A,D,E at suitable complex values of their coupling constants. Considering their Thermodynamic Bethe Ansatz equations, we give analytic arguments in support of a conjectured renormalisation group flow visiting the neighbourhood of each W_g minimal model in turn.
Dynamical interpretation of nonrelativistic conformal groups
International Nuclear Information System (INIS)
Andrzejewski, K.; Gonera, J.
2013-01-01
It is shown that N-Galilean conformal algebra with N odd and nontrivial central charge is the maximal symmetry algebra for higher derivative free theory both on classical and quantum levels. By maximal symmetry algebra the Lie algebra of the maximal group of space–time symmetry transformations is understood which preserves higher order free dynamics
An application of vector coherent state theory to the SO95) proton-neutron quasi-spin algebra
International Nuclear Information System (INIS)
Berej, W.
2002-01-01
Vector coherent state theory (VCS), developed for computing Lie group and Lie algebra representations and coupling coefficients, has been used for many groups of interest an actual physics applications. It is shown that VCS construction of a rotor type can be performed for the SO(5) ∼ Sp(4) quasi-spin group where the relevant physical subgroup SU(2) x U(1) is generalized by the isospin operators and the number of particle operators [ru
A Review of Group Systems Theory
Connors, Joanie V.; Caple, Richard B.
2005-01-01
The ability to see interpersonal and group processes beyond the individual level is an essential skill for group therapists (Crouch, Bloch & Wanlass, 1994; Dies, 1994; Fuhriman & Burlingame, 1994). In addition to interpersonal therapy models (e.g., Sullivan and Yalom), there are a number of systems theory models that offer a broad array of…
Sen, Avijit; Sen, Sangita; Samanta, Pradipta Kumar; Mukherjee, Debashis
2015-04-05
We present here a comprehensive account of the formulation and pilot applications of the second-order perturbative analogue of the recently proposed unitary group adapted state-specific multireference coupled cluster theory (UGA-SSMRCC), which we call as the UGA-SSMRPT2. We also discuss the essential similarities and differences between the UGA-SSMRPT2 and the allied SA-SSMRPT2. Our theory, like its parent UGA-SSMRCC formalism, is size-extensive. However, because of the noninvariance of the theory with respect to the transformation among the active orbitals, it requires the use of localized orbitals to ensure size-consistency. We have demonstrated the performance of the formalism with a set of pilot applications, exploring (a) the accuracy of the potential energy surface (PES) of a set of small prototypical difficult molecules in their various low-lying states, using natural, pseudocanonical and localized orbitals and compared the respective nonparallelity errors (NPE) and the mean average deviations (MAD) vis-a-vis the full CI results with the same basis; (b) the efficacy of localized active orbitals to ensure and demonstrate manifest size-consistency with respect to fragmentation. We found that natural orbitals lead to the best overall PES, as evidenced by the NPE and MAD values. The MRMP2 results for individual states and of the MCQDPT2 for multiple states displaying avoided curve crossings are uniformly poorer as compared with the UGA-SSMRPT2 results. The striking aspect of the size-consistency check is the complete insensitivity of the sum of fragment energies with given fragment spin-multiplicities, which are obtained as the asymptotic limit of super-molecules with different coupled spins. © 2015 Wiley Periodicals, Inc.
Disconnected forms of the standard group
International Nuclear Information System (INIS)
McInnes, B.
1996-10-01
Recent work in quantum gravity has led to a revival of interest in the concept of disconnected gauge groups. Here we explain how to classify all of the (non-trivial) groups which have the same Lie algebra as the ''standard group'', SU(3) x SU(2) x U(1), without requiring connectedness. The number of possibilities is surprisingly large. We also discuss the geometry of the ''Kiskis effect'', the ambiguity induced by non-trivial spacetime topology in such gauge theories. (author). 12 refs
Differential geometry of groups in string theory
International Nuclear Information System (INIS)
Schmidke, W.B. Jr.
1990-09-01
Techniques from differential geometry and group theory are applied to two topics from string theory. The first topic studied is quantum groups, with the example of GL (1|1). The quantum group GL q (1|1) is introduced, and an exponential description is derived. The algebra and coproduct are determined using the invariant differential calculus method introduced by Woronowicz and generalized by Wess and Zumino. An invariant calculus is also introduced on the quantum superplane, and a representation of the algebra of GL q (1|1) in terms of the super-plane coordinates is constructed. The second topic follows the approach to string theory introduced by Bowick and Rajeev. Here the ghost contribution to the anomaly of the energy-momentum tensor is calculated as the Ricci curvature of the Kaehler quotient space Diff(S 1 )/S 1 . We discuss general Kaehler quotient spaces and derive an expression for their Ricci curvatures. Application is made to the string and superstring diffeomorphism groups, considering all possible choices of subgroup. The formalism is extended to associated holomorphic vector bundles, where the Ricci curvature corresponds to the anomaly for different ghost sea levels. 26 refs
Three Conceptual Replication Studies in Group Theory
Melhuish, Kathleen
2018-01-01
Many studies in mathematics education research occur with a nonrepresentative sample and are never replicated. To challenge this paradigm, I designed a large-scale study evaluating student conceptions in group theory that surveyed a national, representative sample of students. By replicating questions previously used to build theory around student…
International Nuclear Information System (INIS)
Kerner, R.
1983-01-01
The mathematical background for a graded extension of gauge theories is investigated. After discussing the general properties of graded Lie algebras and what may serve as a model for a graded Lie group, the graded fiber bundle is constructed. Its basis manifold is supposed to be the so-called superspace, i.e. the product of the Minkowskian space-time with the Grassmann algebra spanned by the anticommuting Lorentz spinors; the vertical subspaces tangent to the fibers are isomorphic with the graded extension of the SU(N) Lie algebra. The connection and curvature are defined then on this bundle; the two different gradings are either independent of each other, or may be unified in one common grading, which is equivalent to the choice of the spin-statistics dependence. The Yang-Mills lagrangian is investigated in the simplified case. The conformal symmetry breaking is discussed, as well as some other physical consequences of the model. (orig.)
Introduction to orthogonal, symplectic and unitary representations of finite groups
Riehm, Carl R
2011-01-01
Orthogonal, symplectic and unitary representations of finite groups lie at the crossroads of two more traditional subjects of mathematics-linear representations of finite groups, and the theory of quadratic, skew symmetric and Hermitian forms-and thus inherit some of the characteristics of both. This book is written as an introduction to the subject and not as an encyclopaedic reference text. The principal goal is an exposition of the known results on the equivalence theory, and related matters such as the Witt and Witt-Grothendieck groups, over the "classical" fields-algebraically closed, rea
Workshop on Topology and Geometric Group Theory
Fowler, James; Lafont, Jean-Francois; Leary, Ian
2016-01-01
This book presents articles at the interface of two active areas of research: classical topology and the relatively new field of geometric group theory. It includes two long survey articles, one on proofs of the Farrell–Jones conjectures, and the other on ends of spaces and groups. In 2010–2011, Ohio State University (OSU) hosted a special year in topology and geometric group theory. Over the course of the year, there were seminars, workshops, short weekend conferences, and a major conference out of which this book resulted. Four other research articles complement these surveys, making this book ideal for graduate students and established mathematicians interested in entering this area of research.
Applied group theory selected readings in physics
Cracknell, Arthur P
1968-01-01
Selected Readings in Physics: Applied Group Theory provides information pertinent to the fundamental aspects of applied group theory. This book discusses the properties of symmetry of a system in quantum mechanics.Organized into two parts encompassing nine chapters, this book begins with an overview of the problem of elastic vibrations of a symmetric structure. This text then examines the numbers, degeneracies, and symmetries of the normal modes of vibration. Other chapters consider the conditions under which a polyatomic molecule can have a stable equilibrium configuration when its electronic
Counting Semisimple Orbits of Finite Lie Algebras by Genus
Fulman, Jason
1999-01-01
The adjoint action of a finite group of Lie type on its Lie algebra is studied. A simple formula is conjectured for the number of split semisimple orbits of a given genus. This conjecture is proved for type A, and partial results are obtained for other types. For type A a probabilistic interpretation is given in terms of card shuffling.
A geometrical interpretation of renormalisation group flow
International Nuclear Information System (INIS)
Dolan, B.P.
1993-05-01
The renormalisation group (RG) equation in D-dimensional Euclidean space, R D , is analysed from a geometrical point of view. A general form of the RG equation is derived which is applicable to composite operators as well tensor operators (on R D ) which may depend on the Euclidean metric. It is argued that physical N-point amplitudes should be interpreted as rank N co-variant tensors on the space of couplings, G, and that the RG equation can be viewed as an equation for Lie transport on G with respect to the vector field generated by the β-functions of the theory. In one sense it is nothing more than the definition of a Lie derivative. The source of the anomalous dimensions can be interpreted as being due to the change of the basis vectors on G under Lie transport. The RG equation acts as a bridge between Euclidean space and coupling constant space in that the effect on amplitudes of a diffeomorphism of R D (that of dilations) is completely equivalent to a diffeomorphism of G generated by the β-functions of the theory. A form of the RG equation for operators is also given. These ideas are developed in detail for the example of massive λΦ 4 theory in 4 dimensions. (orig.)
Bases in Lie and quantum algebras
International Nuclear Information System (INIS)
Ballesteros, A; Celeghini, E; Olmo, M A del
2008-01-01
Applications of algebras in physics are related to the connection of measurable observables to relevant elements of the algebras, usually the generators. However, in the determination of the generators in Lie algebras there is place for some arbitrary conventions. The situation is much more involved in the context of quantum algebras, where inside the quantum universal enveloping algebra, we have not enough primitive elements that allow for a privileged set of generators and all basic sets are equivalent. In this paper we discuss how the Drinfeld double structure underlying every simple Lie bialgebra characterizes uniquely a particular basis without any freedom, completing the Cartan program on simple algebras. By means of a perturbative construction, a distinguished deformed basis (we call it the analytical basis) is obtained for every quantum group as the analytical prolongation of the above defined Lie basis of the corresponding Lie bialgebra. It turns out that the whole construction is unique, so to each quantum universal enveloping algebra is associated one and only one bialgebra. In this way the problem of the classification of quantum algebras is moved to the classification of bialgebras. In order to make this procedure more clear, we discuss in detail the simple cases of su(2) and su q (2).
The eyes don't have it: lie detection and Neuro-Linguistic Programming.
Directory of Open Access Journals (Sweden)
Richard Wiseman
Full Text Available Proponents of Neuro-Linguistic Programming (NLP claim that certain eye-movements are reliable indicators of lying. According to this notion, a person looking up to their right suggests a lie whereas looking up to their left is indicative of truth telling. Despite widespread belief in this claim, no previous research has examined its validity. In Study 1 the eye movements of participants who were lying or telling the truth were coded, but did not match the NLP patterning. In Study 2 one group of participants were told about the NLP eye-movement hypothesis whilst a second control group were not. Both groups then undertook a lie detection test. No significant differences emerged between the two groups. Study 3 involved coding the eye movements of both liars and truth tellers taking part in high profile press conferences. Once again, no significant differences were discovered. Taken together the results of the three studies fail to support the claims of NLP. The theoretical and practical implications of these findings are discussed.
Fu, Genyue; Xu, Fen; Cameron, Catherine Ann; Heyman, Gail; Lee, Kang
2008-01-01
This study examined cross-cultural differences and similarities in children’s moral understanding of individual- or collective-oriented lies and truths. Seven-, 9-, and 11-year-old Canadian and Chinese children were read stories about story characters facing moral dilemmas about whether to lie or tell the truth to help a group but harm an individual or vice versa. Participants chose to lie or to tell the truth as if they were the character (Experiments 1 and 2) and categorized and evaluated the story characters’ truthful and untruthful statements (Experiments 3 and 4). Most children in both cultures labeled lies as lies and truths as truths. The major cultural differences lay in choices and moral evaluations. Chinese children chose lying to help a collective but harm an individual, and they rated it less negatively than lying with opposite consequences. Chinese children rated truth telling to help an individual but harm a group less positively than the alternative. Canadian children did the opposite. These findings suggest that cross-cultural differences in emphasis on groups versus individuals affect children’s choices and moral judgments about truth and deception. PMID:17352539
A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation
Somma, Rolando D.
2016-06-01
We present a product formula to approximate the exponential of a skew-Hermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from well-known results. We apply our results to construct product formulas useful for the quantum simulation of some continuous-variable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by the energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.
Graph theory and the Virasoro master equation
International Nuclear Information System (INIS)
Obers, N.A.J.
1991-01-01
A brief history of affine Lie algebra, the Virasoro algebra and its culmination in the Virasoro master equation is given. By studying ansaetze of the master equation, the author obtains exact solutions and gains insight in the structure of large slices of affine-Virasoro space. He finds an isomorphism between the constructions in the ansatz SO(n) diag , which is a set of unitary, generically irrational affine-Virasoro constructions on SO(n), and the unlabeled graphs of order n. On the one hand, the conformal constructions, are classified by the graphs, while, conversely, a group-theoretic and conformal field-theoretic identification is obtained for every graph of graph theory. He also defines a class of magic Lie group bases in which the Virasoro master equation admits a simple metric ansatz {g metric }, whose structure is visible in the high-level expansion. When a magic basis is real on compact g, the corresponding g metric is a large system of unitary, generically irrational conformal field theories. Examples in this class include the graph-theory ansatz SO(n) diag in the Cartesian basis of SO(n), and the ansatz SU(n) metric in the Pauli-like basis of SU(n). Finally, he defines the 'sine-area graphs' of SU(n), which label the conformal field theories of SU(n) metric , and he notes that, in similar fashion, each magic basis of g defines a generalized graph theory on g which labels the conformal field theories of g metric
Differential geometry of group lattices
International Nuclear Information System (INIS)
Dimakis, Aristophanes; Mueller-Hoissen, Folkert
2003-01-01
In a series of publications we developed ''differential geometry'' on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that first-order differential calculi (over the algebra of functions) on a discrete set are in bijective correspondence with digraph structures where the vertices are given by the elements of the set. A particular class of digraphs are Cayley graphs, also known as group lattices. They are determined by a discrete group G and a finite subset S. There is a distinguished subclass of ''bicovariant'' Cayley graphs with the property ad(S)S subset of S. We explore the properties of differential calculi which arise from Cayley graphs via the above correspondence. The first-order calculi extend to higher orders and then allow us to introduce further differential geometric structures. Furthermore, we explore the properties of ''discrete'' vector fields which describe deterministic flows on group lattices. A Lie derivative with respect to a discrete vector field and an inner product with forms is defined. The Lie-Cartan identity then holds on all forms for a certain subclass of discrete vector fields. We develop elements of gauge theory and construct an analog of the lattice gauge theory (Yang-Mills) action on an arbitrary group lattice. Also linear connections are considered and a simple geometric interpretation of the torsion is established. By taking a quotient with respect to some subgroup of the discrete group, generalized differential calculi associated with so-called Schreier diagrams are obtained
Applications of Lie algebras in the solution of dynamic problems
International Nuclear Information System (INIS)
Fellay, G.
1983-01-01
The purpose of this paper is to give some insight into the Lie-algebras and their applications. The first part introduces the elementary properties of such algebras, e.g. nilpotency, solvability, etc. The second part shows how to use the demonstrated theory for solving differential equations with time-dependent coefficients. (Auth.)
Lie symmetry analysis and soliton solutions of time-fractional K(m, n ...
Indian Academy of Sciences (India)
2016-12-03
Dec 3, 2016 ... Factional differential equations are increasingly used to model problems in physics, such as fluid mechan- ics, biology, viscoelasticity, engineering etc. [1–4]. In .... According to the Lie theory, applying the prolongation. Pr.
Quantum groups and algebraic geometry in conformal field theory
International Nuclear Information System (INIS)
Smit, T.J.H.
1989-01-01
The classification of two-dimensional conformal field theories is described with algebraic geometry and group theory. This classification is necessary in a consistent formulation of a string theory. (author). 130 refs.; 4 figs.; schemes
Group theoretical methods and wavelet theory: coorbit theory and applications
Feichtinger, Hans G.
2013-05-01
Before the invention of orthogonal wavelet systems by Yves Meyer1 in 1986 Gabor expansions (viewed as discretized inversion of the Short-Time Fourier Transform2 using the overlap and add OLA) and (what is now perceived as) wavelet expansions have been treated more or less at an equal footing. The famous paper on painless expansions by Daubechies, Grossman and Meyer3 is a good example for this situation. The description of atomic decompositions for functions in modulation spaces4 (including the classical Sobolev spaces) given by the author5 was directly modeled according to the corresponding atomic characterizations by Frazier and Jawerth,6, 7 more or less with the idea of replacing the dyadic partitions of unity of the Fourier transform side by uniform partitions of unity (so-called BUPU's, first named as such in the early work on Wiener-type spaces by the author in 19808). Watching the literature in the subsequent two decades one can observe that the interest in wavelets "took over", because it became possible to construct orthonormal wavelet systems with compact support and of any given degree of smoothness,9 while in contrast the Balian-Low theorem is prohibiting the existence of corresponding Gabor orthonormal bases, even in the multi-dimensional case and for general symplectic lattices.10 It is an interesting historical fact that* his construction of band-limited orthonormal wavelets (the Meyer wavelet, see11) grew out of an attempt to prove the impossibility of the existence of such systems, and the final insight was that it was not impossible to have such systems, and in fact quite a variety of orthonormal wavelet system can be constructed as we know by now. Meanwhile it is established wisdom that wavelet theory and time-frequency analysis are two different ways of decomposing signals in orthogonal resp. non-orthogonal ways. The unifying theory, covering both cases, distilling from these two situations the common group theoretical background lead to the
An Lp−Lq version of Hardy's theorem for spherical Fourier transform on semisimple Lie groups
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S. Ben Farah
2004-01-01
Full Text Available We consider a real semisimple Lie group G with finite center and K a maximal compact subgroup of G. We prove an Lp−Lq version of Hardy's theorem for the spherical Fourier transform on G. More precisely, let a, b be positive real numbers, 1≤p, q≤∞, and f a K-bi-invariant measurable function on G such that ha−1f∈Lp(G and eb‖λ‖2ℱ(f∈Lq(+* (ha is the heat kernel on G. We establish that if ab≥1/4 and p or q is finite, then f=0 almost everywhere. If ab<1/4, we prove that for all p, q, there are infinitely many nonzero functions f and if ab=1/4 with p=q=∞, we have f=const ha.
Quantum mechanics, group theory, and C60
International Nuclear Information System (INIS)
Rioux, F.
1994-01-01
The recent discovery of a new allotropic form of carbon and its production in macroscopic amounts has generated a tremendous amount of research activity in chemistry, physics, and material science. It has also provided educators with an exciting new vehicle for breathing fresh life into some old, well-established methods and principles. Recently, for example, Boo demonstrated the power of group theory in classifying existing and hypothetical fullerenes by their symmetries. In a similar spirit this note describes a model for the electronic structure of C 60 based on the most elementary principles of quantum mechanics and group theory
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Connes, A.; Kreimer, D.
2000-01-01
This paper gives a complete selfcontained proof of our result (1999) showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra H which is commutative asan algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra G whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of H. We show then that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop γ(z) element of G, z element of C, where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z=D of the holomorphic part γ + of the Birkhoff decomposition of γ. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. (orig.)
Theory Loves Practice: A Teacher Researcher Group
Hochtritt, Lisa; Thulson, Anne; Delaney, Rachael; Dornbush, Talya; Shay, Sarah
2014-01-01
Once a month, art educators from the Denver metro area have been gathering together in the spirit of inquiry to explore issues of the perceived theory and daily practice divide. The Theory Loves Practice (TLP) group was started in 2010 by Professors Rachael Delaney and Anne Thulson from Metropolitan State University of Denver (MSU) and now has 40…
K-theory for group C*-algebras and semigroup C*-algebras
Cuntz, Joachim; Li, Xin; Yu, Guoliang
2017-01-01
This book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions.
Energy Technology Data Exchange (ETDEWEB)
Schwichtenberg, Jakob
2017-09-01
The following topics are dealt with: Special relativity theory, theory of Lie groups, the Lagrang formalism for field theories, quantum operators, quantum wave equations, the theory of interactions, quantum mechanics, quantum field theory, classical mechanics, electrodynamics. (HSI)
Elliptic hypergeometric functions and the representation theory
International Nuclear Information System (INIS)
Spiridonov, V.P.
2011-01-01
Full text: (author)Elliptic hypergeometric functions were discovered around ten years ago. They represent the top level known generalization of the Euler beta integral and Euler-Gauss 2 F 1 hypergeometric function. In general form they are defined by contour integrals involving elliptic gamma functions. We outline the structure of the simplest examples of such functions and discuss their relations to the representation theory of the classical Lie groups and their various deformations. In one of the constructions elliptic hypergeometric integrals describe purely group-theoretical objects having the physical meaning of superconformal indices of four-dimensional supersymmetric gauge field theories
The dynamical Yang-Baxter equation, representation theory, and quantum integrable systems
Etingof, Pavel
2005-01-01
The text is based on an established graduate course given at MIT that provides an introduction to the theory of the dynamical Yang-Baxter equation and its applications, which is an important area in representation theory and quantum groups. The book, which contains many detailed proofs and explicit calculations, will be accessible to graduate students of mathematics, who are familiar with the basics of representation theory of semisimple Lie algebras.
Lie detection based on nonverbal expressions - study of the Czech Republic Police employees
Directory of Open Access Journals (Sweden)
Hedvika Boukalová
2014-12-01
Full Text Available Lie detection based on nonverbal behavior is not a standard method, it is an intuitive process, applied by lay persons, but also professionals. Some of the major sources (e.g. widespread Interrogation Manual by F. Inbau et al., 2004 offer clear recommendations about the nonverbal behavior of liars to investigators of serious crime. These findings are not supported by the research, moreover they can lead to lowering the ability to detect lie (Blair, Kooi 2004. Another topic is mapping the skills of professionals (police officers, members of the secret services and non-specialists to detect lies by nonverbal signs. Across the studies (with few exceptions a low performance in the task of detecting lies by nonverbal expressions (Ekman P., 1996; Vrij, 2004 and others is found. The levels of success are usually around the level of chance. The potential reasons for such results are analyzed (e.g. Blair, Kooi, 2004. However a group of psychologists led by P. Ekman and M. O'Sullivan (O'Sullivan, 2007 managed to find in their years lasting research a group of people whose ability to detect lies is well above the population average. This group is diverse in terms of age, interests and professions, all of them come from the USA. There were certain common features found in this group and also a focus on similar phenomena in the detection of lying. The main goal and research question is to find out: what is the success rate of differentiation between lies and truths in this specific professional group of Czech population, is it the same or different from the results reported in the context of available resources. The research will focus on the ability of respondents to determine the truth or deceit on the basis of non-verbal and paraverbal expressions of observed subjects, with focus on specific professional groups - mainly police workers. We assume, that the police officers are frequently in the contact with people, who are not willing to reveal critical
The topology of Double Field Theory
Hassler, Falk
2018-04-01
We describe the doubled space of Double Field Theory as a group manifold G with an arbitrary generalized metric. Local information from the latter is not relevant to our discussion and so G only captures the topology of the doubled space. Strong Constraint solutions are maximal isotropic submanifold M in G. We construct them and their Generalized Geometry in Double Field Theory on Group Manifolds. In general, G admits different physical subspace M which are Poisson-Lie T-dual to each other. By studying two examples, we reproduce the topology changes induced by T-duality with non-trivial H-flux which were discussed by Bouwknegt, Evslin and Mathai [1].
A course in finite group representation theory
Webb, Peter
2016-01-01
This graduate-level text provides a thorough grounding in the representation theory of finite groups over fields and rings. The book provides a balanced and comprehensive account of the subject, detailing the methods needed to analyze representations that arise in many areas of mathematics. Key topics include the construction and use of character tables, the role of induction and restriction, projective and simple modules for group algebras, indecomposable representations, Brauer characters, and block theory. This classroom-tested text provides motivation through a large number of worked examples, with exercises at the end of each chapter that test the reader's knowledge, provide further examples and practice, and include results not proven in the text. Prerequisites include a graduate course in abstract algebra, and familiarity with the properties of groups, rings, field extensions, and linear algebra.
Lie-admissible structure of Hamilton's original equations with external terms
International Nuclear Information System (INIS)
Santilli, R.M.
1991-09-01
As a necessary additional step in preparation of our operator studies of closed nonhamiltonian systems, in this note we consider the algebraic structure of the original equations proposed by Lagrange and Hamilton, those with external terms representing precisely the contact nonpotential forces of the interior dynamical problem. We show that the brackets of the theory violate the conditions to characterize any algebra. Nevertheless, when properly written, they characterize a covering of the Lie-isotopic algebras called Lie-admissible algebras. It is indicated that a similar occurrence exists for conventional operator treatments, e.g. for nonconservative nuclear cases characterized by nonhermitean Hamiltonians. This occurrence then prevents a rigorous treatment of basic notions, such as that of angular momentum and spin spin, which are centrally dependent on the existence of a consistent algebraic structure. The emergence of the Lie-admissible algebras is therefore expected to be unavoidable for any rigorous operator treatment of open systems with nonlinear, nonlocal and nonhamiltonian external forces. (author). 14 refs, 1 fig
Representation theory a first course
Fulton, William
1991-01-01
The primary goal of these lectures is to introduce a beginner to the finite dimensional representations of Lie groups and Lie algebras. Since this goal is shared by quite a few other books, we should explain in this Preface how our approach differs, although the potential reader can probably see this better by a quick browse through the book. Representation theory is simple to define: it is the study of the ways in which a given group may act on vector spaces. It is almost certainly unique, however, among such clearly delineated subjects, in the breadth of its interest to mathematicians. This is not surprising: group actions are ubiquitous in 20th century mathematics, and where the object on which a group acts is not a vector space, we have learned to replace it by one that is {e. g. , a cohomology group, tangent space, etc. }. As a consequence, many mathematicians other than specialists in the field {or even those who think they might want to be} come in contact with the subject in various ways. It is for ...
Klein Topological Field Theories from Group Representations
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Sergey A. Loktev
2011-07-01
Full Text Available We show that any complex (respectively real representation of finite group naturally generates a open-closed (respectively Klein topological field theory over complex numbers. We relate the 1-point correlator for the projective plane in this theory with the Frobenius-Schur indicator on the representation. We relate any complex simple Klein TFT to a real division ring.
The transverse index theorem for proper cocompact actions of Lie groupoids
Pflaum, M.J.; Posthuma, H.; Tang, X.
2015-01-01
Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we
Applications of group theory to combinatorics
Koolen, Jack; Xu, Ming-Yao; Xu, Mingyao
2008-01-01
Each paper gives an overview of the current state of the art of the given subject and is aimed at researchers and graduate students who use combinatorics and group theory.-John van Bon, Nieuw Archief voor Wiskunde, December 2011.
Medicine, lies and deceptions.
Benn, P
2001-04-01
This article offers a qualified defence of the view that there is a moral difference between telling lies to one's patients, and deceiving them without lying. However, I take issue with certain arguments offered by Jennifer Jackson in support of the same conclusion. In particular, I challenge her claim that to deny that there is such a moral difference makes sense only within a utilitarian framework, and I cast doubt on the aptness of some of her examples of non-lying deception. But I argue that lies have a greater tendency to damage trust than does non-lying deception, and suggest that since many doctors do believe there is a moral boundary between the two types of deception, encouraging them to violate that boundary may have adverse general effects on their moral sensibilities.
Holman, Gordon D.
1989-01-01
The primary purpose of the Theory and Modeling Group meeting was to identify scientists engaged or interested in theoretical work pertinent to the Max '91 program, and to encourage theorists to pursue modeling which is directly relevant to data which can be expected to result from the program. A list of participants and their institutions is presented. Two solar flare paradigms were discussed during the meeting -- the importance of magnetic reconnection in flares and the applicability of numerical simulation results to solar flare studies.
Phase diagrams of exceptional and supersymmetric lattice gauge theories
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Wellegehausen, Bjoern-Hendrik
2012-07-10
In this work different strongly-coupled gauge theories with and without fundamental matter have been studied on the lattice with an emphasis on the confinement problem and the QCD phase diagram at nonvanishing net baryon density as well as on possible supersymmetric extensions of the standard model of particle physics. In gauge theories with a non-trivial centre symmetry, as for instance SU(3)-Yang-Mills theory, confinement is intimately related to the centre of the gauge group, and the Polyakov loop serves as an order parameter for confinement. In QCD, this centre symmetry is explicitly broken by quarks in the fundamental representation of the gauge group. But still quarks and gluons are confined in mesons, baryons and glueballs at low temperatures and small densities, suggesting that centre symmetry is not responsible for the phenomenon of confinement. Therefore it is interesting to study pure gauge theories without centre symmetry. In this work this has been done by replacing the gauge group SU(3) of the strong interaction with the exceptional Lie group G{sub 2}, that has a trivial centre. To investigate G{sub 2} gauge theory on the lattice, a new and highly efficient update algorithm has been developed, based on a local HMC algorithm. Employing this algorithm, the proposed and already investigated first order phase transition from a confined to a deconfined phase has been confirmed, showing that indeed a first order phase transition without symmetry breaking or an order parameter is possible. In this context, also the deconfinement phase transition of the exceptional Lie groups F4 and E6 in three spacetime dimensions has been studied. It has been shown that both theories also possess a first order phase transition.
Phase diagrams of exceptional and supersymmetric lattice gauge theories
International Nuclear Information System (INIS)
Wellegehausen, Bjoern-Hendrik
2012-01-01
In this work different strongly-coupled gauge theories with and without fundamental matter have been studied on the lattice with an emphasis on the confinement problem and the QCD phase diagram at nonvanishing net baryon density as well as on possible supersymmetric extensions of the standard model of particle physics. In gauge theories with a non-trivial centre symmetry, as for instance SU(3)-Yang-Mills theory, confinement is intimately related to the centre of the gauge group, and the Polyakov loop serves as an order parameter for confinement. In QCD, this centre symmetry is explicitly broken by quarks in the fundamental representation of the gauge group. But still quarks and gluons are confined in mesons, baryons and glueballs at low temperatures and small densities, suggesting that centre symmetry is not responsible for the phenomenon of confinement. Therefore it is interesting to study pure gauge theories without centre symmetry. In this work this has been done by replacing the gauge group SU(3) of the strong interaction with the exceptional Lie group G 2 , that has a trivial centre. To investigate G 2 gauge theory on the lattice, a new and highly efficient update algorithm has been developed, based on a local HMC algorithm. Employing this algorithm, the proposed and already investigated first order phase transition from a confined to a deconfined phase has been confirmed, showing that indeed a first order phase transition without symmetry breaking or an order parameter is possible. In this context, also the deconfinement phase transition of the exceptional Lie groups F4 and E6 in three spacetime dimensions has been studied. It has been shown that both theories also possess a first order phase transition.
Group Chaos Theory: A Metaphor and Model for Group Work
Rivera, Edil Torres; Wilbur, Michael; Frank-Saraceni, James; Roberts-Wilbur, Janice; Phan, Loan T.; Garrett, Michael T.
2005-01-01
Group phenomena and interactions are described through the use of the chaos theory constructs and characteristics of sensitive dependence on initial conditions, phase space, turbulence, emergence, self-organization, dissipation, iteration, bifurcation, and attractors and fractals. These constructs and theoretical tenets are presented as applicable…
The derivation of the conventional basis for the classical Lie algebra generators
International Nuclear Information System (INIS)
Karadayi, H.R.
1982-01-01
The explicit construction of the classical Lie algebra generators in the conventional Gell-Mann basis is derived for all irreducible unitary representations of all classical groups. The main framework is based on a description of the simple roots of the classical Lie algebras such that the inter-relations implied by the Cartan matrix of the group among these simple roots are explicit within this description. (author)
N-particle effective generators of the Poincare group derived from a field theory
International Nuclear Information System (INIS)
Krueger, A.; Gloeckle, W.
1999-01-01
In quantum mechanics the principle of relativity is guaranteed by unitary operators being associated with inhomogeneous Lorentz transformations ensuring that quantum mechanical expectation values remain unchanged. In field theory the ten generators of inhomogeneous Lorentz transformations can be derived from a scalar Lagrangian density describing the physical system of interest. They obey the well known Poincare Lie algebra. For interacting systems some of the generators become operators allowing for particle production or annihilation so that the generators act on the full Fock space. However, given a field theory on the whole Fock space we prove that it is possible to construct generators acting on a subspace with a finite number of particles by one and the same unitary transformation of all generators leaving the Poincare algebra valid. In this manner it is in principle possible to derive a relativistically invariant theory of interacting particles on a Hilbert space with a finite number of particles from a field theoretical Lagrangian. Refs. 3 (author)
Computing nilpotent quotients in finitely presented Lie rings
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Csaba Schneider
1997-12-01
Full Text Available A nilpotent quotient algorithm for finitely presented Lie rings over Z (and Q is described. The paper studies the graded and non-graded cases separately. The algorithm computes the so-called nilpotent presentation for a finitely presented, nilpotent Lie ring. A nilpotent presentation consists of generators for the abelian group and the products expressed as linear combinations for pairs formed by generators. Using that presentation the word problem is decidable in L. Provided that the Lie ring L is graded, it is possible to determine the canonical presentation for a lower central factor of L. Complexity is studied and it is shown that optimising the presentation is NP-hard. Computational details are provided with examples, timing and some structure theorems obtained from computations. Implementation in C and GAP interface are available.
25 Years of Quantum Groups: from Definition to Classification
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A. Stolin
2008-01-01
Full Text Available In mathematics and theoretical physics, quantum groups are certain non-commutative, non-cocommutative Hopf algebras, which first appeared in the theory of quantum integrable models and later they were formalized by Drinfeld and Jimbo. In this paper we present a classification scheme for quantum groups, whose classical limit is a polynomial Lie algebra. As a consequence we obtain deformed XXX and XXZ Hamiltonians.
Directory of Open Access Journals (Sweden)
Avraham eMerzel
2015-10-01
Full Text Available Do we feel bound by our own misrepresentations? Does one act of cheating compel the cheater to make subsequent choices that maintain the false image even at a cost? To answer these questions we employed a two-task paradigm such that in the first task the participants could benefit from false reporting of private observations whereas in the second they could benefit from making a prediction in line with their actual, rather than their previously reported observations. Thus, for those participants who inflated their report during the first task, sticking with that report for the second task was likely to lead to a loss, whereas deviating from it would imply that they had lied. Data from three experiments (total N=116 indicate that, having lied, participants were ready to suffer future loss rather than admit, even if implicitly, that they had lied.
When is a lie acceptable? Work and private life lying acceptance depends on its beneficiary.
Cantarero, Katarzyna; Szarota, Piotr; Stamkou, Eftychia; Navas, Marisol; Dominguez Espinosa, Alejandra Del Carmen
2018-01-01
In this article we show that when analyzing attitude towards lying in a cross-cultural setting, both the beneficiary of the lie (self vs other) and the context (private life vs. professional domain) should be considered. In a study conducted in Estonia, Ireland, Mexico, The Netherlands, Poland, Spain, and Sweden (N = 1345), in which participants evaluated stories presenting various types of lies, we found usefulness of relying on the dimensions. Results showed that in the joint sample the most acceptable were other-oriented lies concerning private life, then other-oriented lies in the professional domain, followed by egoistic lies in the professional domain; and the least acceptance was shown for egoistic lies regarding one's private life. We found a negative correlation between acceptance of a behavior and the evaluation of its deceitfulness.
Lie algebraical aspects of quantum statistics
International Nuclear Information System (INIS)
Palev, T.D.
1976-01-01
It is shown that the secon quantization axioms can, in principle, be satisfied with creation and annihilation operators generating (in the case of n pairs of such operators) the Lie algebra Asub(n) of the group SL(n+1). A concept of the Fock space is introduced. The matrix elements of the operators are found
Introduction to representation theory
Etingof, Pavel; Hensel, Sebastian; Liu, Tiankai; Schwendner, Alex
2011-01-01
Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum field theory. The goal of this book is to give a "holistic" introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints. The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic k...
A cohomological characterization of Leibniz central extensions of Lie algebras
International Nuclear Information System (INIS)
Hu Naihong; Pei Yufeng; Liu Dong
2006-12-01
Motivated by Pirashvili's spectral sequences on a Leibniz algebra, some notions such as invariant symmetric bilinear forms, dual space derivations and the Cartan-Koszul homomorphism are connected together to give a description of the second Leibniz cohomology groups with trivial coefficients of Lie algebras (as Leibniz objects), which leads to a concise approach to determining one-dimensional Leibniz central extensions of Lie algebras. As applications, we contain the discussions for some interesting classes of infinite-dimensional Lie algebras. In particular, our results include the cohomological version of Gao's main Theorem for Kac-Moody algebras and answer a question. (author)
Vrij, Aldert; Taylor, Paul J.; Picornell, Isabel; Oxburgh, Gavin; Myklebust, Trond; Grant, Tim; Milne, Rebecca
2015-01-01
In this chapter, we discuss verbal lie detection and will argue that speech content can be revealing about deception. Starting with a section discussing the, in our view, myth that non-verbal behaviour would be more revealing about deception than speech, we then provide an overview of verbal lie
Theoretical progress at CNDC theory group
International Nuclear Information System (INIS)
Lu Zhongdao
1993-01-01
In 1992, CNDC (Chinese Nuclear Data Center) theory group has made progress in model study, code making and data calculations for low energy nuclear reaction, intermediate and high energy nuclear reaction. It has also made progress in parameter library establishment. The brief explanations are presented
Heterotic α ’-corrections in Double Field Theory
Bedoya, OscarInstituto de Astronomía y Física del Espacio (CONICET-UBA), Ciudad Universitaria, Buenos Aires, Argentina; Marqués, Diego(Instituto de Astronomía y Física del Espacio (CONICET-UBA), Ciudad Universitaria, Buenos Aires, Argentina); Núñez, Carmen(Instituto de Astronomía y Física del Espacio (CONICET-UBA), Ciudad Universitaria, Buenos Aires, Argentina)
2014-01-01
We extend the generalized flux formulation of Double Field Theory to include all the first order bosonic contributions to the α′ expansion of the heterotic string low energy effective theory. The generalized tangent space and duality group are enhanced by α′ corrections, and the gauge symmetries are generated by the usual (gauged) generalized Lie derivative in the extended space. The generalized frame receives derivative corrections through the spin connection with torsion, which is incorpora...
Algebraic K-theory of crystallographic groups the three-dimensional splitting case
Farley, Daniel Scott
2014-01-01
The Farrell-Jones isomorphism conjecture in algebraic K-theory offers a description of the algebraic K-theory of a group using a generalized homology theory. In cases where the conjecture is known to be a theorem, it gives a powerful method for computing the lower algebraic K-theory of a group. This book contains a computation of the lower algebraic K-theory of the split three-dimensional crystallographic groups, a geometrically important class of three-dimensional crystallographic group, representing a third of the total number. The book leads the reader through all aspects of the calculation. The first chapters describe the split crystallographic groups and their classifying spaces. Later chapters assemble the techniques that are needed to apply the isomorphism theorem. The result is a useful starting point for researchers who are interested in the computational side of the Farrell-Jones isomorphism conjecture, and a contribution to the growing literature in the field.
Conformal field theory, triality and the Monster group
International Nuclear Information System (INIS)
Dolan, L.; Goddard, P.; Montague, P.
1990-01-01
From an even self-dual N-dimensional lattice, Λ, it is always possible to construct two (chiral) conformal field theories, an untwisted theory H (Λ), and a Z 2 -twisted theory H (Λ), constructed using the reflection twist. (N must be a multiple of 8 and the theories are modular invariant if it is a multiple of 24.) Similarly, from a doubly-even self-dual binary code C, it is possible to construct two even self-dual lattices, an untwisted one Λ C and a twisted one anti Λ C . It is shown that H(Λ C ) always has a triality structure, and that this triality induces first an isomorphism H(anti Λ C )≅H(Λ C ) and, through this, a triality of H(anti Λ C ). In the case where C is the Golay code, anti Λ C is the Leech lattice and the induced triality is the extra symmetry necessary to generate the Monster group from (an extension of) Conway's group. Thus it is demonstrated that triality is a generic symmetry. The induced isomorphism accounts for all 9 of the coincidences between the 48 conformal field theories H(Λ) and H(Λ) with N=24. (orig.)
Studies on representation of the Lorentz group and gauge theory
International Nuclear Information System (INIS)
Hanitriarivo, R.
2002-01-01
This work is focused on studies about the representation of the Lorentz group and gauge theory. The mathematical tools required for the different studies are presented, as well as for the representation of the Lorentz group and for the gauge theory. Representation of the Lorentz group gives the possible types of fields and wave functions that describe particles: fermions are described by spinors and bosons are described by scalar or vector. Each of these entities (spinors, scalars, vectors) are characterized by their behavior under the action of Lorentz transformations.Gauge theory is used to describe the interactions between particles. [fr
Group theory approach to scattering
International Nuclear Information System (INIS)
Wu, J.
1985-01-01
For certain physical systems, there exists a dynamical group which contains the operators connecting states with the same energy but belonging to potentials with different strengths. This group is called the potential group of that system. The SO(2,1) potential groups structure is introduced to describe physical systems with mixed spectra, such as Morse and Poeschl-teller potentials. The discrete spectrum describes bound states and the continuous spectrum describes bound states and the continuous spectrum describes scattering states. A solvable class of one-dimensional potentials given by Natanzon belongs to this structure with an SO(2,2) potential group. The potential group structure provides us with an algebraic procedure generating the recursion relations for the scattering matrix, which can be formulated in a purely algebraic fashion, divorced from any differential realization. This procedure, when applied to the three-dimensional scattering problem with SO(3,1) symmetry, generates the scattering matrix of the Coulomb problem. Preliminary phenomenological models for elastic scattering in a heavy-ion collision are constructed on the basis. The results obtained here can be regarded as an important extension of the group theory techniques to scattering problems similar to that developed for bound state problems
Nonflexible Lie-admissible algebras
International Nuclear Information System (INIS)
Myung, H.C.
1978-01-01
We discuss the structure of Lie-admissible algebras which are defined by nonflexible identities. These algebras largely arise from the antiflexible algebras, 2-varieties and associator dependent algebras. The nonflexible Lie-admissible algebras in our discussion are in essence byproducts of the study of nonassociative algebras defined by identities of degree 3. The main purpose is to discuss the classification of simple Lie-admissible algebras of nonflexible type
Group contractions in quantum field theory
International Nuclear Information System (INIS)
Concini, C. De; Vitiello, G.
1979-01-01
General theorems are given for SU(n) and SO(n). A projective geometry argument is also presented with disclosure of the occurrence a group contraction mechanism as a geometric consequence of spontaneous breakdown of symmetry. It is also shown that a contraction of the conformal group gives account of the number of degrees of freedom of an n-pseudoparticle system in an Euclidean SU(2) gauge invariant Yang-Mills theory, in agreement with the result obtained by algebraic geometry methods. Low-energy theorems and ordered states symmetry patterns are observable manifestations of group contractions. These results seem to support the conjecture that the transition from quantum to classical physics involves a group contraction mechanism. (author)
Structure preserving transformations for Newtonian Lie-admissible equations
International Nuclear Information System (INIS)
Cantrijn, F.
1979-01-01
Recently, a new formulation of non-conservative mechanics has been presented in terms of Hamilton-admissible equations which constitute a generalization of the conventional Hamilton equations. The algebraic structure entering the Hamilton-admissible description of a non-conservative system is that of a Lie-admissible algebra. The corresponding geometrical treatment is related to the existence of a so-called symplectic-admissible form. The transformation theory for Hamilton-admissible systems is currently investigated. The purpose of this paper is to describe one aspect of this theory by identifying the class of transformations which preserve the structure of Hamilton-admissible equations. Necessary and sufficient conditions are established for a transformation to be structure preserving. Some particular cases are discussed and an example is worked out
A generalization of the Lie derivative
International Nuclear Information System (INIS)
Dolan, P.
1984-01-01
If X=xisup(i)deltasub(i) and Y=etasup(i)deltasub(i) are vector fields then it is well-known that the Lie derivative Poundsub(X)Y equivalent to [X,Y] (xisup(s)deltasub(s) etasup(s)deltasub(s)xisup(i))deltasub(i) is also a vector field under general coordinate transformations. A generalization of this result, due to previous workers, allows a definition of Poundsub(F)G, where F,G are arbitrary contravariant tensor fields. The formulae are linear in the first partial derivatives of F and G. An application to the theory of Killing-Yano tensor fields on Riemannian manifolds is given. (author)
Akivis, M A
2011-01-01
This book describes the life and achievements of the great French mathematician, Elie Cartan. Here readers will find detailed descriptions of Cartan's discoveries in Lie groups and algebras, associative algebras, differential equations, and differential geometry, as well of later developments stemming from his ideas. There is also a biographical sketch of Cartan's life. A monumental tribute to a towering figure in the history of mathematics, this book will appeal to mathematicians and historians alike.
Isomorphism of Intransitive Linear Lie Equations
Directory of Open Access Journals (Sweden)
Jose Miguel Martins Veloso
2009-11-01
Full Text Available We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the intransitive Lie algebra we recover the linear Lie equation, unless of formal isomorphism. The intransitive Lie algebra gives the structure functions introduced by É. Cartan.
Directory of Open Access Journals (Sweden)
Cantarero Katarzyna
2017-06-01
Full Text Available Lay perceptions of lying are argued to consist of a lie prototype. The latter was found to entail the intention to deceive, belief in falsity and falsity (Coleman & Kay, 1981. We proposed and found that the perceptions of the benefits of others are also an important factor that influences the extent, to which an act of intentional misleading someone to foster a false belief is labeled as a lie. Drawing from the intuitionist model of moral judgments (Haidt, 2001 we assumed that moral judgment of the behaviour would mediate the relationship. In Study 1 we analyzed data coming from a crosscultural project and found that perceived intention to benefit others was negatively related to lie labeling and that this relationship was mediated by the moral judgment of that act. In Study 2 we found that manipulating the benefits of others influenced the extent, to which an act of intentional misleading in order to foster a false belief is labeled as a lie and that, again, this relationship is mediated by the moral judgment of that act.
Dimensional reduction of exceptional E6,E8 gauge groups and flavour chirality
International Nuclear Information System (INIS)
Koca, M.
1984-01-01
Ten-dimensional Yang - Mills gauge theories based on the exceptional groups E 6 and E 8 are reduced to four-dimensional flavour-chiral Yang - Mills - Higgs theories where the extra six dimensions are identified with the compact G 2 /SU(3) and SO(7)/SO(6) coset spaces. A ten-dimensional E 8 theory leads to three families of SU(5), one of which lies in the 144-dimensional representation of SO(10)
Renormalization group theory impact on experimental magnetism
Köbler, Ulrich
2010-01-01
Spin wave theory of magnetism and BCS theory of superconductivity are typical theories of the time before renormalization group (RG) theory. The two theories consider atomistic interactions only and ignore the energy degrees of freedom of the continuous (infinite) solid. Since the pioneering work of Kenneth G. Wilson (Nobel Prize of physics in 1982) we know that the continuous solid is characterized by a particular symmetry: invariance with respect to transformations of the length scale. Associated with this symmetry are particular field particles with characteristic excitation spectra. In diamagnetic solids these are the well known Debye bosons. This book reviews experimental work on solid state physics of the last five decades and shows in a phenomenological way that the dynamics of ordered magnets and conventional superconductors is controlled by the field particles of the infinite solid and not by magnons and Cooper pairs, respectively. In the case of ordered magnets the relevant field particles are calle...
An introduction to the theory of groups
Alexandroff, Paul; Petersen, GM
2012-01-01
This introductory exposition of group theory by an eminent Russian mathematician is particularly suited to undergraduates. Includes a wealth of simple examples, primarily geometrical, and end-of-chapter exercises. 1959 edition.
Exact and microscopic one-instanton calculations in N=2 supersymmetric Yang-Mills theories
International Nuclear Information System (INIS)
Ito, K.; Sasakura, N.
1997-01-01
We study the low-energy effective theory in N=2 super Yang-Mills theories by microscopic and exact approaches. We calculate the one-instanton correction to the prepotential for any simple Lie group from the microscopic approach. We also study the Picard-Fuchs equations and their solutions in the semi-classical regime for classical gauge groups with rank r≤3. We find that for gauge groups G=A r , B r , C r (r≤3) the microscopic results agree with those from the exact solutions. (orig.)
Classical gauge theories on the coadjoint orbits of infinite dimensional groups
International Nuclear Information System (INIS)
Grabowski, M.P.; Virginia Polytechnic Inst. and State Univ., Blacksburg; Tze Chiahsiung
1991-01-01
We reformulate several classical gauge theories on the coadjoint orbits of the semidirect product of the gauge group and the Weyl group. The construction is given for the Yang-Mills theories in arbitrary spacetime dimension d, Chern-Simons topological theory (d=3) and higher dimensional topological models of Horowitz (d≥4). (orig.)
Lie Algebras and Integrable Systems
International Nuclear Information System (INIS)
Zhang Yufeng; Mei Jianqin
2012-01-01
A 3 × 3 matrix Lie algebra is first introduced, its subalgebras and the generated Lie algebras are obtained, respectively. Applications of a few Lie subalgebras give rise to two integrable nonlinear hierarchies of evolution equations from their reductions we obtain the nonlinear Schrödinger equations, the mKdV equations, the Broer-Kaup (BK) equation and its generalized equation, etc. The linear and nonlinear integrable couplings of one integrable hierarchy presented in the paper are worked out by casting a 3 × 3 Lie subalgebra into a 2 × 2 matrix Lie algebra. Finally, we discuss the elliptic variable solutions of a generalized BK equation. (general)
Zanette, Sarah; Gao, Xiaoqing; Brunet, Megan; Bartlett, Marian Stewart; Lee, Kang
2016-10-01
The current study used computer vision technology to examine the nonverbal facial expressions of children (6-11years old) telling antisocial and prosocial lies. Children in the antisocial lying group completed a temptation resistance paradigm where they were asked not to peek at a gift being wrapped for them. All children peeked at the gift and subsequently lied about their behavior. Children in the prosocial lying group were given an undesirable gift and asked if they liked it. All children lied about liking the gift. Nonverbal behavior was analyzed using the Computer Expression Recognition Toolbox (CERT), which employs the Facial Action Coding System (FACS), to automatically code children's facial expressions while lying. Using CERT, children's facial expressions during antisocial and prosocial lying were accurately and reliably differentiated significantly above chance-level accuracy. The basic expressions of emotion that distinguished antisocial lies from prosocial lies were joy and contempt. Children expressed joy more in prosocial lying than in antisocial lying. Girls showed more joy and less contempt compared with boys when they told prosocial lies. Boys showed more contempt when they told prosocial lies than when they told antisocial lies. The key action units (AUs) that differentiate children's antisocial and prosocial lies are blink/eye closure, lip pucker, and lip raise on the right side. Together, these findings indicate that children's facial expressions differ while telling antisocial versus prosocial lies. The reliability of CERT in detecting such differences in facial expression suggests the viability of using computer vision technology in deception research. Copyright © 2016 Elsevier Inc. All rights reserved.
Diagrammatic group theory in quark models
International Nuclear Information System (INIS)
Canning, G.P.
1977-05-01
A simple and systematic diagrammatic method is presented for calculating the numerical factors arising from group theory in quark models: dimensions, casimir invariants, vector coupling coefficients and especially recoupling coefficients. Some coefficients for the coupling of 3 quark objects are listed for SU(n) and SU(2n). (orig.) [de
Bakhurst, D
1992-01-01
This article challenges Jennifer Jackson's recent defence of doctors' rights to deceive patients. Jackson maintains there is a general moral difference between lying and intentional deception: while doctors have a prima facie duty not to lie, there is no such obligation to avoid deception. This paper argues 1) that an examination of cases shows that lying and deception are often morally equivalent, and 2) that Jackson's position is premised on a species of moral functionalism that misconstrue...
Mapping spaces and automorphism groups of toric noncommutative spaces
Barnes, Gwendolyn E.; Schenkel, Alexander; Szabo, Richard J.
2017-09-01
We develop a sheaf theory approach to toric noncommutative geometry which allows us to formalize the concept of mapping spaces between two toric noncommutative spaces. As an application, we study the `internalized' automorphism group of a toric noncommutative space and show that its Lie algebra has an elementary description in terms of braided derivations.
Manifestations of group covariance in a metric theory
International Nuclear Information System (INIS)
Halpern, L.
1984-01-01
The requirement to present Dirac's Large Number Hypothesis in one system of units in which the resulting modifications to Einstein's theory are exhibited, led to the construction of generalizations of General Relativity based rigorously on the geometry of semisimple groups. The foundations of such a theory are discussed and some of the possible interpretations are presented. (author)
The structure of compact groups a primer for the student, a handbook for the expert
Hofmann, Karl H
2013-01-01
Dealing with subject matter of compact groups that is frequently cited in fields like algebra, topology, functional analysis, and theoretical physics, this book - now in its third revised and augmented edition - has been conceived with the dual purpose of providing a text book for upper level graduate courses or seminars, and of serving as a source book for research specialists who need to apply the structure and representation theory of compact groups. After a gentle introduction to compact groups and their representation theory, the book presents self-contained courses on linear Lie groups,
G-theory: The generator of M-theory and supersymmetry
Sepehri, Alireza; Pincak, Richard
2018-04-01
In string theory with ten dimensions, all Dp-branes are constructed from D0-branes whose action has two-dimensional brackets of Lie 2-algebra. Also, in M-theory, with 11 dimensions, all Mp-branes are built from M0-branes whose action contains three-dimensional brackets of Lie 3-algebra. In these theories, the reason for difference between bosons and fermions is unclear and especially in M-theory there is not any stable object like stable M3-branes on which our universe would be formed on it and for this reason it cannot help us to explain cosmological events. For this reason, we construct G-theory with M dimensions whose branes are formed from G0-branes with N-dimensional brackets. In this theory, we assume that at the beginning there is nothing. Then, two energies, which differ in their signs only, emerge and produce 2M degrees of freedom. Each two degrees of freedom create a new dimension and then M dimensions emerge. M-N of these degrees of freedom are removed by symmetrically compacting half of M-N dimensions to produce Lie-N-algebra. In fact, each dimension produces a degree of freedom. Consequently, by compacting M-N dimensions from M dimensions, N dimensions and N degrees of freedom is emerged. These N degrees of freedoms produce Lie-N-algebra. During this compactification, some dimensions take extra i and are different from other dimensions, which are known as time coordinates. By this compactification, two types of branes, Gp and anti-Gp-branes, are produced and rank of tensor fields which live on them changes from zero to dimension of brane. The number of time coordinates, which are produced by negative energy in anti-Gp-branes, is more sensible to number of times in Gp-branes. These branes are compactified anti-symmetrically and then fermionic superpartners of bosonic fields emerge and supersymmetry is born. Some of gauge fields play the role of graviton and gravitino and produce the supergravity. The question may arise that what is the physical reason
General relativity and gauge gravity theories of higher order
International Nuclear Information System (INIS)
Konopleva, N.P.
1998-01-01
It is a short review of today's gauge gravity theories and their relations with Einstein General Relativity. The conceptions of construction of the gauge gravity theories with higher derivatives are analyzed. GR is regarded as the gauge gravity theory corresponding to the choice of G ∞4 as the local gauge symmetry group and the symmetrical tensor of rank two g μν as the field variable. Using the mathematical technique, single for all fundamental interactions (namely variational formalism for infinite Lie groups), we can obtain Einstein's theory as the gauge theory without any changes. All other gauge approaches lead to non-Einstein theories of gravity. But above-mentioned mathematical technique permits us to construct the gauge gravity theory of higher order (for instance SO (3,1)-gravity) so that all vacuum solutions of Einstein equations are the solutions of the SO (3,1)-gravity theory. The structure of equations of SO(3,1)-gravity becomes analogous to Weeler-Misner geometrodynamics one
Quantum groups, quantum categories and quantum field theory
Fröhlich, Jürg
1993-01-01
This book reviews recent results on low-dimensional quantum field theories and their connection with quantum group theory and the theory of braided, balanced tensor categories. It presents detailed, mathematically precise introductions to these subjects and then continues with new results. Among the main results are a detailed analysis of the representation theory of U (sl ), for q a primitive root of unity, and a semi-simple quotient thereof, a classfication of braided tensor categories generated by an object of q-dimension less than two, and an application of these results to the theory of sectors in algebraic quantum field theory. This clarifies the notion of "quantized symmetries" in quantum fieldtheory. The reader is expected to be familiar with basic notions and resultsin algebra. The book is intended for research mathematicians, mathematical physicists and graduate students.
A study of collective coordinates and dynamical groups in nuclear theory
International Nuclear Information System (INIS)
Papadopolos, Z.
1983-01-01
Lie-algebraic techniques for the group action on manifolds given as a direct product of coset spaces and group manifolds are developed. The microscopic realisation of the Mass Quadrupole Collective Model (MQC) in the S0(3)xSO(n) and GLsub(+)(3, R)xSO(n) schemes is studied. The problem of the separation of the kinetic energy and the velocity field into a collective and an intrinsic part is analyzed. Different coordinate schemes in phase space for the U(n)-invariant collective motion and the U(3) dynamical group are introduced. In the GL(3,C)xU(n) scheme, the invariant volume element in the new coordinates and a completely orthonormal basis is constructed. (orig.) [de
Wigner's little group as a gauge generator in linearized gravity theories
International Nuclear Information System (INIS)
Scaria, Tomy; Chakraborty, Biswajit
2002-01-01
We show that the translational subgroup of Wigner's little group for massless particles in 3 + 1 dimensions generates gauge transformation in linearized Einstein gravity. Similarly, a suitable representation of the one-dimensional translational group T(1) is shown to generate gauge transformation in the linearized Einstein-Chern-Simons theory in 2 + 1 dimensions. These representations are derived systematically from appropriate representations of translational groups which generate gauge transformations in gauge theories living in spacetime of one higher dimension by the technique of dimensional descent. The unified picture thus obtained is compared with a similar picture available for vector gauge theories in 3 + 1 and 2 + 1 dimensions. Finally, the polarization tensor of the Einstein-Pauli-Fierz theory in 2 + 1 dimensions is shown to split into the polarization tensors of a pair of Einstein-Chern-Simons theories with opposite helicities suggesting a doublet structure for the Einstein-Pauli-Fierz theory
International Nuclear Information System (INIS)
Edelen, D.G.B.
1986-01-01
Homogeneous scaling of the group space of the Poincare group, P 10 , is shown to induce scalings of all geometric quantities associated with the local action of P 10 . The field equations for both the translation and the Lorentz rotation compensating fields reduce to O(1) equations if the scaling parameter is set equal to the general relativistic gravitational coupling constant 8πGc -4 . Standard expansions of all field variables in power series in the scaling parameter give the following results. The zeroth-order field equations are exactly the classical field equations for matter fields on Minkowski space subject to local action of an internal symmetry group (classical gauge theory). The expansion process is shown to break P 10 -gauge covariance of the theory, and hence solving the zeroth-order field equations imposes an implicit system of P 10 -gauge conditions. Explicit systems of field equations are obtained for the first- and higher-order approximations. The first-order translation field equations are driven by the momentum-energy tensor of the matter and internal compensating fields in the zeroth order (classical gauge theory), while the first-order Lorentz rotation field equations are driven by the spin currents of the same classical gauge theory. Field equations for the first-order gravitational corrections to the matter fields and the gauge fields for the internal symmetry group are obtained. Direct Poincare gauge theory is thus shown to satisfy the first two of the three-part acid test of any unified field theory. Satisfaction of the third part of the test, at least for finite neighborhoods, seems probable
Computations in finite-dimensional Lie algebras
Directory of Open Access Journals (Sweden)
A. M. Cohen
1997-12-01
Full Text Available This paper describes progress made in context with the construction of a general library of Lie algebra algorithms, called ELIAS (Eindhoven Lie Algebra System, within the computer algebra package GAP. A first sketch of the package can be found in Cohen and de Graaf[1]. Since then, in a collaborative effort with G. Ivanyos, the authors have continued to develop algorithms which were implemented in ELIAS by the second author. These activities are part of a bigger project, called ACELA and financed by STW, the Dutch Technology Foundation, which aims at an interactive book on Lie algebras (cf. Cohen and Meertens [2]. This paper gives a global description of the main ways in which to present Lie algebras on a computer. We focus on the transition from a Lie algebra abstractly given by an array of structure constants to a Lie algebra presented as a subalgebra of the Lie algebra of n×n matrices. We describe an algorithm typical of the structure analysis of a finite-dimensional Lie algebra: finding a Levi subalgebra of a Lie algebra.
Infrared fixed point of SU(2) gauge theory with six flavors
Leino, Viljami; Rummukainen, Kari; Suorsa, Joni; Tuominen, Kimmo; Tähtinen, Sara
2018-06-01
We compute the running of the coupling in SU(2) gauge theory with six fermions in the fundamental representation of the gauge group. We find strong evidence that this theory has an infrared stable fixed point at strong coupling and measure also the anomalous dimension of the fermion mass operator at the fixed point. This theory therefore likely lies close to the boundary of the conformal window and will display novel infrared dynamics if coupled with the electroweak sector of the Standard Model.
Dilogarithm identities in conformal field theory and group homology
International Nuclear Information System (INIS)
Dupont, J.L.
1994-01-01
Recently, Rogers' dilogarithm identities have attracted much attention in the setting of conformal field theory as well as lattice model calculations. One of the connecting threads is an identity of Richmond-Szekeres that appeared in the computation of central charges in conformal field theory. We show that the Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin (equivalent to an identity found earlier by Lewin) can be interpreted as a lift of a generator of the third integral homology of a finite cyclic subgroup sitting inside the projective special linear group of all 2x2 real matrices viewed as a discrete group. This connection allows us to clarify a few of the assertions and conjectures stated in the work of Nahm-Recknagel-Terhoven concerning the role of algebraic K-theory and Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related to hyperbolic 3-manifolds as suggested but is more appropriately related to the group manifold of the universal covering group of the projective special linear group of all 2x2 real matrices viewed as a topological group. This also resolves the weaker version of the conjecture as formulated by Kirillov. We end with a summary of a number of open conjectures on the mathematical side. (orig.)
Minimal Representations and Reductive Dual Pairs in Conformal Field Theory
International Nuclear Information System (INIS)
Todorov, Ivan
2010-01-01
A minimal representation of a simple non-compact Lie group is obtained by 'quantizing' the minimal nilpotent coadjoint orbit of its Lie algebra. It provides context for Roger Howe's notion of a reductive dual pair encountered recently in the description of global gauge symmetry of a (4-dimensional) conformal observable algebra. We give a pedagogical introduction to these notions and point out that physicists have been using both minimal representations and dual pairs without naming them and hence stand a chance to understand their theory and to profit from it.
Applications of Lie-group methods to the equations of magnetohydrodynamics
International Nuclear Information System (INIS)
Mandrekas, J.
1987-01-01
The invariance properties of various sets of magnetohydrodynamic (MHD) equations are studied using techniques from the theory of differential forms. Equations considered include the ideal MHD equations in different geometries and with different magnetic field configurations, the MHD equations in the presence of gravitational forces due to self-attraction or external fields, and the MHD equations including finite thermal conductivity and magnetic viscosity. The knowledge of the group structure of these equations is then used to introduce similarity variables to these equations. For each choice of similarity variables, the original set of partial differential equations is transformed into a set of ordinary differential equations and the most general form of the initial conditions is determined. Three cases are studied in detail and the corresponding sets of ordinary differential equations are solved numerically: the problem of a blast wave in an inhomogeneous atmosphere, the problem of a piston moving according to a power law in time, and the problem of a piston moving according to an exponential law in time
Debey, E.; De Houwer, J.; Verschuere, B.
2014-01-01
Cognitive models of deception focus on the conflict-inducing nature of the truth activation during lying. Here we tested the counterintuitive hypothesis that the truth can also serve a functional role in the act of lying. More specifically, we examined whether the construction of a lie can involve a
Teaching Group Theory Using Rubik's Cubes
Cornock, Claire
2015-01-01
Being situated within a course at the applied end of the spectrum of maths degrees, the pure mathematics modules at Sheffield Hallam University have an applied spin. Pure topics are taught through consideration of practical examples such as knots, cryptography and automata. Rubik's cubes are used to teach group theory within a final year pure…
Groups, generators, syzygies, and orbits in invariant theory
Popov, V L
2011-01-01
The history of invariant theory spans nearly a century and a half, with roots in certain problems from number theory, algebra, and geometry appearing in the work of Gauss, Jacobi, Eisenstein, and Hermite. Although the connection between invariants and orbits was essentially discovered in the work of Aronhold and Boole, a clear understanding of this connection had not been achieved until recently, when invariant theory was in fact subsumed by a general theory of algebraic groups. Written by one of the major leaders in the field, this book provides an excellent, comprehensive exposition of invariant theory. Its point of view is unique in that it combines both modern and classical approaches to the subject. The introductory chapter sets the historical stage for the subject, helping to make the book accessible to nonspecialists.
Symmetry an introduction to group theory and its applications
McWeeny, Roy
2002-01-01
Well-organized volume develops ideas of group and representation theory in progressive fashion. Emphasis on finite groups describing symmetry of regular polyhedra and of repeating patterns, plus geometric illustrations.
Geometrical theory of spin motion
International Nuclear Information System (INIS)
Halpern, L.
1983-01-01
A discussion of the fundamental interrelation of geometry and physical laws with Lie groups leads to a reformulation and heuristic modification of the principle of inertia and the principle of equivalence, which is based on the simple De Sitter group instead of the Poincare group. The resulting law of motion allows a unified formulation for structureless and spinning test particles. A metrical theory of gravitation is constructed with the modified principle, which is structured after the geometry of the manifold of the De Sitter group. The theory is equivalent to a particular Kaluza-Klein theory in ten dimensions with the Lorentz group as gauge group. A restricted version of this theory excludes torsion. It is shown by a reformulation of the energy momentum complex that this version is equivalent to general relativity with a cosmologic term quadratic in the curvature tensor and in which the existence of spinning particle fields is inherent from first principles. The equations of the general theory with torsion are presented and it is shown in a special case how the boundary conditions for the torsion degree of freedom have to be chosen such as to treat orbital and spin angular momenta on an equal footing. The possibility of verification of the resulting anomalous spin-spin interaction is mentioned and a model imposed by the group topology of SO(3, 2) is outlined in which the unexplained discrepancy between the magnitude of the discrete valued coupling constants and the gravitational constant in Kaluza-Klein theories is resolved by the identification of identical fermions as one orbit. The mathematical structure can be adapted to larger groups to include other degrees of freedom. 41 references
Yang-Mills theory for non-semisimple groups
Nuyts, J; Nuyts, Jean; Wu, Tai Tsun
2003-01-01
For semisimple groups, possibly multiplied by U(1)'s, the number of Yang-Mills gauge fields is equal to the number of generators of the group. In this paper, it is shown that, for non-semisimple groups, the number of Yang-Mills fields can be larger. These additional Yang-Mills fields are not irrelevant because they appear in the gauge transformations of the original Yang-Mills fields. Such non-semisimple Yang-Mills theories may lead to physical consequences worth studying. The non-semisimple group with only two generators that do not commute is studied in detail.
On Lie point symmetry of classical Wess-Zumino-Witten model
International Nuclear Information System (INIS)
Maharana, Karmadeva
2001-06-01
We perform the group analysis of Witten's equations of motion for a particle moving in the presence of a magnetic monopole, and also when constrained to move on the surface of a sphere, which is the classical example of Wess-Zumino-Witten model. We also consider variations of this model. Our analysis gives the generators of the corresponding Lie point symmetries. The Lie symmetry corresponding to Kepler's third law is obtained in two related examples. (author)
International Nuclear Information System (INIS)
Blau, M.; Thompson, G.
1995-01-01
We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaat-Heckman theorem, and the Weyl integral formula respectively. In each case, we first introduce the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups), and describe the finite dimensional integration formulae. We then discuss some applications to path integrals and give an overview of the relevant literature. The applications we deal with include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two-dimensional Yang-Mills theory. (author). 83 refs
Quantum gravity with matter and group field theory
International Nuclear Information System (INIS)
Krasnov, Kirill
2007-01-01
A generalization of the matrix model idea to quantum gravity in three and higher dimensions is known as group field theory (GFT). In this paper we study generalized GFT models that can be used to describe 3D quantum gravity coupled to point particles. The generalization considered is that of replacing the group leading to pure quantum gravity by the twisted product of the group with its dual-the so-called Drinfeld double of the group. The Drinfeld double is a quantum group in that it is an algebra that is both non-commutative and non-cocommutative, and special care is needed to define group field theory for it. We show how this is done, and study the resulting GFT models. Of special interest is a new topological model that is the 'Ponzano-Regge' model for the Drinfeld double. However, as we show, this model does not describe point particles. Motivated by the GFT considerations, we consider a more general class of models that are defined not using GFT, but the so-called chain mail techniques. A general model of this class does not produce 3-manifold invariants, but has an interpretation in terms of point particle Feynman diagrams
Unitary Representations of Gauge Groups
Huerfano, Ruth Stella
I generalize to the case of gauge groups over non-trivial principal bundles representations that I. M. Gelfand, M. I. Graev and A. M. Versik constructed for current groups. The gauge group of the principal G-bundle P over M, (G a Lie group with an euclidean structure, M a compact, connected and oriented manifold), as the smooth sections of the associated group bundle is presented and studied in chapter I. Chapter II describes the symmetric algebra associated to a Hilbert space, its Hilbert structure, a convenient exponential and a total set that later play a key role in the construction of the representation. Chapter III is concerned with the calculus needed to make the space of Lie algebra valued 1-forms a Gaussian L^2-space. This is accomplished by studying general projective systems of finitely measurable spaces and the corresponding systems of sigma -additive measures, all of these leading to the description of a promeasure, a concept modeled after Bourbaki and classical measure theory. In the case of a locally convex vector space E, the corresponding Fourier transform, family of characters and the existence of a promeasure for every quadratic form on E^' are established, so the Gaussian L^2-space associated to a real Hilbert space is constructed. Chapter III finishes by exhibiting the explicit Hilbert space isomorphism between the Gaussian L ^2-space associated to a real Hilbert space and the complexification of its symmetric algebra. In chapter IV taking as a Hilbert space H the L^2-space of the Lie algebra valued 1-forms on P, the gauge group acts on the motion group of H defining in an straight forward fashion the representation desired.
Poisson-Lie T-duality open strings and D-branes
Klimcik, C.
1996-01-01
Global issues of the Poisson-Lie T-duality are addressed. It is shown that oriented open strings propagating on a group manifold G are dual to D-brane - anti-D-brane pairs propagating on the dual group manifold \\ti G. The D-branes coincide with the symplectic leaves of the standard Poisson structure induced on the dual group \\ti G by the dressing action of the group G. T-duality maps the momentum of the open string into the mutual distance of the D-branes in the pair. The whole picture is then extended to the full modular space M(D) of the Poisson-Lie equivalent \\si-models which is the space of all Manin triples of a given Drinfeld double.T-duality rotates the zero modes of pairs of D-branes living on targets belonging to M(D). In this more general case the D-branes are preimages of symplectic leaves in certain Poisson homogeneous spaces of their targets and, as such, they are either all even or all odd dimensional.
Purposes and Effects of Lying.
Hample, Dale
Three exploratory studies were aimed at describing the purposes of lies and the consequences of lying. Data were collected through a partly open-ended questionnaire, a content analysis of several tape-recorded interviews, and a large-scale survey. The results showed that two of every three lies were told for selfish reasons, while three of every…
Group field theory and simplicial quantum gravity
International Nuclear Information System (INIS)
Oriti, D
2010-01-01
We present a new group field theory for 4D quantum gravity. It incorporates the constraints that give gravity from BF theory and has quantum amplitudes with the explicit form of simplicial path integrals for first-order gravity. The geometric interpretation of the variables and of the contributions to the quantum amplitudes is manifest. This allows a direct link with other simplicial gravity approaches, like quantum Regge calculus, in the form of the amplitudes of the model, and dynamical triangulations, which we show to correspond to a simple restriction of the same.
Teaching the Truth about Lies to Psychology Students: The Speed Lying Task
Pearson, Matthew R.; Richardson, Thomas A.
2013-01-01
To teach the importance of deception in everyday social life, an in-class activity called the "Speed Lying Task" was given in an introductory social psychology class. In class, two major research findings were replicated: Individuals detected deception at levels no better than expected by chance and lie detection confidence was unrelated…
Reference group theory with implications for information studies: a theoretical essay
Directory of Open Access Journals (Sweden)
E. Murell Dawson
2001-01-01
Full Text Available This article explores the role and implications of reference group theory in relation to the field of library and information science. Reference group theory is based upon the principle that people take the standards of significant others as a basis for making self-appraisals, comparisons, and choices regarding need and use of information. Research that applies concepts of reference group theory to various sectors of library and information studies can provide data useful in enhancing areas such as information-seeking research, special populations, and uses of information. Implications are promising that knowledge gained from like research can be beneficial in helping information professionals better understand the role theory plays in examining ways in which people manage their information and social worlds.
The low-lying quartet electronic states of group 14 diatomic borides XB (X = C, Si, Ge, Sn, Pb)
Pontes, Marcelo A. P.; de Oliveira, Marcos H.; Fernandes, Gabriel F. S.; Da Motta Neto, Joaquim D.; Ferrão, Luiz F. A.; Machado, Francisco B. C.
2018-04-01
The present work focuses in the characterization of the low-lying quartet electronic and spin-orbit states of diatomic borides XB, in which X is an element of group 14 (C, Si, Ge, Sn, PB). The wavefunction was obtained at the CASSCF/MRCI level with a quintuple-ζ quality basis set. Scalar relativistic effects were also taken into account. A systematic and comparative analysis of the spectroscopic properties for the title molecular series was carried out, showing that the (1)4Π→X4Σ- transition band is expected to be measurable by emission spectroscopy to the GeB, SnB and PbB molecules, as already observed for the lighter CB and SiB species.
Paldus, J.; Li, X.
1992-10-01
Following a brief outline of various developments and exploitations of the unitary group approach (UGA), and its extension referred to as Clifford algebra UGA (CAUGA), in molecular electronic structure calculations, we present a summary of a recently introduced implementation of CAUGA for the valence bond (VB) method based on the Pariser-Parr-Pople (PPP)-type Hamiltonian. The existing applications of this PPP-VB approach have been limited to groundstates of various π-electron systems or, at any rate, to the lowest states of a given multiplicity. In this paper the method is applied to the low-lying excited states of several archetypal models, namely cyclobutadiene and benzene, representing antiaromatic and aromatic systems, hexatriene, representing linear polyenic systems and, finally, naphthalene, representing polyacenes.
Bakhurst, D
1992-06-01
This article challenges Jennifer Jackson's recent defence of doctors' rights to deceive patients. Jackson maintains there is a general moral difference between lying and intentional deception: while doctors have a prima facie duty not to lie, there is no such obligation to avoid deception. This paper argues 1) that an examination of cases shows that lying and deception are often morally equivalent, and 2) that Jackson's position is premised on a species of moral functionalism that misconstrues the nature of moral obligation. Against Jackson, it is argued that both lying and intentional deception are wrong where they infringe a patient's right to autonomy or his/her right to be treated with dignity. These rights represent 'deontological constraints' on action, defining what we must not do whatever the functional value of the consequences. Medical ethics must recognise such constraints if it is to contribute to the moral integrity of medical practice.
Energy Technology Data Exchange (ETDEWEB)
Heilmann, D.B.
2007-02-15
The two-plane HUBBARD model, which is a model for some electronic properties of undoped YBCO superconductors as well as displays a MOTT metal-to-insulator transition and a metal-to-band insulator transition, is studied within Dynamical Mean-Field Theory using HIRSCH-FYE Monte Carlo. In order to find the different transitions and distinguish the types of insulator, we calculate the single-particle spectral densities, the self-energies and the optical conductivities. We conclude that there is a continuous transition from MOTT to band insulator. In the second part, ground state properties of a diagonally disordered HUBBARD model is studied using a generalisation of Path Integral Renormalisation Group, a variational method which can also determine low-lying excitations. In particular, the distribution of antiferromagnetic properties is investigated. We conclude that antiferromagnetism breaks down in a percolation-type transition at a critical disorder, which is not changed appreciably by the inclusion of correlation effects, when compared to earlier studies. Electronic and excitation properties at the system sizes considered turn out to primarily depend on the geometry. (orig.)
International Nuclear Information System (INIS)
Heilmann, D.B.
2007-02-01
The two-plane HUBBARD model, which is a model for some electronic properties of undoped YBCO superconductors as well as displays a MOTT metal-to-insulator transition and a metal-to-band insulator transition, is studied within Dynamical Mean-Field Theory using HIRSCH-FYE Monte Carlo. In order to find the different transitions and distinguish the types of insulator, we calculate the single-particle spectral densities, the self-energies and the optical conductivities. We conclude that there is a continuous transition from MOTT to band insulator. In the second part, ground state properties of a diagonally disordered HUBBARD model is studied using a generalisation of Path Integral Renormalisation Group, a variational method which can also determine low-lying excitations. In particular, the distribution of antiferromagnetic properties is investigated. We conclude that antiferromagnetism breaks down in a percolation-type transition at a critical disorder, which is not changed appreciably by the inclusion of correlation effects, when compared to earlier studies. Electronic and excitation properties at the system sizes considered turn out to primarily depend on the geometry. (orig.)
System theory as applied differential geometry. [linear system
Hermann, R.
1979-01-01
The invariants of input-output systems under the action of the feedback group was examined. The approach used the theory of Lie groups and concepts of modern differential geometry, and illustrated how the latter provides a basis for the discussion of the analytic structure of systems. Finite dimensional linear systems in a single independent variable are considered. Lessons of more general situations (e.g., distributed parameter and multidimensional systems) which are increasingly encountered as technology advances are presented.
Renormalization group evolution of the universal theories EFT
International Nuclear Information System (INIS)
Wells, James D.; Zhang, Zhengkang
2016-01-01
The conventional oblique parameters analyses of precision electroweak data can be consistently cast in the modern framework of the Standard Model effective field theory (SMEFT) when restrictions are imposed on the SMEFT parameter space so that it describes universal theories. However, the usefulness of such analyses is challenged by the fact that universal theories at the scale of new physics, where they are matched onto the SMEFT, can flow to nonuniversal theories with renormalization group (RG) evolution down to the electroweak scale, where precision observables are measured. The departure from universal theories at the electroweak scale is not arbitrary, but dictated by the universal parameters at the matching scale. But to define oblique parameters, and more generally universal parameters at the electroweak scale that directly map onto observables, additional prescriptions are needed for the treatment of RG-induced nonuniversal effects. We perform a RG analysis of the SMEFT description of universal theories, and discuss the impact of RG on simplified, universal-theories-motivated approaches to fitting precision electroweak and Higgs data.
A non-Lie algebraic framework and its possible merits for symmetry descriptions
International Nuclear Information System (INIS)
Ktorides, C.N.
1975-01-01
A nonassociative algebraic construction is introduced which bears a relation to a Lie algebra L paralleling the relation between an associative enveloping algebra and L. The key ingredient of this algebraic construction is the presence of two parameters which relate it to the enveloping algebra of L. The analog of the Poincare--Birkhoff--Witt theorem is proved for the new algebra. Possibilities of physical relevance are also considered. It is noted that, if fully developed, the mathematical framework suggested by this new algebra should be non-Lie. Subsequently, a certain scheme resulting from specific considerations connected with this (non-Lie) algebraic structure is found to bear striking resemblance to a recent phenomenological theory proposed for explaining CP violation by the K 0 system. Some relevant speculations are also made in view of certain recent trends of thought in elementary particle physics. Finally, in an appendix, a Gell-Mann--Okubo-like mass formula for the new algebra is derived for an SU (3) octet
Krichever-Novikov type algebras theory and applications
Schlichenmaier, Martin
2014-01-01
Krichever and Novikov introduced certain classes of infinite dimensionalLie algebrasto extend the Virasoro algebra and its related algebras to Riemann surfaces of higher genus. The author of this book generalized and extended them toa more general setting needed by the applications. Examples of applications are Conformal Field Theory, Wess-Zumino-Novikov-Witten models, moduli space problems, integrable systems, Lax operator algebras, and deformation theory of Lie algebra. Furthermore they constitute an important class of infinite dimensional Lie algebras which due to their geometric origin are
Verschuere, B.; Spruyt, A.; Meijer, E.H.; Otgaar, H.
2011-01-01
Brain imaging studies suggest that truth telling constitutes the default of the human brain and that lying involves intentional suppression of the predominant truth response. By manipulating the truth proportion in the Sheffield lie test, we investigated whether the dominance of the truth response
Quantum group and quantum symmetry
International Nuclear Information System (INIS)
Chang Zhe.
1994-05-01
This is a self-contained review on the theory of quantum group and its applications to modern physics. A brief introduction is given to the Yang-Baxter equation in integrable quantum field theory and lattice statistical physics. The quantum group is primarily introduced as a systematic method for solving the Yang-Baxter equation. Quantum group theory is presented within the framework of quantum double through quantizing Lie bi-algebra. Both the highest weight and the cyclic representations are investigated for the quantum group and emphasis is laid on the new features of representations for q being a root of unity. Quantum symmetries are explored in selected topics of modern physics. For a Hamiltonian system the quantum symmetry is an enlarged symmetry that maintains invariance of equations of motion and allows a deformation of the Hamiltonian and symplectic form. The configuration space of the integrable lattice model is analyzed in terms of the representation theory of quantum group. By means of constructing the Young operators of quantum group, the Schroedinger equation of the model is transformed to be a set of coupled linear equations that can be solved by the standard method. Quantum symmetry of the minimal model and the WZNW model in conformal field theory is a hidden symmetry expressed in terms of screened vertex operators, and has a deep interplay with the Virasoro algebra. In quantum group approach a complete description for vibrating and rotating diatomic molecules is given. The exact selection rules and wave functions are obtained. The Taylor expansion of the analytic formulas of the approach reproduces the famous Dunham expansion. (author). 133 refs, 20 figs
The geometry and physics of Abelian gauge groups in F-theory
Energy Technology Data Exchange (ETDEWEB)
Keitel, Jan
2015-07-14
In this thesis we study the geometry and the low-energy effective physics associated with Abelian gauge groups in F-theory compactifications. To construct suitable torus-fibered Calabi-Yau manifolds, we employ the framework of toric geometry. By identifying appropriate building blocks of Calabi-Yau manifolds that can be studied independently, we devise a method to engineer large numbers of manifolds that give rise to a specified gauge group and achieve a partial classification of toric gauge groups. Extending our analysis from gauge groups to matter spectra, we prove that the matter content of the most commonly studied F-theory set-ups is rather constrained. To circumvent such limitations, we introduce an algorithm to analyze torus-fibrations defined as complete intersections and present several novel kinds of F-theory compactifications. Finally, we show how torus-fibrations without section are linked to fibrations with multiple sections through a network of successive geometric transitions. In order to investigate the low-energy effective physics resulting from our compactifications, we apply M- to F-theory duality. After determining the effective action of F-theory with Abelian gauge groups in six dimensions, we compare the loop-corrected Chern-Simons terms to topological quantities of the compactification manifold to read off the massless matter content. Under certain assumptions, we show that all gravitational and mixed anomalies are automatically canceled in F-theory. Furthermore, we compute the low-energy effective action of F-theory compactifications without section and suggest that the absence of a section signals the presence of an additional massive Abelian gauge field. Adjusting our analysis to four dimensions, we show that remnants of this massive gauge field survive as discrete symmetries that impose selection rules on the Yukawa couplings of the effective theory.
The geometry and physics of Abelian gauge groups in F-theory
International Nuclear Information System (INIS)
Keitel, Jan
2015-01-01
In this thesis we study the geometry and the low-energy effective physics associated with Abelian gauge groups in F-theory compactifications. To construct suitable torus-fibered Calabi-Yau manifolds, we employ the framework of toric geometry. By identifying appropriate building blocks of Calabi-Yau manifolds that can be studied independently, we devise a method to engineer large numbers of manifolds that give rise to a specified gauge group and achieve a partial classification of toric gauge groups. Extending our analysis from gauge groups to matter spectra, we prove that the matter content of the most commonly studied F-theory set-ups is rather constrained. To circumvent such limitations, we introduce an algorithm to analyze torus-fibrations defined as complete intersections and present several novel kinds of F-theory compactifications. Finally, we show how torus-fibrations without section are linked to fibrations with multiple sections through a network of successive geometric transitions. In order to investigate the low-energy effective physics resulting from our compactifications, we apply M- to F-theory duality. After determining the effective action of F-theory with Abelian gauge groups in six dimensions, we compare the loop-corrected Chern-Simons terms to topological quantities of the compactification manifold to read off the massless matter content. Under certain assumptions, we show that all gravitational and mixed anomalies are automatically canceled in F-theory. Furthermore, we compute the low-energy effective action of F-theory compactifications without section and suggest that the absence of a section signals the presence of an additional massive Abelian gauge field. Adjusting our analysis to four dimensions, we show that remnants of this massive gauge field survive as discrete symmetries that impose selection rules on the Yukawa couplings of the effective theory.
Freestall maintenance: effects on lying behavior of dairy cattle.
Drissler, M; Gaworski, M; Tucker, C B; Weary, D M
2005-07-01
In a series of 3 experiments, we documented how sand-bedding depth and distribution changed within freestalls after new bedding was added and the effect of these changes on lying behavior. In experiment 1, we measured changes in bedding depth over a 10-d period at 43 points in 24 freestalls. Change in depth of sand was the greatest the day after new sand was added and decreased over time. Over time, the stall surface became concave, and the deepest part of the stall was at the center. Based on the results of experiment 1, we measured changes in lying behavior when groups of cows had access to freestalls with sand bedding that was 0, 3.5, 5.2, or 6.2 cm at the deepest point, below the curb, while other dimensions remained fixed. We found that daily lying time was 1.15 h shorter in stalls with the lowest levels of bedding compared with stalls filled with bedding. Indeed, for every 1-cm decrease in bedding, cows spent 11 min less time lying down during each 24-h period. In a third experiment, we imposed 4 treatments that reflected the variation in sand depth within stalls: 0, 6.2, 9.9, and 13.7 cm below the curb. Again, lying times reduced with decreasing bedding, such that cows using the stalls with the least amount of bedding (13.7 cm below curb) spent 2.33 h less time per day lying down than when housed with access to freestalls filled with sand (0 cm below curb).
Iachello, Francesco
2015-01-01
This course-based primer provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, it concisely presents the basic concepts of Lie algebras, their representations and their invariants. The second part includes a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book highlights a number of examples that help to illustrate the abstract algebraic definitions and includes a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators...
Bakhurst, D
1992-01-01
This article challenges Jennifer Jackson's recent defence of doctors' rights to deceive patients. Jackson maintains there is a general moral difference between lying and intentional deception: while doctors have a prima facie duty not to lie, there is no such obligation to avoid deception. This paper argues 1) that an examination of cases shows that lying and deception are often morally equivalent, and 2) that Jackson's position is premised on a species of moral functionalism that misconstrues the nature of moral obligation. Against Jackson, it is argued that both lying and intentional deception are wrong where they infringe a patient's right to autonomy or his/her right to be treated with dignity. These rights represent 'deontological constraints' on action, defining what we must not do whatever the functional value of the consequences. Medical ethics must recognise such constraints if it is to contribute to the moral integrity of medical practice. PMID:1619626
Lie and conditional symmetries of the three-component diffusive Lotka–Volterra system
International Nuclear Information System (INIS)
Cherniha, Roman; Davydovych, Vasyl’
2013-01-01
Lie and Q-conditional symmetries of the classical three-component diffusive Lotka–Volterra system in the case of one space variable are studied. The group-classification problems for finding Lie symmetries and Q-conditional symmetries of the first type are completely solved. Notably, non-Lie symmetries (Q-conditional symmetry operators) for a multi-component nonlinear reaction–diffusion system are constructed for the first time. The results are compared with those derived for the two-component diffusive Lotka–Volterra system. The conditional symmetry obtained for the non-Lie reduction of the three-component system used for modeling competition between three species in population dynamics is applied and the relevant exact solutions are found. Particularly, the exact solution describing different scenarios of competition between three species is constructed. (paper)
Equivariant K-theory, groupoids and proper actions
Cantarero, Jose
2008-01-01
In this paper we define complex equivariant K-theory for actions of Lie groupoids using finite-dimensional vector bundles. For a Bredon-compatible Lie groupoid, this defines a periodic cohomology theory on the category of finite equivariant CW-complexes. We also establish an analogue of the completion theorem of Atiyah and Segal. Some examples are discussed.
Knots, topology and quantum field theories
International Nuclear Information System (INIS)
Lusanna, L.
1989-01-01
The title of the workshop, Knots, Topology and Quantum Field Theory, accurate reflected the topics discussed. There have been important developments in mathematical and quantum field theory in the past few years, which had a large impact on physicist thinking. It is historically unusual and pleasing that these developments are taking place as a result of an intense interaction between mathematical physicists and mathematician. On the one hand, topological concepts and methods are playing an increasingly important lead to novel mathematical concepts: for instance, the study of quantum groups open a new chapter in the deformation theory of Lie algebras. These developments at present will lead to new insights into the theory of elementary particles and their interactions. In essence, the talks dealt with three, broadly defined areas of theoretical physics. One was topological quantum field theories, the other the problem of quantum groups and the third one certain aspects of more traditional field theories, such as, for instance, quantum gravity. These topics, however, are interrelated and the general theme of the workshop defies rigid classification; this was evident from the cross references to be found in almo all the talks
Noether and Lie symmetries for charged perfect fluids
International Nuclear Information System (INIS)
Kweyama, M C; Govinder, K S; Maharaj, S D
2011-01-01
We study the underlying nonlinear partial differential equation that governs the behaviour of spherically symmetric charged fluids in general relativity. We investigate the conditions for the equation to admit a first integral or be reduced to quadratures using symmetry methods for differential equations. A general Noether first integral is found. We also undertake a comprehensive group analysis of the underlying equation using Lie point symmetries. The existence of a Lie symmetry is subject to solving an integro-differential equation in general; we investigate the conditions under which it can be reduced to quadratures. Earlier results for uncharged fluids and particular first integrals for charged matter are regained as special cases of our treatment.
Kern, R S; Green, M F; Fiske, A P; Kee, K S; Lee, J; Sergi, M J; Horan, W P; Subotnik, K L; Sugar, C A; Nuechterlein, K H
2009-04-01
Interpersonal communication problems are common among persons with schizophrenia and may be linked, in part, to deficits in theory of mind (ToM), the ability to accurately perceive the attitudes, beliefs and intentions of others. Particular difficulties might be expected in the processing of counterfactual information such as sarcasm or lies. The present study included 50 schizophrenia or schizo-affective out-patients and 44 demographically comparable healthy adults who were administered Part III of The Awareness of Social Inference Test (TASIT; a measure assessing comprehension of sarcasm versus lies) as well as measures of positive and negative symptoms and community functioning. TASIT data were analyzed using a 2 (group: patients versus healthy adults) x 2 (condition: sarcasm versus lie) repeated-measures ANOVA. The results show significant effects for group, condition, and the group x condition interaction. Compared to controls, patients performed significantly worse on sarcasm but not lie scenes. Within-group contrasts showed that patients performed significantly worse on sarcasm versus lie scenes; controls performed comparably on both. In patients, performance on TASIT showed a significant correlation with positive, but not negative, symptoms. The group and interaction effects remained significant when rerun with a subset of patients with low-level positive symptoms. The findings for a relationship between TASIT performance and community functioning were essentially negative. The findings replicate a prior demonstration of difficulty in the comprehension of sarcasm using a different test, but are not consistent with previous studies showing global ToM deficits in schizophrenia.
Lie algebra of conformal Killing–Yano forms
International Nuclear Information System (INIS)
Ertem, Ümit
2016-01-01
We provide a generalization of the Lie algebra of conformal Killing vector fields to conformal Killing–Yano forms. A new Lie bracket for conformal Killing–Yano forms that corresponds to slightly modified Schouten–Nijenhuis bracket of differential forms is proposed. We show that conformal Killing–Yano forms satisfy a graded Lie algebra in constant curvature manifolds. It is also proven that normal conformal Killing–Yano forms in Einstein manifolds also satisfy a graded Lie algebra. The constructed graded Lie algebras reduce to the graded Lie algebra of Killing–Yano forms and the Lie algebras of conformal Killing and Killing vector fields in special cases. (paper)
Non-rigid molecular group theory and its applications
International Nuclear Information System (INIS)
Balasubramanian, K.
1982-06-01
The use of generalized wreath product groups as representations of symmetry groups of nonrigid molecules is considered. Generating function techniques are outlined for nuclear spin statistics and character tables of the symmetry groups of nonrigid molecules. Several applications of nonrigid molecular group theory to NMR spectroscopy, rovibronic splitting and nuclear spin statistics of nonrigid molecules, molecular beam deflection and electric resonance experiments of weakly bound Van der Waal complexes, isomerization processes, configuration interaction calculations and the symmetry of crystals with structural distortions are described. 81 references
DSR Theories, Conformal Group and Generalized Commutation Relation
International Nuclear Information System (INIS)
Leiva, Carlos
2006-01-01
In this paper the relationship of DSR theories and Conformal Group is reviewed. On the other hand, the relation between DSR Magueijo Smolin generators and generalized commutation relations is also shown
On the standard model group in F-theory
International Nuclear Information System (INIS)
Choi, Kang-Sin
2014-01-01
We analyze the standard model gauge group SU(3) x SU(2) x U(1) constructed in F-theory. The non-Abelian part SU(3) x SU(2) is described by a surface singularity of Kodaira type. Blow-up analysis shows that the non-Abelian part is distinguished from the naive product of SU(3) and SU(2), but that it should be a rank three group along the chain of E n groups, because it has non-generic gauge symmetry enhancement structure responsible for desirablematter curves. The Abelian part U(1) is constructed from a globally valid two-form with the desired gauge quantum numbers, using a similar method to the decomposition (factorization) method of the spectral cover. This technique makes use of an extra section in the elliptic fiber of the Calabi-Yau manifold, on which F-theory is compactified. Conventional gauge coupling unification of SU(5) is achieved, without requiring a threshold correction from the flux along the hypercharge direction. (orig.)
Quantum group gauge theory on quantum spaces
International Nuclear Information System (INIS)
Brzezinski, T.; Majid, S.
1993-01-01
We construct quantum group-valued canonical connections on quantum homogeneous spaces, including a q-deformed Dirac monopole on the quantum sphere of Podles quantum differential coming from the 3-D calculus of Woronowicz on SU q (2). The construction is presented within the setting of a general theory of quantum principal bundles with quantum group (Hopf algebra) fiber, associated quantum vector bundles and connection one-forms. Both the base space (spacetime) and the total space are non-commutative algebras (quantum spaces). (orig.)
Fractional supersymmetry and infinite dimensional lie algebras
International Nuclear Information System (INIS)
Rausch de Traubenberg, M.
2001-01-01
In an earlier work extensions of supersymmetry and super Lie algebras were constructed consistently starting from any representation D of any Lie algebra g. Here it is shown how infinite dimensional Lie algebras appear naturally within the framework of fractional supersymmetry. Using a differential realization of g this infinite dimensional Lie algebra, containing the Lie algebra g as a sub-algebra, is explicitly constructed
Generally covariant gauge theories
International Nuclear Information System (INIS)
Capovilla, R.
1992-01-01
A new class of generally covariant gauge theories in four space-time dimensions is investigated. The field variables are taken to be a Lie algebra valued connection 1-form and a scalar density. Modulo an important degeneracy, complex [euclidean] vacuum general relativity corresponds to a special case in this class. A canonical analysis of the generally covariant gauge theories with the same gauge group as general relativity shows that they describe two degrees of freedom per space point, qualifying therefore as a new set of neighbors of general relativity. The modification of the algebra of the constraints with respect to the general relativity case is computed; this is used in addressing the question of how general relativity stands out from its neighbors. (orig.)
New topological theories and conjugacy classes of the Weyl group
International Nuclear Information System (INIS)
Hollowood, T.J.; Miramontes, J.L.
1993-01-01
The problem of interpreting a set of W-algebra constraints constructed in terms of an arbitrarily twisted scalar field as recursion relations of some topological theory is addressed. In this picture, the models of topological gravity coupled to A, D or E topological matter, correspond to taking the scalar field twisted by the Coxeter element of the Weyl group. It turns out that not all conjugacy classes of the Weyl group lead to models which allow for such an interpretation. For example, it is shown that for the A algebras there are two possible choices for the conjugacy class, giving a new set of theories in addition to the conventional ones. Furthermore, it is shown how the new series of theories contains the conventional series as a subsector. A tentative interpretation of this new series in terms of intersection theory is presented. (orig.)
Integral dimension in the string theory based on a group manifold
International Nuclear Information System (INIS)
Toyota, N.
1990-01-01
We study string models on a group manifold with Kac-Moody symmetry where the critical dimension d is integer. In particular the possibility of four-dimensional models is investigated. We find that only nine group manifolds with a relevant k level can have four as the critical dimension among an infinite number of compact Lie groups. They are all listed. The models with minimal conformal sectors adding to the Kac-Moody sector are investigated. In the cases with minimal conformal sector, there are only two groups, SU(5) and SO(43), that can give d=4. Among the cases with some tensoring products of minimal conformal sectors we discuss a few special cases with k=0 and k=1. The cases based on N=1 super Kac-Moody algebra are also studied. Finally we discuss the possibility of the enlargement of gauge symmetry. (orig.)
Fu, Genyue; Xu, Fen; Cameron, Catherine Ann; Leyman, Gail; Lee, Kang
2007-01-01
This study examined cross-cultural differences and similarities in children's moral understanding of individual- or collective-oriented lies and truths. Seven-, 9-, and 11-year-old Canadian and Chinese children were read stories about story characters facing moral dilemmas about whether to lie or tell the truth to help a group but harm an…
Quantum stochastic calculus and representations of Lie superalgebras
Eyre, Timothy M W
1998-01-01
This book describes the representations of Lie superalgebras that are yielded by a graded version of Hudson-Parthasarathy quantum stochastic calculus. Quantum stochastic calculus and grading theory are given concise introductions, extending readership to mathematicians and physicists with a basic knowledge of algebra and infinite-dimensional Hilbert spaces. The develpment of an explicit formula for the chaotic expansion of a polynomial of quantum stochastic integrals is particularly interesting. The book aims to provide a self-contained exposition of what is known about Z_2-graded quantum stochastic calculus and to provide a framework for future research into this new and fertile area.
Moduli space of self-dual connections in dimension greater than four for abelian Gauge groups
Cappelle, Natacha
2018-01-01
In 1954, C. Yang and R. Mills created a Gauge Theory for strong interaction of Elementary Particles. More generally, they proved that it is possible to define a Gauge Theory with an arbitrary compact Lie group as Gauge group. Within this context, it is interesting to find critical values of a functional defined on the space of connections: the Yang-Mills functional. If the based manifold is four dimensional, there exists a natural notion of (anti-)self-dual 2-form, which gives a natural notio...
Invariants of triangular Lie algebras
International Nuclear Information System (INIS)
Boyko, Vyacheslav; Patera, Jiri; Popovych, Roman
2007-01-01
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of Boyko et al (2006 J. Phys. A: Math. Gen.39 5749 (Preprint math-ph/0602046)), developed further in Boyko et al (2007 J. Phys. A: Math. Theor.40 113 (Preprint math-ph/0606045)), is used to determine the invariants. A conjecture of Tremblay and Winternitz (2001 J. Phys. A: Math. Gen.34 9085), concerning the number of independent invariants and their form, is corroborated
International Nuclear Information System (INIS)
Abad, J.; Esteve, J.G.; Pacheco, A.F.
1985-01-01
An approximation technique to construct the low-lying energy eigenstates of any bosonic field theory on the lattice is proposed. It is based on the SLAC blocking method, after performing a finite-spin approximation to the individual degrees of freedom of the problem. General expressions for any polynomial self-interacting theory are given. Numerical results for phi 2 and phi 4 theories in 1+1 dimensions are offered; they exhibit a fast convergence rate. The complete low-lying energy spectrum of the phi 4 theory in 1+1 dimensions is calculated
International Nuclear Information System (INIS)
Hohm, Olaf; Zwiebach, Barton
2017-01-01
We review and develop the general properties of L_∞ algebras focusing on the gauge structure of the associated field theories. Motivated by the L_∞ homotopy Lie algebra of closed string field theory and the work of Roytenberg and Weinstein describing the Courant bracket in this language we investigate the L_∞ structure of general gauge invariant perturbative field theories. We sketch such formulations for non-abelian gauge theories, Einstein gravity, and for double field theory. We find that there is an L_∞ algebra for the gauge structure and a larger one for the full interacting field theory. Theories where the gauge structure is a strict Lie algebra often require the full L_∞ algebra for the interacting theory. The analysis suggests that L_∞ algebras provide a classification of perturbative gauge invariant classical field theories. (copyright 2017 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Scaling algebras and renormalization group in algebraic quantum field theory
International Nuclear Information System (INIS)
Buchholz, D.; Verch, R.
1995-01-01
For any given algebra of local observables in Minkowski space an associated scaling algebra is constructed on which renormalization group (scaling) transformations act in a canonical manner. The method can be carried over to arbitrary spacetime manifolds and provides a framework for the systematic analysis of the short distance properties of local quantum field theories. It is shown that every theory has a (possibly non-unique) scaling limit which can be classified according to its classical or quantum nature. Dilation invariant theories are stable under the action of the renormalization group. Within this framework the problem of wedge (Bisognano-Wichmann) duality in the scaling limit is discussed and some of its physical implications are outlined. (orig.)
Non commutative geometry methods for group C*-algebras
International Nuclear Information System (INIS)
Do Ngoc Diep.
1996-09-01
This book is intended to provide a quick introduction to the subject. The exposition is scheduled in the sequence, as possible for more understanding for beginners. The author exposed a K-theoretic approach to study group C * -algebras: started in the elementary part, with one example of description of the structure of C * -algebra of the group of affine transformations of the real straight line, continued then for some special classes of solvable and nilpotent Lie groups. In the second advanced part, he introduced the main tools of the theory. In particular, the conception of multidimensional geometric quantization and the index of group C * -algebras were created and developed. (author). Refs
Space-time versus world-sheet renormalization group equation in string theory
International Nuclear Information System (INIS)
Brustein, R.; Roland, K.
1991-05-01
We discuss the relation between space-time renormalization group equation for closed string field theory and world-sheet renormalization group equation for first-quantized strings. Restricting our attention to massless states we argue that there is a one-to-one correspondence between the fixed point solutions of the two renormalization group equations. In particular, we show how to extract the Fischler-Susskind mechanism from the string field theory equation in the case of the bosonic string. (orig.)
BRST-operator for quantum Lie algebra and differential calculus on quantum groups
International Nuclear Information System (INIS)
Isaev, A.P.; Ogievetskij, O.V.
2001-01-01
For A Hopf algebra one determined structure of differential complex in two dual external Hopf algebras: A external expansion and in A* dual algebra external expansion. The Heisenberg double of these two Hopf algebras governs the differential algebra for the Cartan differential calculus on A algebra. The forst differential complex is the analog of the de Rame complex. The second complex coincide with the standard complex. Differential is realized as (anti)commutator with Q BRST-operator. Paper contains recursion relation that determines unequivocally Q operator. For U q (gl(N)) Lie quantum algebra one constructed BRST- and anti-BRST-operators and formulated the theorem of the Hodge expansion [ru