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...
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.
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
The relation between quantum W algebras and Lie algebras
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Boer, J. de; Tjin, T.
1994-01-01
By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrary sl 2 embeddings we show that a large set W of quantum W algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set W contains many known W algebras such as W N and W 3 (2) . Our formalism yields a completely algorithmic method for calculating the W algebra generators and their operator product expansions, replacing the cumbersome construction of W algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that any W algebra in W can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Therefore any realization of this semisimple affine Lie algebra leads to a realization of the W algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolutions for all algebras in W. Some examples are explicitly worked out. (orig.)
Nonflexible Lie-admissible algebras
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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
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
Recoupling Lie algebra and universal ω-algebra
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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
Ternary q-Virasoro-Witt Hom-Nambu-Lie algebras
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Ammar, F; Makhlouf, A; Silvestrov, S
2010-01-01
In this paper we construct ternary q-Virasoro-Witt algebras which q-deform the ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos using su(1, 1) enveloping algebra techniques. The ternary Virasoro-Witt algebras constructed by Curtright, Fairlie and Zachos depend on a parameter and are not Nambu-Lie algebras for all but finitely many values of this parameter. For the parameter values for which the ternary Virasoro-Witt algebras are Nambu-Lie, the corresponding ternary q-Virasoro-Witt algebras constructed in this paper are also Hom-Nambu-Lie because they are obtained from the ternary Nambu-Lie algebras using the composition method. For other parameter values this composition method does not yield a Hom-Nambu-Lie algebra structure for q-Virasoro-Witt algebras. We show however, using a different construction, that the ternary Virasoro-Witt algebras of Curtright, Fairlie and Zachos, as well as the general ternary q-Virasoro-Witt algebras we construct, carry a structure of the ternary Hom-Nambu-Lie algebra for all values of the involved parameters.
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.
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.
Differential Hopf algebra structures on the universal enveloping algebra ofa Lie algebra
N.W. van den Hijligenberg; R. Martini
1995-01-01
textabstractWe discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra
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...
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.
Continuum analogues of contragredient Lie algebras
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Saveliev, M.V.; Vershik, A.M.
1989-03-01
We present an axiomatic formulation of a new class of infinite-dimensional Lie algebras - the generalizations of Z-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras ''continuum Lie algebras''. The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential Cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered. (author). 9 refs
Computations in finite-dimensional Lie algebras
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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.
From Rota-Baxter algebras to pre-Lie algebras
International Nuclear Information System (INIS)
An Huihui; Ba, Chengming
2008-01-01
Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics. Rota-Baxter algebras provide a construction of pre-Lie algebras from associative algebras. In this paper, we give all Rota-Baxter operators of weight 1 on complex associative algebras in dimension ≤3 and their corresponding pre-Lie algebras
Automorphic Lie algebras with dihedral symmetry
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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)
Differential Hopf algebra structures on the universal enveloping algebra of a Lie algebra
van den Hijligenberg, N.W.; van den Hijligenberg, N.W.; Martini, Ruud
1995-01-01
We discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of
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...
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...
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
Differential Hopf algebra structures on the Universal Enveloping Algebra of a Lie Algebra
van den Hijligenberg, N.W.; van den Hijligenberg, N.; Martini, Ruud
1995-01-01
We discuss a method to construct a De Rham complex (differential algebra) of Poincaré–Birkhoff–Witt type on the universal enveloping algebra of a Lie algebra g. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebrastructure of U(g).
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
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.
Classification of simple flexible Lie-admissible algebras
International Nuclear Information System (INIS)
Okubo, S.; Myung, H.C.
1979-01-01
Let A be a finite-dimensional flexible Lie-admissible algebra over the complex field such that A - is a simple Lie algebra. It is shown that either A is itself a Lie algebra isomorphic to A - or A - is a Lie algebra of type A/sub n/ (n greater than or equal to 2). In the latter case, A is isomorphic to the algebra defined on the space of (n + 1) x (n + 1) traceless matrices with multiplication given by x * y = μxy + (1 - μ)yx - (1/(n + 100 Tr (xy) E where μ is a fixed scalar, xy denotes the matrix operators in Lie algebras which has been studied in theoretical physics. We also discuss a broader class of Lie algebras over arbitrary field of characteristic not equal to 2, called quasi-classical, which includes semisimple as well as reductive Lie algebras. For this class of Lie algebras, we can introduce a multiplication which makes the adjoint operator space into an associative algebra. When L is a Lie algebra with nondegenerate killing form, it is shown that the adjoint operator algebra of L in the adjoint representation becomes a commutative associative algebra with unit element and its dimension is 1 or 2 if L is simple over the complex field. This is related to the known result that a Lie algebra of type A/sub n/ (n greater than or equal to 2) alone has a nonzero completely symmetric adjoint operator in the adjoint representation while all other algebras have none. Finally, Lie-admissible algebras associated with bilinear form are investigated
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
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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
Vertex ring-indexed Lie algebras
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Fairlie, David; Zachos, Cosmas
2005-01-01
Infinite-dimensional Lie algebras are introduced, which are only partially graded, and are specified by indices lying on cyclotomic rings. They may be thought of as generalizations of the Onsager algebra, but unlike it, or its sl(n) generalizations, they are not subalgebras of the loop algebras associated with sl(n). In a particular interesting case associated with sl(3), their indices lie on the Eisenstein integer triangular lattice, and these algebras are expected to underlie vertex operator combinations in CFT, brane physics, and graphite monolayers
Lie Quasi-Bialgebras and Cohomology of Lie algebra
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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
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)
Introduction to vertex algebras, Borcherds algebras and the Monster Lie algebras
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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.)
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.
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)
On Deformations and Contractions of Lie Algebras
Directory of Open Access Journals (Sweden)
Marc de Montigny
2006-05-01
Full Text Available In this contributed presentation, we discuss and compare the mutually opposite procedures of deformations and contractions of Lie algebras. We suggest that with appropriate combinations of both procedures one may construct new Lie algebras. We first discuss low-dimensional Lie algebras and illustrate thereby that whereas for every contraction there exists a reverse deformation, the converse is not true in general. Also we note that some Lie algebras belonging to parameterized families are singled out by the irreversibility of deformations and contractions. After reminding that global deformations of the Witt, Virasoro, and affine Kac-Moody algebras allow one to retrieve Lie algebras of Krichever-Novikov type, we contract the latter to find new infinite dimensional Lie algebras.
The Adapted Ordering Method for Lie algebras and superalgebras and their generalizations
Energy Technology Data Exchange (ETDEWEB)
Gato-Rivera, Beatriz [Instituto de Matematicas y Fisica Fundamental, CSIC, Serrano 123, Madrid 28006 (Spain); NIKHEF-H, Kruislaan 409, NL-1098 SJ Amsterdam (Netherlands)
2008-02-01
In 1998 the Adapted Ordering Method was developed for the representation theory of the superconformal algebras in two dimensions. It allows us to determine maximal dimensions for a given type of space of singular vectors, to identify all singular vectors by only a few coefficients, to spot subsingular vectors and to set the basis for constructing embedding diagrams. In this paper we present the Adapted Ordering Method for general Lie algebras and superalgebras and their generalizations, provided they can be triangulated. We also review briefly the results obtained for the Virasoro algebra and for the N = 2 and Ramond N = 1 superconformal algebras.
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
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)
The vacuum preserving Lie algebra of a classical W-algebra
International Nuclear Information System (INIS)
Feher, L.; Tsutsui, I.
1993-07-01
We simplify and generalize an argument due to Bowcock and Watts showing that one can associate a finite Lie algebra (the 'classical vacuum preserving algebra') containing the Moebius sl(2) subalgebra to any classical W-algebra. Our construction is based on a kinematical analysis of the Poisson brackets of quasi-fields. In the case of the W S G -subalgebra S of a simple Lie algebra G, we exhibit a natural isomorphism between this finite Lie algebra and G whereby the Moebius sl(2) is identified with S. (orig.)
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.
The Centroid of a Lie Triple Algebra
Directory of Open Access Journals (Sweden)
Xiaohong Liu
2013-01-01
Full Text Available General results on the centroids of Lie triple algebras are developed. Centroids of the tensor product of a Lie triple algebra and a unitary commutative associative algebra are studied. Furthermore, the centroid of the tensor product of a simple Lie triple algebra and a polynomial ring is completely determined.
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
Energy Technology Data Exchange (ETDEWEB)
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.
New examples of continuum graded Lie algebras
International Nuclear Information System (INIS)
Savel'ev, M.V.
1989-01-01
Several new examples of continuum graded Lie algebras which provide an additional elucidation of these algebras are given. Here, in particular, the Kac-Moody algebras, the algebra S 0 Diff T 2 of infinitesimal area-preserving diffeomorphisms of the torus T 2 , the Fairlie, Fletcher and Zachos sine-algebras, etc., are described as special cases of the cross product Lie algebras. 8 refs
Lie Algebraic Treatment of Linear and Nonlinear Beam Dynamics
Energy Technology Data Exchange (ETDEWEB)
Alex J. Dragt; Filippo Neri; Govindan Rangarajan; David Douglas; Liam M. Healy; Robert D. Ryne
1988-12-01
The purpose of this paper is to present a summary of new methods, employing Lie algebraic tools, for characterizing beam dynamics in charged-particle optical systems. These methods are applicable to accelerator design, charged-particle beam transport, electron microscopes, and also light optics. The new methods represent the action of each separate element of a compound optical system, including all departures from paraxial optics, by a certain operator. The operators for the various elements can then be concatenated, following well-defined rules, to obtain a resultant operator that characterizes the entire system. This paper deals mostly with accelerator design and charged-particle beam transport. The application of Lie algebraic methods to light optics and electron microscopes is described elsewhere (1, see also 44). To keep its scope within reasonable bounds, they restrict their treatment of accelerator design and charged-particle beam transport primarily to the use of Lie algebraic methods for the description of particle orbits in terms of transfer maps. There are other Lie algebraic or related approaches to accelerator problems that the reader may find of interest (2). For a general discussion of linear and nonlinear problems in accelerator physics see (3).
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).
On an infinite-dimensional Lie algebra of Virasoro-type
International Nuclear Information System (INIS)
Pei Yufeng; Bai Chengming
2012-01-01
In this paper, we study an infinite-dimensional Lie algebra of Virasoro-type which is realized as an affinization of a two-dimensional Novikov algebra. It is a special deformation of the Lie algebra of differential operators on a circle of order at most 1. There is an explicit construction of a vertex algebra associated with the Lie algebra. We determine all derivations of this Lie algebra in terms of some derivations and centroids of the corresponding Novikov algebra. The universal central extension of this Lie algebra is also determined. (paper)
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
Analytic transfer maps for Lie algebraic design codes
International Nuclear Information System (INIS)
van Zeijts, J.; Neri, F.; Dragt, A.J.
1990-01-01
Lie algebraic methods provide a powerful tool for modeling particle transport through Hamiltonian systems. Briefly summarized, Lie algebraic design codes work as follows: first the time t flow generated by a Hamiltonian system is represented by a Lie algebraic map acting on the initial conditions. Maps are generated for each element in the lattice or beamline under study. Next all these maps are concatenated into a one-turn or one-pass map that represents the complete dynamics of the system. Finally, the resulting map is analyzed and design decisions are made based on the linear and nonlinear entries in the map. The authors give a short description of how to find Lie algebraic transfer maps in analytic form, for inclusion in accelerator design codes. As an example they find the transfer map, through third order, for the combined-function quadrupole magnet, and use such magnets to correct detrimental third-order aberrations in a spot forming system
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)
Dragt, A.J.
1987-01-01
A review is given of elementary Lie algebraic methods for treating Hamiltonian systems. This review is followed by a brief exposition of advanced Lie algebraic methods including resonance bases and conjugacy theorems. Finally, applications are made to the design of third-order achromats for use in accelerators, to the design of subangstroem resolution electron microscopes, and to the classification and study of high order aberrations in light optics. (orig.)
Expansion of the Lie algebra and its applications
International Nuclear Information System (INIS)
Guo Fukui; Zhang Yufeng
2006-01-01
We take the Lie algebra A1 as an example to illustrate a detail approach for expanding a finite dimensional Lie algebra into a higher-dimensional one. By making use of the late and its resulting loop algebra, a few linear isospectral problems with multi-component potential functions are established. It follows from them that some new integrable hierarchies of soliton equations are worked out. In addition, various Lie algebras may be constructed for which the integrable couplings of soliton equations are obtained by employing the expanding technique of the the Lie algebras
Biderivations of finite dimensional complex simple Lie algebras
Tang, Xiaomin
2016-01-01
In this paper, we prove that a biderivation of a finite dimensional complex simple Lie algebra without the restriction of skewsymmetric is inner. As an application, the biderivation of a general linear Lie algebra is presented. In particular, we find a class of a non-inner and non-skewsymmetric biderivations. Furthermore, we also get the forms of linear commuting maps on the finite dimensional complex simple Lie algebra or general linear Lie algebra.
Casimir elements of epsilon Lie algebras
International Nuclear Information System (INIS)
Scheunert, M.
1982-10-01
The classical framework for investigating the Casimir elements of a Lie algebra is generalized to the case of an epsilon Lie algebra L. We construct the standard L-module isomorphism of the epsilon-symmetric algebra of L onto its enveloping algebra and we introduce the Harish-Chandra homomorphism. In case the generators of L can be written in a canonical two-index form, we construct the associated standard sequence of Casimir elements and derive a formula for their eigenvalues in an arbitrary highest weight module. (orig.)
Representations of Lie algebras and partial differential equations
Xu, Xiaoping
2017-01-01
This book provides explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic codes, combinatorics and algebraic varieties, summarizing the author’s works and his joint works with his former students. Further, it presents various oscillator generalizations of the classical representation theorem on harmonic polynomials, and highlights new functors from the representation category of a simple Lie algebra to that of another simple Lie algebra. Partial differential equations play a key role in solving certain representation problems. The weight matrices of the minimal and adjoint representations over the simple Lie algebras of types E and F are proved to generate ternary orthogonal linear codes with large minimal distances. New multi-variable hypergeometric functions related to the root systems of simple Lie algebras are introduced in connection with quantum many-body systems in one dimension. In addition, the book identifies certai...
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)
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
Internally connected graphs and the Kashiwara-Vergne Lie algebra
Felder, Matteo
2018-02-01
It is conjectured that the Kashiwara-Vergne Lie algebra \\widehat{krv}_2 is isomorphic to the direct sum of the Grothendieck-Teichmüller Lie algebra grt_1 and a one-dimensional Lie algebra. In this paper, we use the graph complex of internally connected graphs to define a nested sequence of Lie subalgebras of \\widehat{krv}_2 whose intersection is grt_1 , thus giving a way to interpolate between these two Lie algebras.
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...
Internally connected graphs and the Kashiwara-Vergne Lie algebra
Felder, Matteo
2016-01-01
It is conjectured that the Kashiwara-Vergne Lie algebra $\\widehat{\\mathfrak{krv}}_2$ is isomorphic to the direct sum of the Grothendieck-Teichm\\"uller Lie algebra $\\mathfrak{grt}_1$ and a one-dimensional Lie algebra. In this paper, we use the graph complex of internally connected graphs to define a nested sequence of Lie subalgebras of $\\widehat{\\mathfrak{krv}}_2$ whose intersection is $\\mathfrak{grt}_1$, thus giving a way to interpolate between these two Lie algebras.
Auxiliary representations of Lie algebras and the BRST constructions
International Nuclear Information System (INIS)
Burdik, C.; Pashnev, A.I.; Tsulaya, M.M.
2000-01-01
The method of construction of auxiliary representations for a given Lie algebra is discussed in the framework of the BRST approach. The corresponding BRST charge turns out to be nonhermitian. This problem is solved by the introduction of the additional kernel operator in the definition of the scalar product in the Fock space. The existence of the kernel operator is proved for any Lie algebra
Lie algebra in quantum physics by means of computer algebra
Kikuchi, Ichio; Kikuchi, Akihito
2017-01-01
This article explains how to apply the computer algebra package GAP (www.gap-system.org) in the computation of the problems in quantum physics, in which the application of Lie algebra is necessary. The article contains several exemplary computations which readers would follow in the desktop PC: such as, the brief review of elementary ideas of Lie algebra, the angular momentum in quantum mechanics, the quark eight-fold way model, and the usage of Weyl character formula (in order to construct w...
Cartan calculus on quantum Lie algebras
International Nuclear Information System (INIS)
Schupp, P.; Watts, P.; Zumino, B.
1993-01-01
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions au into one big algebra, the ''Cartan Calculus.''
An algorithm for analysis of the structure of finitely presented Lie algebras
Directory of Open Access Journals (Sweden)
Vladimir P. Gerdt
1997-12-01
Full Text Available We consider the following problem: what is the most general Lie algebra satisfying a given set of Lie polynomial equations? The presentation of Lie algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. That problem is of great practical importance, covering applications ranging from mathematical physics to combinatorial algebra. Some particular applications are constructionof prolongation algebras in the Wahlquist-Estabrook method for integrability analysis of nonlinear partial differential equations and investigation of Lie algebras arising in different physical models. The finite presentations also indicate a way to q-quantize Lie algebras. To solve this problem, one should perform a large volume of algebraic transformations which is sharply increased with growth of the number of generators and relations. For this reason, in practice one needs to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented Lie algebra and its commutator table, and its implementation in the C language. Some computer results illustrating our algorithmand its actual implementation are also presented.
Quartic trace identity for exceptional Lie algebras
International Nuclear Information System (INIS)
Okubo, S.
1979-01-01
Let X be a representation matrix of generic element x of a simple Lie algebra in generic irreducible representation ]lambda] of the Lie algebra. Then, for all exceptional Lie algebras as well as A 1 and A 2 , we can prove the validity of a quartic trace identity Tr(X 4 ) =K (lambda)[Tr(X 2 )] 2 , where the constant K (lambda) depends only upon the irreducible representation ]lambda], and its explicit form is calculated. Some applications of second and fourth order indices have also been discussed
W-realization of Lie algebras. Application to so(4,2) and Poincare algebras
International Nuclear Information System (INIS)
Barbarin, F.; Ragoucy, E.; Sorba, P.
1996-05-01
The property of some finite W-algebras to appear as the commutant of a particular subalgebra in a simple Lie algebra G is exploited for the obtention of new G-realizations from a 'canonical' differential one. The method is applied to the conformal algebra so(4,2) and therefore yields also results for its Poincare subalgebra. Unitary irreducible representations of these algebras are recognized in this approach, which is naturally compared -or associated to - the induced representation technique. (author)
W-realization of Lie algebras. Application to so(4,2) and Poincare algebras
Energy Technology Data Exchange (ETDEWEB)
Barbarin, F.; Ragoucy, E.; Sorba, P.
1996-05-01
The property of some finite W-algebras to appear as the commutant of a particular subalgebra in a simple Lie algebra G is exploited for the obtention of new G-realizations from a `canonical` differential one. The method is applied to the conformal algebra so(4,2) and therefore yields also results for its Poincare subalgebra. Unitary irreducible representations of these algebras are recognized in this approach, which is naturally compared -or associated to - the induced representation technique. (author). 12 refs.
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.)
A survey on stability and rigidity results for Lie algebras
Crainic, Marius; Schätz, Florian; Struchiner, Ivan
2014-01-01
We give simple and unified proofs of the known stability and rigidity results for Lie algebras, Lie subalgebras and Lie algebra homomorphisms. Moreover, we investigate when a Lie algebra homomorphism is stable under all automorphisms of the codomain (including outer automorphisms).
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
Dimension of the c-nilpotent multiplier of Lie algebras
Indian Academy of Sciences (India)
Abstract. The purpose of this paper is to derive some inequalities for dimension of the c-nilpotent multiplier of finite dimensional Lie algebras and their factor Lie algebras. We further obtain an inequality between dimensions of c-nilpotent multiplier of Lie algebra L and tensor product of a central ideal by its abelianized factor ...
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.
Associative and Lie deformations of Poisson algebras
Remm, Elisabeth
2011-01-01
Considering a Poisson algebra as a non associative algebra satisfying the Markl-Remm identity, we study deformations of Poisson algebras as deformations of this non associative algebra. This gives a natural interpretation of deformations which preserves the underlying associative structure and we study deformations which preserve the underlying Lie algebra.
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)
Bicovariant quantum algebras and quantum Lie algebras
International Nuclear Information System (INIS)
Schupp, P.; Watts, P.; Zumino, B.
1993-01-01
A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun(G q ) to U q g, given by elements of the pure braid group. These operators - the 'reflection matrix' Y= triple bond L + SL - being a special case - generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation for Y in SO q (N). (orig.)
On squares of representations of compact Lie algebras
International Nuclear Information System (INIS)
Zeier, Robert; Zimborás, Zoltán
2015-01-01
We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the sum of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems
On squares of representations of compact Lie algebras
Energy Technology Data Exchange (ETDEWEB)
Zeier, Robert, E-mail: robert.zeier@ch.tum.de [Department Chemie, Technische Universität München, Lichtenbergstrasse 4, 85747 Garching (Germany); Zimborás, Zoltán, E-mail: zimboras@gmail.com [Department of Computer Science, University College London, Gower St., London WC1E 6BT (United Kingdom)
2015-08-15
We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the sum of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems.
Homotopy Lie algebras associated with a proto-bialgebra
International Nuclear Information System (INIS)
Bangoura, Momo
2003-10-01
Motivated by the search for examples of homotopy Lie algebras, to any Lie proto-bialgebra structure on a finite-dimensional vector space F, we associate two homotopy Lie algebra structures defined on the suspension of the exterior algebra of F and that of its dual F*, respectively, with a 0-ary map corresponding to the image of the empty set. In these algebras, all n-ary brackets for n ≥ 4 vanish. More generally, to any element of odd degree in Λ(F*+F), we associate a set of n-ary skew-symmetric mappings on the suspension of ΛF (resp. Λ F*), which satisfy the generalized Jacobi identities if the given element is of square zero. (author)
Invariants of generalized Lie algebras
International Nuclear Information System (INIS)
Agrawala, V.K.
1981-01-01
Invariants and invariant multilinear forms are defined for generalized Lie algebras with arbitrary grading and commutation factor. Explicit constructions of invariants and vector operators are given by contracting invariant forms with basic elements of the generalized Lie algebra. The use of the matrix of a linear map between graded vector spaces is emphasized. With the help of this matrix, the concept of graded trace of a linear operator is introduced, which is a rich source of multilinear forms of degree zero. To illustrate the use of invariants, a characteristic identity similar to that of Green is derived and a few Racah coefficients are evaluated in terms of invariants
Lie n-derivations on 7 -subspace lattice algebras
Indian Academy of Sciences (India)
all x ∈ K and all A ∈ Alg L. Based on this result, a complete characterization of linear n-Lie derivations on Alg L is obtained. Keywords. J -subspace lattice algebras; Lie derivations; Lie n-derivations; derivations. 2010 Mathematics Subject Classification. 47B47, 47L35. 1. Introduction. Let A be an algebra. Recall that a linear ...
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.
Simple Lie algebras and Dynkin diagrams
International Nuclear Information System (INIS)
Buccella, F.
1983-01-01
The following theorem is studied: in a simple Lie algebra of rank p there are p positive roots such that all the other n-3p/2 positive roots are linear combinations of them with integer non negative coefficients. Dykin diagrams are built by representing the simple roots with circles and drawing a junction between the roots. Five exceptional algebras are studied, focusing on triple junction algebra, angular momentum algebra, weights of the representation, antisymmetric tensors, and subalgebras
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.
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.
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
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)
On the intersection of irreducible components of the space of finite-dimensional Lie algebras
International Nuclear Information System (INIS)
Gorbatsevich, Vladimir V
2012-01-01
The irreducible components of the space of n-dimensional Lie algebras are investigated. The properties of Lie algebras belonging to the intersection of all the irreducible components of this kind are studied (these Lie algebras are said to be basic or founding Lie algebras). It is proved that all Lie algebras of this kind are nilpotent and each of these Lie algebras has an Abelian ideal of codimension one. Specific examples of founding Lie algebras of arbitrary dimension are described and, to describe the Lie algebras in general, we state a conjecture. The concept of spectrum of a Lie algebra is considered and some of the most elementary properties of the spectrum are studied. Bibliography: 6 titles.
Campoamor-Stursberg, R.
2018-03-01
A procedure for the construction of nonlinear realizations of Lie algebras in the context of Vessiot-Guldberg-Lie algebras of first-order systems of ordinary differential equations (ODEs) is proposed. The method is based on the reduction of invariants and projection of lowest-dimensional (irreducible) representations of Lie algebras. Applications to the description of parameterized first-order systems of ODEs related by contraction of Lie algebras are given. In particular, the kinematical Lie algebras in (2 + 1)- and (3 + 1)-dimensions are realized simultaneously as Vessiot-Guldberg-Lie algebras of parameterized nonlinear systems in R3 and R4, respectively.
The applications of a higher-dimensional Lie algebra and its decomposed subalgebras.
Yu, Zhang; Zhang, Yufeng
2009-01-15
With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 x 6 matrix Lie algebra smu(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra smu(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras smu(6) and E is used to directly construct integrable couplings.
A program for constructing finitely presented Lie algebras and superalgebras
International Nuclear Information System (INIS)
Gerdt, V.P.; Kornyak, V.V.
1997-01-01
The purpose of this paper is to describe a C program FPLSA for investigating finitely presented Lie algebras and superalgebras. The underlying algorithm is based on constructing the complete set of relations called also standard basis or Groebner basis of ideal of free Lie (super) algebra generated by the input set of relations. The program may be used, in particular, to compute the Lie (super)algebra basis elements and its structure constants, to classify the finitely presented algebras depending on the values of parameters in the relations, and to construct the Hilbert series. These problems are illustrated by examples. (orig.)
Vibrational spectroscopy of SnBr4 and CCl4 using Lie algebraic ...
Indian Academy of Sciences (India)
experimentalists because of the development of new laser spectroscopic techniques. Wulfman played a ... used Lie algebraic methods to study the spectra of molecules (vibron model) using. U(4) algebra. ..... to vibrations of gas molecules.
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
The applications of a higher-dimensional Lie algebra and its decomposed subalgebras
International Nuclear Information System (INIS)
Yu Zhang; Zhang Yufeng
2009-01-01
With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 x 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings
The applications of a higher-dimensional Lie algebra and its decomposed subalgebras
Yu, Zhang; Zhang, Yufeng
2009-01-01
With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings. PMID:20084092
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...
Study of some properties of partial differential equations by Lie algebra method
International Nuclear Information System (INIS)
Chongdar, A.K.; Ludu, A.
1990-05-01
In this note we present a system of optimal subalgebras of the Lie algebra obtained in course of investigating hypergeometric polynomial. In addition to this we have obtained some reduced equation and invariants of the P.D.E. obtained under certain transformation while studying hypergeometric polynomial by Weisner's method. Some topological properties of the solutions of P.D.E. are pointed out by using the extended jet bundle formalism. Some applications of our work on plasma physics and hydrodynamics are also cited. (author). 8 refs
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.)
On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations
International Nuclear Information System (INIS)
Zhang Yu-Feng; Tam, Honwah
2016-01-01
In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A_1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz–Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A_1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. (paper)
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
Two Types of Expanding Lie Algebra and New Expanding Integrable Systems
International Nuclear Information System (INIS)
Dong Huanhe; Yang Jiming; Wang Hui
2010-01-01
From a new Lie algebra proposed by Zhang, two expanding Lie algebras and its corresponding loop algebras are obtained. Two expanding integrable systems are produced with the help of the generalized zero curvature equation. One of them has complex Hamiltion structure with the help of generalized Tu formula (GTM). (general)
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 ...
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.
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...
2-variable Laguerre matrix polynomials and Lie-algebraic techniques
International Nuclear Information System (INIS)
Khan, Subuhi; Hassan, Nader Ali Makboul
2010-01-01
The authors introduce 2-variable forms of Laguerre and modified Laguerre matrix polynomials and derive their special properties. Further, the representations of the special linear Lie algebra sl(2) and the harmonic oscillator Lie algebra G(0,1) are used to derive certain results involving these polynomials. Furthermore, the generating relations for the ordinary as well as matrix polynomials related to these matrix polynomials are derived as applications.
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
Zhao, Shouwei
2011-06-01
A Lie algebraic condition for global exponential stability of linear discrete switched impulsive systems is presented in this paper. By considering a Lie algebra generated by all subsystem matrices and impulsive matrices, when not all of these matrices are Schur stable, we derive new criteria for global exponential stability of linear discrete switched impulsive systems. Moreover, simple sufficient conditions in terms of Lie algebra are established for the synchronization of nonlinear discrete systems using a hybrid switching and impulsive control. As an application, discrete chaotic system's synchronization is investigated by the proposed method.
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
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
The BRST complex and the cohomology of compact lie algebras
International Nuclear Information System (INIS)
Holten, J.W. van
1990-02-01
The authors construct the BRST and anti-BRST operator for a compact Lie algebra which is a direct sum of abelian and simple ideals. Two different inner products are defined on the ghost space and the hermiticity propeties of the ghost and BRST operators with respect to these inner products are discussed. A decomposition theorem for ghost states is derived and the cohomology of the BRST complex is shown to reduce to the standard Lie-algebra cohomology. The authors show that the cohomology classes of the Lie algebra are given by all invariant anti-symmetric tensors and explain how thse can be obtained as zero-modes of an invariant operator in the representation space of the ghosts. Explicit examples are given. (author) 24 refs
Boson realizations of Lie algebras with applications to nuclear physics
International Nuclear Information System (INIS)
Klein, A.; Marshalek, E.R.
1991-01-01
The concept of boson realization (or mapping) of Lie algebras appeared first in nuclear physics in 1962 as the idea of expanding bilinear forms in fermion creation and annihilation operators in Taylor series of boson operators, with the object of converting the study of nuclear vibrational motion into a problem of coupled oscillators. The physical situations of interest are quite diverse, depending, for instance, on whether excitations for fixed- or variable-particle number are being studied, on how total angular momentum is decomposed into orbital and spin parts, and on whether isotopic spin and other intrinsic degrees of freedom enter. As a consequence, all of the semisimple algebras other than the exceptional ones have proved to be of interest at one time or another, and all are studied in this review. Though the salient historical facts are presented in the introduction, in the body of the review the progression is (generally) from the simplest algebras to the more complex ones. With a sufficiently broad view of the physics requirements, the mathematical problem is the realization of an arbitrary representation of a Lie algebra in a subspace of a suitably chosen Hilbert space of bosons (Heisenberg-Weyl algebra). Indeed, if one includes the study of odd nuclei, one is forced to consider the mappings to spaces that are direct-product spaces of bosons and (quasi)fermions. Though all the methods that have been used for these problems are reviewed, emphasis is placed on a relatively new algebraic method that has emerged over the past decade. Many of the classic results are rederived, and some new results are obtained for odd systems
Lectures on Lie algebras and their representations: 1
International Nuclear Information System (INIS)
Dobrev, V.K.
1988-05-01
The paper is based on sixteen lectures given by the author in April-June 1988 at the International Centre for Theoretical Physics, Trieste. It covers the basic material on the structure, classification and representations of Lie algebras G associated with a (generalized) Cartan matrix, or Kac-Moody algebras for short. 16 refs, tabs
Lie algebraic discussion for affinity based information diffusion in social networks
Shang, Yilun
2017-11-01
In this paper we develop a dynamical information diffusion model which features the affinity of people with information disseminated in social networks. Four types of agents, i.e., susceptible, informed, known, and refractory ones, are involved in the system, and the affinity mechanism composing of an affinity threshold which represents the fitness of information to be propagated is incorporated. The model can be generally described by a time-inhomogeneous Markov chain, which is governed by its master (Kolmogorov) equation. Based on the Wei-Norman method, we derive analytical solutions of the model by constructing a low-dimensional Lie algebra. Numerical examples are provided to illustrate the obtained theoretical results. This study provides useful insights into the closed-form solutions of complex social dynamics models through the Lie algebra method.
Generating higher-order Lie algebras by expanding Maurer-Cartan forms
International Nuclear Information System (INIS)
Caroca, R.; Merino, N.; Salgado, P.; Perez, A.
2009-01-01
By means of a generalization of the Maurer-Cartan expansion method, we construct a procedure to obtain expanded higher-order Lie algebras. The expanded higher-order Maurer-Cartan equations for the case G=V 0 +V 1 are found. A dual formulation for the S-expansion multialgebra procedure is also considered. The expanded higher-order Maurer-Cartan equations are recovered from S-expansion formalism by choosing a special semigroup. This dual method could be useful in finding a generalization to the case of a generalized free differential algebra, which may be relevant for physical applications in, e.g., higher-spin gauge theories.
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...
Newton equation for canonical, Lie-algebraic, and quadratic deformation of classical space
International Nuclear Information System (INIS)
Daszkiewicz, Marcin; Walczyk, Cezary J.
2008-01-01
The Newton equation describing particle motion in a constant external field force on canonical, Lie-algebraic, and quadratic space-time is investigated. We show that for canonical deformation of space-time the dynamical effects are absent, while in the case of Lie-algebraic noncommutativity, when spatial coordinates commute to the time variable, the additional acceleration of the particle is generated. We also indicate that in the case of spatial coordinates commuting in a Lie-algebraic way, as well as for quadratic deformation, there appear additional velocity and position-dependent forces
International Nuclear Information System (INIS)
Feng, H.; Zheng, Y.; Ding, S.
2007-01-01
Infrared multiphoton vibrational excitation of the linear triatomic molecule has been studied using the quadratic anharmonic Lie-algebra model, unitary transformations, and Magnus approximation. An explicit Lie-algebra expression for the vibrational transition probability is obtained by using a Lie-algebra approach. This explicit Lie-algebra expressions for time-evolution operator and vibrational transition probabilities make the computation clearer and easier. The infrared multiphoton vibrational excitation of the DCN linear tri-atomic molecule is discussed as an example
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
q-Derivatives, quantization methods and q-algebras
International Nuclear Information System (INIS)
Twarock, Reidun
1998-01-01
Using the example of Borel quantization on S 1 , we discuss the relation between quantization methods and q-algebras. In particular, it is shown that a q-deformation of the Witt algebra with generators labeled by Z is realized by q-difference operators. This leads to a discrete quantum mechanics. Because of Z, the discretization is equidistant. As an approach to a non-equidistant discretization of quantum mechanics one can change the Witt algebra using not the number field Z as labels but a quadratic extension of Z characterized by an irrational number τ. This extension is denoted as quasi-crystal Lie algebra, because this is a relation to one-dimensional quasicrystals. The q-deformation of this quasicrystal Lie algebra is discussed. It is pointed out that quasicrystal Lie algebras can be considered also as a 'deformed' Witt algebra with a 'deformation' of the labeling number field. Their application to the theory is discussed
Semi-direct sums of Lie algebras and continuous integrable couplings
International Nuclear Information System (INIS)
Ma Wenxiu; Xu Xixiang; Zhang Yufeng
2006-01-01
A relation between semi-direct sums of Lie algebras and integrable couplings of continuous soliton equations is presented, and correspondingly, a feasible way to construct integrable couplings is furnished. A direct application to the AKNS spectral problem leads to a novel hierarchy of integrable couplings of the AKNS hierarchy of soliton equations. It is also indicated that the study of integrable couplings using semi-direct sums of Lie algebras is an important step towards complete classification of integrable systems
Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability
Directory of Open Access Journals (Sweden)
Muhammad Ayub
2013-01-01
the case of k≥3. We discuss the singular invariant representations of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. Furthermore, we give an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprises of two approaches, namely, division into four types I, II, III, and IV and that of integrability of the invariant representations. We prove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra, then, its general solution can be obtained from a partially linear, partially coupled or reduced invariantly represented system of equations. A natural extension of this result is provided for a system of two kth-order (k≥3 ODEs. We present illustrative examples of familiar integrable physical systems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories.
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
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...
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.
On split Lie algebras with symmetric root systems
Indian Academy of Sciences (India)
ideal of L, satisfying [Ij ,Ik] = 0 if j = k. Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its minimal ideals, each one being a simple split Lie algebra with a symmetric root system and having all its nonzero roots connected. Keywords. Infinite dimensional Lie ...
Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations
Directory of Open Access Journals (Sweden)
Rutwig Campoamor-Stursberg
2016-03-01
Full Text Available A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebra. In particular, well-known types, such as the Milne-Pinney or Kummer-Schwarz equations, are recovered as special cases of this classification. The analogous problem for systems of second-order differential equations in the real plane is considered for a special case that enlarges the generalized Ermakov systems.
The application of Lie algebra techniques to beam transport design
International Nuclear Information System (INIS)
Irwin, J.
1990-01-01
Using a final focus system for high-energy linear colliders as an example of a beam transport system, we illustrate for each element, and for the interplay of elements, the connection of Lie algebra techniques with usual optical analysis methods. Our analysis describes, through fourth order, the calculation and compensation of all important aberrations. (orig.)
Structure of Lie point and variational symmetry algebras for a class of odes
Ndogmo, J. C.
2018-04-01
It is known for scalar ordinary differential equations, and for systems of ordinary differential equations of order not higher than the third, that their Lie point symmetry algebras is of maximal dimension if and only if they can be reduced by a point transformation to the trivial equation y(n)=0. For arbitrary systems of ordinary differential equations of order n ≥ 3 reducible by point transformations to the trivial equation, we determine the complete structure of their Lie point symmetry algebras as well as that for their variational, and their divergence symmetry algebras. As a corollary, we obtain the maximal dimension of the Lie point symmetry algebra for any system of linear or nonlinear ordinary differential equations.
International Nuclear Information System (INIS)
Varaksin, O.L.; Firstov, V.V.; Shapovalov, A.V.
1995-01-01
The study is continued on noncommutative integration of linear partial differential equations in application to the exact integration of quantum-mechanical equations in a Riemann space. That method gives solutions to the Klein-Gordon equation when the set of noncommutative symmetry operations for that equation forms a quadratic algebra consisting of one second-order operator and of first-order operators forming a Lie algebra. The paper is a continuation of, where a single nontrivial example is used to demonstrate noncommutative integration of the Klein-Gordon equation in a Riemann space not permitting variable separation
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)
Le Van Hop.
1989-12-01
The combinatorics computation is used to describe the Casimir operators of the symplectic Lie Algebra. This result is applied for determining the Center of the enveloping Algebra of the semidirect Product of the Heisenberg Lie Algebra and the symplectic Lie Algebra. (author). 10 refs
Canonical realizations of B2 approximately C2 Lie algebras
International Nuclear Information System (INIS)
Iosifescu, M.; Scutaru, H.
1982-12-01
Canonical realizations associated to subrepresentations of ad x ad, for B 2 apppoximately C 2 semisimple Lie algebras, have been determined. An algebraic foundation has been obtained for the constraints satisfied by the dinamical variables of the classical limit of the generalized Helium problem. (authors)
Test elements of direct sums and free products of free Lie algebras
Indian Academy of Sciences (India)
Abstract. We give a characterization of test elements of a direct sum of free Lie algebras in terms of test elements of the factors. In addition, we construct certain types of test elements and we prove that in a free product of free Lie algebras, product of the homogeneous test elements of the factors is also a test element.
Test elements of direct sums and free products of free Lie algebras
Indian Academy of Sciences (India)
We give a characterization of test elements of a direct sum of free Lie algebras in terms of test elements of the factors. In addition, we construct certain types of test elements and we prove that in a free product of free Lie algebras, product of the homogeneous test elements of the factors is also a test element.
A discrete variational identity on semi-direct sums of Lie algebras
Energy Technology Data Exchange (ETDEWEB)
M, Wenxiu [Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700 (United States)
2007-12-14
The discrete variational identity under general bilinear forms on semi-direct sums of Lie algebras is established. The constant {gamma} involved in the variational identity is determined through the corresponding solution to the stationary discrete zero-curvature equation. An application of the resulting variational identity to a class of semi-direct sums of Lie algebras in the Volterra lattice case furnishes Hamiltonian structures for the associated integrable couplings of the Volterra lattice hierarchy.
A discrete variational identity on semi-direct sums of Lie algebras
International Nuclear Information System (INIS)
M, Wenxiu
2007-01-01
The discrete variational identity under general bilinear forms on semi-direct sums of Lie algebras is established. The constant γ involved in the variational identity is determined through the corresponding solution to the stationary discrete zero-curvature equation. An application of the resulting variational identity to a class of semi-direct sums of Lie algebras in the Volterra lattice case furnishes Hamiltonian structures for the associated integrable couplings of the Volterra lattice hierarchy
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-algebra approach to symmetry breaking
International Nuclear Information System (INIS)
Anderson, J.T.
1981-01-01
A formal Lie-algebra approach to symmetry breaking is studied in an attempt to reduce the arbitrariness of Lagrangian (Hamiltonian) models which include several free parameters and/or ad hoc symmetry groups. From Lie algebra it is shown that the unbroken Lagrangian vacuum symmetry can be identified from a linear function of integers which are Cartan matrix elements. In broken symmetry if the breaking operators form an algebra then the breaking symmetry (or symmetries) can be identified from linear functions of integers characteristic of the breaking symmetries. The results are applied to the Dirac Hamiltonian of a sum of flavored fermions and colored bosons in the absence of dynamical symmetry breaking. In the partially reduced quadratic Hamiltonian the breaking-operator functions are shown to consist of terms of order g 2 , g, and g 0 in the color coupling constants and identified with strong (boson-boson), medium strong (boson-fermion), and fine-structure (fermion-fermion) interactions. The breaking operators include a boson helicity operator in addition to the familiar fermion helicity and ''spin-orbit'' terms. Within the broken vacuum defined by the conventional formalism, the field divergence yields a gauge which is a linear function of Cartan matrix integers and which specifies the vacuum symmetry. We find that the vacuum symmetry is chiral SU(3) x SU(3) and the axial-vector-current divergence gives a PCAC -like function of the Cartan matrix integers which reduces to PCAC for SU(2) x SU(2) breaking. For the mass spectra of the nonets J/sup P/ = 0 - ,1/2 + ,1 - the integer runs through the sequence 3,0,-1,-2, which indicates that the breaking subgroups are the simple Lie groups. Exact axial-vector-current conservation indicates a breaking sum rule which generates octet enhancement. Finally, the second-order breaking terms are obtained from the second-order spin tensor sum of the completely reduced quartic Hamiltonian
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.)
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)
Application of differential-and-Lie-algebraic techniques to the orbit dynamics of cyclotrons
International Nuclear Information System (INIS)
Davies, W.G.; Douglas, S.R.; Pusch, G.D.; Lee-Whiting, G.E.
1991-01-01
A new orbit-dynamics code, DACYC, is being developed for the TASCC superconducting cyclotron. DACYC makes use of differential algebra and Lie Algebra to calculate and analyze partial, one-and/or multi-turn maps to very high order. Accurate, three-dimensional, analytic models of the magnetic and RF fields are used, which satisfy Maxwell's equations exactly. The maps can be analyzed with normal-form methods or to produce linear or high-order phase-space plots
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.
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
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
Lie Algebras for Constructing Nonlinear Integrable Couplings
International Nuclear Information System (INIS)
Zhang Yufeng
2011-01-01
Two new explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Giachetti-Johnson (GJ) hierarchy and the Yang hierarchy are obtained, respectively. By employing the variational identity their Hamiltonian structures are also generated. The approach presented in the paper can also provide nonlinear integrable couplings of other soliton hierarchies of evolution equations. (general)
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
International Nuclear Information System (INIS)
Prykarpatsky, A.K.; Blackmore, D.L.; Bogolubov, N.N. Jr.
2007-05-01
The infinite-dimensional operator Lie algebras of the related integrable nonlocal differential-difference dynamical systems are treated as their hidden symmetries. As a result of their dimerization the Lax type representations for both local differential-difference equations and nonlocal ones are obtained. An alternative approach to the Lie-algebraic interpretation of the integrable local differential-difference systems is also proposed. The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the centrally extended Lie algebra of integro-differential operators with matrix-valued coefficients coupled with suitable eigenfunctions and adjoint eigenfunctions evolutions of associated spectral problems is obtained by means of a specially constructed Baecklund transformation. The Hamiltonian description for the corresponding set of additional symmetry hierarchies is represented. The relation of these hierarchies with Lax type integrable (3+1)-dimensional nonlinear dynamical systems and their triple Lax type linearizations is analyzed. The Lie-algebraic structures, related with centrally extended current operator Lie algebras are discussed with respect to constructing new nonlinear integrable dynamical systems on functional manifolds and super-manifolds. Special Poisson structures and related with them factorized integrable operator dynamical systems having interesting applications in modern mathematical physics, quantum computing mathematics and other fields are constructed. The previous purely computational results are explained within the approach developed. (author)
Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces
Meljanac, Stjepan; Krešić–Jurić, Saša; Martinić, Tea
2017-07-01
This paper investigates bicovariant differential calculus on noncommutative spaces of the Lie algebra type. For a given Lie algebra g0, we construct a Lie superalgebra g =g0⊕g1 containing noncommutative coordinates and one-forms. We show that g can be extended by a set of generators TAB whose action on the enveloping algebra U (g ) gives the commutation relations between monomials in U (g0 ) and one-forms. Realizations of noncommutative coordinates, one-forms, and the generators TAB as formal power series in a semicompleted Weyl superalgebra are found. In the special case dim(g0 ) =dim(g1 ) , we also find a realization of the exterior derivative on U (g0 ) . The realizations of these geometric objects yield a bicovariant differential calculus on U (g0 ) as a deformation of the standard calculus on the Euclidean space.
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...
International Nuclear Information System (INIS)
Dobrev, V.K.
1992-01-01
We review and explain a canonical procedure for the q-deformation of the real forms G of complex Lie (super-) algebras associated with (generalized) Cartan matrices. Our procedure gives different q-deformations for the non-conjugate Cartan subalgebras of G. We give several in detail the q-deformed Lorentz and conformal (super-) algebras. The q-deformed conformal algebra contains as a subalgebra a q-deformed Poincare algebra and as Hopf subalgebras two conjugate 11-generator q-deformed Weyl algebras. The q-deformed Lorentz algebra in Hopf subalgebra of both Weyl algebras. (author). 24 refs
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
Identification of dynamical Lie algebras for finite-level quantum control systems
Energy Technology Data Exchange (ETDEWEB)
Schirmer, S.G.; Pullen, I.C.H.; Solomon, A.I. [Quantum Processes Group and Department of Applied Maths, Open University, Milton Keynes (United Kingdom)]. E-mails: S.G.Schirmer@open.ac.uk; I.C.H.Pullen@open.ac.uk; A.I.Solomon@open.ac.uk
2002-03-08
The problem of identifying the dynamical Lie algebras of finite-level quantum systems subject to external control is considered, with special emphasis on systems that are not completely controllable. In particular, it is shown that the dynamical Lie algebra for an N-level system with symmetrically coupled transitions, such as a system with equally spaced energy levels and uniform transition dipole moments, is a subalgebra of so(N) if N=2l+1, and a subalgebra of sp(l) if N=2l. General criteria for obtaining either so(2l+1) or sp(l) are established. (author)
Versal deformation of the Lie algebra $L_2$
Fialowski, A.; Post, Gerhard F.
1999-01-01
We investigate deformations of the infinite dimensional vector field Lie algebra spanned by the fields $e_i = z^{i+1}d/dz$, where $i \\ge 2 $. The goal is to describe the base of a ``versal'' deformation; such a versal deformation induces all the other nonequivalent deformations and solves the
Versal deformation of the Lie algebra L_2
Post, Gerhard F.; Fialowski, Alice
2001-01-01
We investigate deformations of the infinite-dimensional vector-field Lie algebra spanned by the fields ei = zi + 1d/dz, where i ≥ 2. The goal is to describe the base of a “versal” deformation; such a versal deformation induces all the other nonequivalent deformations and solves the deformation
On the structure of graded transitive Lie algebras
Post, Gerhard F.
2000-01-01
We study finite-dimensional Lie algebras ${\\mathfrak L}$ of polynomial vector fields in $n$ variables that contain the vector fields $\\dfrac{\\partial}{\\partial x_i} \\; (i=1,\\ldots, n)$ and $x_1\\dfrac{\\partial}{\\partial x_1}+ \\dots + x_n\\dfrac{\\partial}{\\partial x_n}$. We show that the maximal ones
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
Real division algebras and other algebras motivated by physics
International Nuclear Information System (INIS)
Benkart, G.; Osborn, J.M.
1981-01-01
In this survey we discuss several general techniques which have been productive in the study of real division algebras, flexible Lie-admissible algebras, and other nonassociative algebras, and we summarize results obtained using these methods. The principal method involved in this work is to view an algebra A as a module for a semisimple Lie algebra of derivations of A and to use representation theory to study products in A. In the case of real division algebras, we also discuss the use of isotopy and the use of a generalized Peirce decomposition. Most of the work summarized here has appeared in more detail in various other papers. The exceptions are results on a class of algebras of dimension 15, motivated by physics, which admit the Lie algebra sl(3) as an algebra of derivations
Bianchi type I cyclic cosmology from Lie-algebraically deformed phase space
International Nuclear Information System (INIS)
Vakili, Babak; Khosravi, Nima
2010-01-01
We study the effects of noncommutativity, in the form of a Lie-algebraically deformed Poisson commutation relations, on the evolution of a Bianchi type I cosmological model with a positive cosmological constant. The phase space variables turn out to correspond to the scale factors of this model in x, y, and z directions. According to the conditions that the structure constants (deformation parameters) should satisfy, we argue that there are two types of noncommutative phase space with Lie-algebraic structure. The exact classical solutions in commutative and type I noncommutative cases are presented. In the framework of this type of deformed phase space, we investigate the possibility of building a Bianchi I model with cyclic scale factors in which the size of the Universe in each direction experiences an endless sequence of contractions and reexpansions. We also obtain some approximate solutions for the type II noncommutative structure by numerical methods and show that the cyclic behavior is repeated as well. These results are compared with the standard commutative case, and similarities and differences of these solutions are discussed.
Guzzo, H.; Hernández, I.; Sánchez-Valenzuela, O. A.
2014-09-01
Finite dimensional semisimple real Lie superalgebras are described via finite dimensional semisimple complex Lie superalgebras. As an application of these results, finite dimensional real Lie superalgebras mathfrak {m}=mathfrak {m}_0 oplus mathfrak {m}_1 for which mathfrak {m}_0 is a simple Lie algebra are classified up to isomorphism.
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.
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.
Green's functions through so(2,1) lie algebra in nonrelativistic quantum mechanics
International Nuclear Information System (INIS)
Boschi-Filho, H.; Vaidya, A.N.
1991-01-01
The authors discuss an algebraic technique to construct the Green's function for systems described by the noncompact so(2,1) Lie algebra. They show that this technique solves the one-dimensional linear oscillator and Coulomb potentials and also generates particular solutions for other one-dimensional potentials. Then they construct explicitly the Green's function for the three-dimensional oscillator and the three-dimensional Coulomb potential, which are generalizations of the one-dimensional cases, and the Coulomb plus an Aharanov-Bohm, potential. They discuss the dynamical algebra involved in each case and also find their wave functions and bound state spectra. Finally they introduce in each case and also find their wave functions and bound state spectra. Finally they introduce a point canonical transformation in the generators of so(2,10) Lie algebra, show that this procedure permits us to solve the one-dimensional Morse potential in addition to the previous cases, and construct its Green's function and find its energy spectrum and wave functions
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)
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.
Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms
Energy Technology Data Exchange (ETDEWEB)
Fu, Chih-Hao; Krasnov, Kirill [School of Mathematical Sciences, The University of Nottingham,University Park, Nottingham NG7 2RD (United Kingdom)
2017-01-17
Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we uncover is the Drinfeld double of the Lie algebra of vector fields. More specifically, we show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contribution of the YM quartic vertex. We then show that this Jacobi-like identity is in fact the Jacobi identity of the Drinfeld double. All our considerations are off-shell. Our construction explains why numerators computed using the Feynman rules satisfy the colour-kinematics at four but not at higher numbers of points. It also suggests a way of modifying the Feynman rules so that the duality can continue to hold for an arbitrary number of gluons. Our construction stops short of producing explicit higher point numerators because of an absence of a certain property at four points. We comment on possible ways of correcting this, but leave the next word in the story to future work.
Generalized Lotka—Volterra systems connected with simple Lie algebras
International Nuclear Information System (INIS)
Charalambides, Stelios A; Damianou, Pantelis A; Evripidou, Charalambos A
2015-01-01
We devise a new method for producing Hamiltonian systems by constructing the corresponding Lax pairs. This is achieved by considering a larger subset of the positive roots than the simple roots of the root system of a simple Lie algebra. We classify all subsets of the positive roots of the root system of type A n for which the corresponding Hamiltonian systems are transformed, via a simple change of variables, to Lotka-Volterra systems. For some special cases of subsets of the positive roots of the root system of type A n , we produce new integrable Hamiltonian systems. (paper)
Generalized Lotka—Volterra systems connected with simple Lie algebras
Charalambides, Stelios A.; Damianou, Pantelis A.; Evripidou, Charalambos A.
2015-06-01
We devise a new method for producing Hamiltonian systems by constructing the corresponding Lax pairs. This is achieved by considering a larger subset of the positive roots than the simple roots of the root system of a simple Lie algebra. We classify all subsets of the positive roots of the root system of type An for which the corresponding Hamiltonian systems are transformed, via a simple change of variables, to Lotka-Volterra systems. For some special cases of subsets of the positive roots of the root system of type An, we produce new integrable Hamiltonian systems.
On Split Lie Algebras with Symmetric Root Systems
Indian Academy of Sciences (India)
... and any I j a well described ideal of , satisfying [ I j , I k ] = 0 if j ≠ k . Under certain conditions, the simplicity of is characterized and it is shown that is the direct sum of the family of its minimal ideals, each one being a simple split Lie algebra with a symmetric root system and having all its nonzero roots connected.
Vector fields and nilpotent Lie algebras
Grayson, Matthew; Grossman, Robert
1987-01-01
An infinite-dimensional family of flows E is described with the property that the associated dynamical system: x(t) = E(x(t)), where x(0) is a member of the set R to the Nth power, is explicitly integrable in closed form. These flows E are of the form E = E1 + E2, where E1 and E2 are the generators of a nilpotent Lie algebra, which is either free, or satisfies some relations at a point. These flows can then be used to approximate the flows of more general types of dynamical systems.
Lie algebra symmetries and quantum phase transitions in nuclei
Indian Academy of Sciences (India)
2014-04-05
Apr 5, 2014 ... 743–755. Lie algebra symmetries and quantum phase transitions in nuclei .... Applications of this CS to QPT in sdgIBM model will be briefly ..... as a linear combination of ˆC2, ˆC3 and ˆC4 of SUsdg(5) and similarly also for the.
Two Kinds of New Lie Algebras for Producing Integrable Couplings
International Nuclear Information System (INIS)
Yan Qingyou; Qi Jianxun
2006-01-01
Two types of Lie algebras are constructed, which are directly used to deduce the two resulting integrable coupling systems with multi-component potential functions. Many other integrable couplings of the known integrable systems may be obtained by the approach.
International Nuclear Information System (INIS)
Govorkov, A.B.
1980-01-01
The density matrix, rather than the wavefunction describing the system of a fixed number of non-relativistic identical particles, is subject to the second quantisation. Here the bilinear operators which move a particle from a given state to another appear and satisfy the Lie algebraic relations of the unitary group SU(rho) when the dimension rho→infinity. The drawing into consideration of the system with a variable number of particles implies the extension of this algebra into one of the simple Lie algebras of classical (orthogonal, symplectic or unitary) groups in the even-dimensional spaces. These Lie algebras correspond to the para-Fermi-, para-Bose- and para-uniquantisation of fields, respectively. (author)
De Sole, Alberto; Kac, Victor G.; Valeri, Daniele
2018-06-01
We prove that any classical affine W-algebra W (g, f), where g is a classical Lie algebra and f is an arbitrary nilpotent element of g, carries an integrable Hamiltonian hierarchy of Lax type equations. This is based on the theories of generalized Adler type operators and of generalized quasideterminants, which we develop in the paper. Moreover, we show that under certain conditions, the product of two generalized Adler type operators is a Lax type operator. We use this fact to construct a large number of integrable Hamiltonian systems, recovering, as a special case, all KdV type hierarchies constructed by Drinfeld and Sokolov.
Continual Lie algebras and noncommutative counterparts of exactly solvable models
Zuevsky, A.
2004-01-01
Noncommutative counterparts of exactly solvable models are introduced on the basis of a generalization of Saveliev-Vershik continual Lie algebras. Examples of noncommutative Liouville and sin/h-Gordon equations are given. The simplest soliton solution to the noncommutative sine-Gordon equation is found.
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)
On flux integrals for generalized Melvin solution related to simple finite-dimensional Lie algebra
Energy Technology Data Exchange (ETDEWEB)
Ivashchuk, V.D. [VNIIMS, Center for Gravitation and Fundamental Metrology, Moscow (Russian Federation); Peoples' Friendship University of Russia (RUDN University), Institute of Gravitation and Cosmology, Moscow (Russian Federation)
2017-10-15
A generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra G is considered. The solution contains a metric, n Abelian 2-forms and n scalar fields, where n is the rank of G. It is governed by a set of n moduli functions H{sub s}(z) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials - the so-called fluxbrane polynomials. These polynomials depend upon integration constants q{sub s}, s = 1,.., n. In the case when the conjecture on the polynomial structure for the Lie algebra G is satisfied, it is proved that 2-form flux integrals Φ{sup s} over a proper 2d submanifold are finite and obey the relations q{sub s} Φ{sup s} = 4πn{sub s}h{sub s}, where the h{sub s} > 0 are certain constants (related to dilatonic coupling vectors) and the n{sub s} are powers of the polynomials, which are components of a twice dual Weyl vector in the basis of simple (co-)roots, s = 1,.., n. The main relations of the paper are valid for a solution corresponding to a finite-dimensional semi-simple Lie algebra G. Examples of polynomials and fluxes for the Lie algebras A{sub 1}, A{sub 2}, A{sub 3}, C{sub 2}, G{sub 2} and A{sub 1} + A{sub 1} are presented. (orig.)
Integrable finite-dimensional systems related to Lie algebras
International Nuclear Information System (INIS)
Olshanetsky, M.A.; Perelomov, A.M.
1979-01-01
Some solvable finite-dimensional classical and quantum systems related to the Lie algebras are considered. The dynamics of these systems is closely related to free motion on symmetric spaces. In specific cases the systems considered describe the one-dimensional n-body problem recently considered by many authors. The review represents from general and universal point of view the results obtained during the last few years. Besides, it contains some results both of physical and mathematical type
Analysis of higher order optical aberrations in the SLC final focus using Lie Algebra techniques
International Nuclear Information System (INIS)
Walker, N.J.; Irwin, J.; Woodley, M.
1993-04-01
The SLC final focus system is designed to have an overall demagnification of 30:1, with a β at the interaction point (β*) of 5 mm, and an energy band pass of ∼0.4%. Strong sextupole pairs are used to cancel the large chromaticity which accrues primarily from the final triplet. Third-order aberrations limit the performance of the system, the dominating terms being U 1266 and U 3466 terms (in the notation of K. Brown). Using Lie Algebra techniques, it is possible to analytically calculate the soave of these terms in addition to understanding their origin. Analytical calculations (using Lie Algebra packages developed in the Mathematica language) are presented of the bandwidth and minimum spot size as a function of divergence at the interaction point (IP). Comparisons of the analytical results from the Lie Algebra maps and results from particle tracking (TURTLE) are also presented
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.
International Nuclear Information System (INIS)
Ibarra-Sierra, V.G.; Sandoval-Santana, J.C.; Cardoso, J.L.; Kunold, A.
2015-01-01
We discuss the one-dimensional, time-dependent general quadratic Hamiltonian and the bi-dimensional charged particle in time-dependent electromagnetic fields through the Lie algebraic approach. Such method consists in finding a set of generators that form a closed Lie algebra in terms of which it is possible to express a quantum Hamiltonian and therefore the evolution operator. The evolution operator is then the starting point to obtain the propagator as well as the explicit form of the Heisenberg picture position and momentum operators. First, the set of generators forming a closed Lie algebra is identified for the general quadratic Hamiltonian. This algebra is later extended to study the Hamiltonian of a charged particle in electromagnetic fields exploiting the similarities between the terms of these two Hamiltonians. These results are applied to the solution of five different examples: the linear potential which is used to introduce the Lie algebraic method, a radio frequency ion trap, a Kanai–Caldirola-like forced harmonic oscillator, a charged particle in a time dependent magnetic field, and a charged particle in constant magnetic field and oscillating electric field. In particular we present exact analytical expressions that are fitting for the study of a rotating quadrupole field ion trap and magneto-transport in two-dimensional semiconductor heterostructures illuminated by microwave radiation. In these examples we show that this powerful method is suitable to treat quadratic Hamiltonians with time dependent coefficients quite efficiently yielding closed analytical expressions for the propagator and the Heisenberg picture position and momentum operators. -- Highlights: •We deal with the general quadratic Hamiltonian and a particle in electromagnetic fields. •The evolution operator is worked out through the Lie algebraic approach. •We also obtain the propagator and Heisenberg picture position and momentum operators. •Analytical expressions for a
Energy Technology Data Exchange (ETDEWEB)
Ibarra-Sierra, V.G.; Sandoval-Santana, J.C. [Departamento de Física, Universidad Autónoma Metropolitana Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, 09340 México D.F. (Mexico); Cardoso, J.L. [Área de Física Teórica y Materia Condensada, Universidad Autónoma Metropolitana Azcapotzalco, Av. San Pablo 180, Col. Reynosa-Tamaulipas, Azcapotzalco, 02200 México D.F. (Mexico); Kunold, A., E-mail: akb@correo.azc.uam.mx [Área de Física Teórica y Materia Condensada, Universidad Autónoma Metropolitana Azcapotzalco, Av. San Pablo 180, Col. Reynosa-Tamaulipas, Azcapotzalco, 02200 México D.F. (Mexico)
2015-11-15
We discuss the one-dimensional, time-dependent general quadratic Hamiltonian and the bi-dimensional charged particle in time-dependent electromagnetic fields through the Lie algebraic approach. Such method consists in finding a set of generators that form a closed Lie algebra in terms of which it is possible to express a quantum Hamiltonian and therefore the evolution operator. The evolution operator is then the starting point to obtain the propagator as well as the explicit form of the Heisenberg picture position and momentum operators. First, the set of generators forming a closed Lie algebra is identified for the general quadratic Hamiltonian. This algebra is later extended to study the Hamiltonian of a charged particle in electromagnetic fields exploiting the similarities between the terms of these two Hamiltonians. These results are applied to the solution of five different examples: the linear potential which is used to introduce the Lie algebraic method, a radio frequency ion trap, a Kanai–Caldirola-like forced harmonic oscillator, a charged particle in a time dependent magnetic field, and a charged particle in constant magnetic field and oscillating electric field. In particular we present exact analytical expressions that are fitting for the study of a rotating quadrupole field ion trap and magneto-transport in two-dimensional semiconductor heterostructures illuminated by microwave radiation. In these examples we show that this powerful method is suitable to treat quadratic Hamiltonians with time dependent coefficients quite efficiently yielding closed analytical expressions for the propagator and the Heisenberg picture position and momentum operators. -- Highlights: •We deal with the general quadratic Hamiltonian and a particle in electromagnetic fields. •The evolution operator is worked out through the Lie algebraic approach. •We also obtain the propagator and Heisenberg picture position and momentum operators. •Analytical expressions for a
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)
International Nuclear Information System (INIS)
Zhang Yufeng; Guo Fukui
2007-01-01
Two types of Lie algebras, which are the subalgebras of the Lie algebra A 2 , A 3 respectively, are presented. The resulting loop algebras are following. As their applications, two different integrable couplings of the Yang hierarchy are obtained, called them the double integrable couplings. The Hamiltonian structure of one of them is worked out by a proper linear isomorphic transformation and the quadratic-form identity
Directory of Open Access Journals (Sweden)
Nader Ali Makboul Hassan
2014-01-01
Full Text Available This paper is an attempt to stress the usefulness of the multi-variable special functions. In this paper, we derive certain generating relations involving 2-indices 5-variables 5-parameters Tricomi functions (2I5V5PTF by using a Lie-algebraic method. Further, we derive certain new and known generating relations involving other forms of Tricomi and Bessel functions as applications.
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
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.
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)
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
Lie-deformed quantum Minkowski spaces from twists: Hopf-algebraic versus Hopf-algebroid approach
Lukierski, Jerzy; Meljanac, Daniel; Meljanac, Stjepan; Pikutić, Danijel; Woronowicz, Mariusz
2018-02-01
We consider new Abelian twists of Poincare algebra describing nonsymmetric generalization of the ones given in [1], which lead to the class of Lie-deformed quantum Minkowski spaces. We apply corresponding twist quantization in two ways: as generating quantum Poincare-Hopf algebra providing quantum Poincare symmetries, and by considering the quantization which provides Hopf algebroid describing class of quantum relativistic phase spaces with built-in quantum Poincare covariance. If we assume that Lorentz generators are orbital i.e. do not describe spin degrees of freedom, one can embed the considered generalized phase spaces into the ones describing the quantum-deformed Heisenberg algebras.
Fixed points of IA-endomorphisms of a free metabelian Lie algebra
Indian Academy of Sciences (India)
Let be a free metabelian Lie algebra of finite rank at least 2. We show the existence of non-trivial fixed points of an -endomorphism of and give an algorithm detecting them. In particular, we prove that the fixed point subalgebra Fix of an -endomorphism of is not finitely generated.
Canonical realizations of the Lie algebra sp(2n,R)
International Nuclear Information System (INIS)
Havlicek, M.; Lassner, W.
1975-01-01
The generators of the Lie algebra of the symplectic group sp(2n,R) are, rezcurently, realied by means of polynomials in the quantum canonical variables qsub(i) and psub(i), i=1,...,d(2n-d);d=1,...,n. These realisations are skew-hermitean, the Casimir operators are realised by constant multiples of identity element and they depend on d free real parameters
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.
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 'near to minimal' canonical realizations of the Lie algebra Csub(n)
International Nuclear Information System (INIS)
Havlicek, M.; Lassner, W.
1975-01-01
It is proved that canonical realizations of the Lie algebra Csub(n) in the quotient division ring Dsub(2(2n-2)) of the Weyl algebra Wsub(2(2n-2)) in 2n-2 quantum canonical pairs are, in a definite sense, related to the standard minimal one in Dsub(2n) contains Dsub(2(2n-2)). Further, in any realization of Csub(n) in Wsub(2(2n-1)) all Casimir operators are realized by multiples of identity element
Partition functions for quantum gravity, black holes, elliptic genera and Lie algebra homologies
Energy Technology Data Exchange (ETDEWEB)
Bonora, L., E-mail: bonora@sissa.it [International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste (Italy); INFN, Sezione di Trieste (Italy); Bytsenko, A.A., E-mail: abyts@uel.br [Departamento de Fisica, Universidade Estadual de Londrina, Caixa Postal 6001, Londrina (Brazil)
2011-11-11
There is a remarkable connection between quantum generating functions of field theory and formal power series associated with dimensions of chains and homologies of suitable Lie algebras. We discuss the homological aspects of this connection with its applications to partition functions of the minimal three-dimensional gravities in the space-time asymptotic to AdS{sub 3}, which also describe the three-dimensional Euclidean black holes, the pure N=1 supergravity, and a sigma model on N-fold generalized symmetric products. We also consider in the same context elliptic genera of some supersymmetric sigma models. These examples can be considered as a straightforward application of the machinery of modular forms and spectral functions (with values in the congruence subgroup of SL(2,Z)) to partition functions represented by means of formal power series that encode Lie algebra properties.
On generalized Melvin solution for the Lie algebra E6
International Nuclear Information System (INIS)
Bolokhov, S.V.; Ivashchuk, V.D.
2017-01-01
A multidimensional generalization of Melvin's solution for an arbitrary simple Lie algebra G is considered. The gravitational model in D dimensions, D ≥ 4, contains n 2-forms and l ≥ n scalar fields, where n is the rank of G. The solution is governed by a set of n functions H s (z) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials (the so-called fluxbrane polynomials). The polynomials H s (z), s = 1,.., 6, for the Lie algebra E 6 are obtained and a corresponding solution for l = n = 6 is presented. The polynomials depend upon integration constants Q s , s = 1,.., 6. They obey symmetry and duality identities. The latter ones are used in deriving asymptotic relations for solutions at large distances. The power-law asymptotic relations for E 6 -polynomials at large z are governed by the integer-valued matrix ν = A -1 (I + P), where A -1 is the inverse Cartan matrix, I is the identity matrix and P is a permutation matrix, corresponding to a generator of the Z 2 -group of symmetry of the Dynkin diagram. The 2-form fluxes Φ s , s = 1,.., 6, are calculated. (orig.)
Linear operator pencils on Lie algebras and Laurent biorthogonal polynomials
International Nuclear Information System (INIS)
Gruenbaum, F A; Vinet, Luc; Zhedanov, Alexei
2004-01-01
We study operator pencils on generators of the Lie algebras sl 2 and the oscillator algebra. These pencils are linear in a spectral parameter λ. The corresponding generalized eigenvalue problem gives rise to some sets of orthogonal polynomials and Laurent biorthogonal polynomials (LBP) expressed in terms of the Gauss 2 F 1 and degenerate 1 F 1 hypergeometric functions. For special choices of the parameters of the pencils, we identify the resulting polynomials with the Hendriksen-van Rossum LBP which are widely believed to be the biorthogonal analogues of the classical orthogonal polynomials. This places these examples under the umbrella of the generalized bispectral problem which is considered here. Other (non-bispectral) cases give rise to some 'nonclassical' orthogonal polynomials including Tricomi-Carlitz and random-walk polynomials. An application to solutions of relativistic Toda chain is considered
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.)
International Nuclear Information System (INIS)
Fernandez Nunez, J.; Garcia Fuertes, W.; Perelomov, A.M.
2003-01-01
We express the Hamiltonian of the quantum trigonometric Calogero-Sutherland model related to the Lie algebra D 4 in terms of a set of Weyl-invariant variables, namely, the characters of the fundamental representations of the Lie algebra. This parametrization allows us to solve for the energy eigenfunctions of the theory and to study properties of the system of orthogonal polynomials associated with them such as recurrence relations and generating functions
Surfaces immersed in Lie algebras associated with elliptic integrals
International Nuclear Information System (INIS)
Grundland, A M; Post, S
2012-01-01
The objective of this work is to adapt the Fokas–Gel’fand immersion formula to ordinary differential equations written in the Lax representation. The formalism of generalized vector fields and their prolongation structure is employed to establish necessary and sufficient conditions for the existence and integration of immersion functions for surfaces in Lie algebras. As an example, a class of second-order, integrable, ordinary differential equations is considered and the most general solutions for the wavefunctions of the linear spectral problem are found. Several explicit examples of surfaces associated with Jacobian and P-Weierstrass elliptic functions are presented. (paper)
International Nuclear Information System (INIS)
Accardi, Luigi; Boukas, Andreas
2008-01-01
The identification of the *-Lie algebra of the renormalized higher powers of White noise (RHPWN) and the analytic continuation of the second quantized centreless Virasoro (or Witt)-Zamolodchikov-w ∞ *-Lie algebra of conformal field theory and high-energy physics, was recently established on results obtained. In the present paper, we show how the RHPWN Fock kernels must be truncated in order to be positive semi-definite and we obtain a Fock representation of the two algebras. We show that the truncated renormalized higher powers of White noise (TRHPWN) Fock spaces of order ≥2 host the continuous binomial and beta processes
International Nuclear Information System (INIS)
Macfarlane, A J; Pfeiffer, Hendryk
2003-01-01
The uniformity, for the family of exceptional Lie algebras g, of the decompositions of the powers of their adjoint representations is now well known for powers up to four. The paper describes an extension of this uniformity for the totally antisymmetrized nth powers up to n = 9, identifying families of representations with integer eigenvalues 5, ..., 9 for the quadratic Casimir operator, in each case providing a formula for the dimensions of the representations in the family as a function of D = dim g. This generalizes previous results for powers j and Casimir eigenvalues j, j ≤ 4. Many intriguing, perhaps puzzling, features of the dimension formulae are discussed and the possibility that they may be valid for a wider class of not necessarily simple Lie algebras is considered
On a parametrization of Baker-Campbell-Hausdorf formula for bosonic superfields in Lie algebra
International Nuclear Information System (INIS)
Gabeskiria, M.A.
1984-01-01
A compact form for the Baker-Cambell-Hausdorf formula has been obtained. Here the dependence of bosonic superfields, with their values on the Crassmann hull G(LAMBDA 2 ) of Lie algebra G, on the generators LAMBDA 2 has been factorized as a single exponent
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.
On generalized Melvin solution for the Lie algebra E{sub 6}
Energy Technology Data Exchange (ETDEWEB)
Bolokhov, S.V. [Peoples' Friendship University of Russia (RUDN University), Moscow (Russian Federation); Ivashchuk, V.D. [VNIIMS, Center for Gravitation and Fundamental Metrology, Moscow (Russian Federation); Peoples' Friendship University of Russia (RUDN University), Moscow (Russian Federation)
2017-10-15
A multidimensional generalization of Melvin's solution for an arbitrary simple Lie algebra G is considered. The gravitational model in D dimensions, D ≥ 4, contains n 2-forms and l ≥ n scalar fields, where n is the rank of G. The solution is governed by a set of n functions H{sub s}(z) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials (the so-called fluxbrane polynomials). The polynomials H{sub s}(z), s = 1,.., 6, for the Lie algebra E{sub 6} are obtained and a corresponding solution for l = n = 6 is presented. The polynomials depend upon integration constants Q{sub s}, s = 1,.., 6. They obey symmetry and duality identities. The latter ones are used in deriving asymptotic relations for solutions at large distances. The power-law asymptotic relations for E{sub 6}-polynomials at large z are governed by the integer-valued matrix ν = A{sup -1}(I + P), where A{sup -1} is the inverse Cartan matrix, I is the identity matrix and P is a permutation matrix, corresponding to a generator of the Z{sub 2}-group of symmetry of the Dynkin diagram. The 2-form fluxes Φ{sup s}, s = 1,.., 6, are calculated. (orig.)
Constraint Lie algebra and local physical Hamiltonian for a generic 2D dilatonic model
International Nuclear Information System (INIS)
Corichi, Alejandro; Karami, Asieh; Rastgoo, Saeed; Vukašinac, Tatjana
2016-01-01
We consider a class of two-dimensional dilatonic models, and revisit them from the perspective of a new set of ‘polar type’ variables. These are motivated by recently defined variables within the spherically symmetric sector of 4D general relativity. We show that for a large class of dilatonic models, including the case with matter, one can perform a series of canonical transformations in such a way that the Poisson algebra of the constraints becomes a Lie algebra. Furthermore, we construct Dirac observables and a reduced Hamiltonian that accounts for the time evolution of the system. (paper)
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
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)
Three-dimensional quantum algebras: a Cartan-like point of view
International Nuclear Information System (INIS)
Ballesteros, A; Celeghini, E; Olmo, M A del
2004-01-01
A perturbative quantization procedure for Lie bialgebras is introduced. The relevance of the choice of a completely symmetrized basis of the quantum universal enveloping algebra is stressed. Sets of elements of the quantum algebra that play a role similar to generators in the case of Lie algebras are considered and a Cartan-like procedure applied to find a representative for each class of quantum algebras. The method is used to construct and classify all three-dimensional complex quantum algebras that are compatible with a given type of coproduct. The quantization of all Lie algebras that, in the classical limit, belong to the most relevant sector in the classification for three-dimensional Lie bialgebras is thus performed. New quantizations of solvable algebras, whose simplicity makes them suitable for possible physical applications, are obtained and already known related quantum algebras recovered
The quantum Rabi model and Lie algebra representations of sl2
International Nuclear Information System (INIS)
Wakayama, Masato; Yamasaki, Taishi
2014-01-01
The aim of the present paper is to understand the spectral problem of the quantum Rabi model in terms of Lie algebra representations of sl 2 (R). We define a second order element of the universal enveloping algebra U(sl 2 ) of sl 2 (R), which, through the image of a principal series representation of sl 2 (R), provides a picture equivalent to the quantum Rabi model drawn by confluent Heun differential equations. By this description, in particular, we give a representation theoretic interpretation of the degenerate part of the spectrum (i.e., Judd's eigenstates) of the Rabi Hamiltonian due to Kuś in 1985, which is a part of the exceptional spectrum parameterized by integers. We also discuss the non-degenerate part of the exceptional spectrum of the model, in addition to the Judd eigenstates, from a viewpoint of infinite dimensional irreducible submodules (or subquotients) of the non-unitary principal series such as holomorphic discrete series representations of sl 2 (R). (paper)
The two-parameter deformation of GL(2), its differential calculus, and Lie algebra
International Nuclear Information System (INIS)
Schirrmacher, A.; Wess, J.
1991-01-01
The Yang-Baxter equation is solved in two dimensions giving rise to a two-parameter deformation of GL(2). The transformation properties of quantum planes are briefly discussed. Non-central determinant and inverse are constructed. A right-invariant differential calculus is presented and the role of the different deformation parameters investigated. While the corresponding Lie algebra relations are simply deformed, the comultiplication exhibits both quantization parameters. (orig.)
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.
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)
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.)
n-ary algebras: a review with applications
International Nuclear Information System (INIS)
De Azcarraga, J A; Izquierdo, J M
2010-01-01
This paper reviews the properties and applications of certain n-ary generalizations of Lie algebras in a self-contained and unified way. These generalizations are algebraic structures in which the two-entry Lie bracket has been replaced by a bracket with n entries. Each type of n-ary bracket satisfies a specific characteristic identity which plays the role of the Jacobi identity for Lie algebras. Particular attention will be paid to generalized Lie algebras, which are defined by even multibrackets obtained by antisymmetrizing the associative products of its n components and that satisfy the generalized Jacobi identity, and to Filippov (or n-Lie) algebras, which are defined by fully antisymmetric n-brackets that satisfy the Filippov identity. 3-Lie algebras have surfaced recently in multi-brane theory in the context of the Bagger-Lambert-Gustavsson model. As a result, Filippov algebras will be discussed at length, including the cohomology complexes that govern their central extensions and their deformations (it turns out that Whitehead's lemma extends to all semisimple n-Lie algebras). When the skewsymmetry of the Lie or n-Lie algebra bracket is relaxed, one is led to a more general type of n-algebras, the n-Leibniz algebras. These will be discussed as well, since they underlie the cohomological properties of n-Lie algebras. The standard Poisson structure may also be extended to the n-ary case. We shall review here the even generalized Poisson structures, whose generalized Jacobi identity reproduces the pattern of the generalized Lie algebras, and the Nambu-Poisson structures, which satisfy the Filippov identity and determine Filippov algebras. Finally, the recent work of Bagger-Lambert and Gustavsson on superconformal Chern-Simons theory will be briefly discussed. Emphasis will be made on the appearance of the 3-Lie algebra structure and on why the A 4 model may be formulated in terms of an ordinary Lie algebra, and on its Nambu bracket generalization. (topical
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Lebedenko, V.M.
1978-01-01
The PR-algebras, i.e. the Lie algebras with commutation relations of [Hsub(i),Hsub(j)]=rsub(ij)Hsub(i)(i< j) type are investigated. On the basis of former results a criterion for the membership of 2-solvable Lie algebras to the PR-algebra class is given. The conditions imposed by the criterion are formulated in the linear algebra language
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
The geometry of lie algebras and broken SO(6) symmetries
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Lawrence, T.R.
2001-10-01
Non-linear realisations of the groups SU(2), SO(1,4) and SO(2,4) are analysed, described by the coset spaces SU(2)/U(1), SO(1,4)/SO(1,3) and SO(2,4)/SO(1,3) x SO(1,1). The Lie algebras of certain special unitary and special orthogonal groups are studied and their projection operators are determined in order to facilitate the above analyses, in particular that of SO(2,4)/SO(l,3) x SO(1,1). The analysis consists of determining the transformation properties of the Goldstone bosons, constructing the most general possible Lagrangian for the realisations and finding the metric of the coset space. (author)
G-identities of non-associative algebras
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Bakhturin, Yu A; Zaitsev, M V; Sehgal, S K
1999-01-01
The main class of algebras considered in this paper is the class of algebras of Lie type. This class includes, in particular, associative algebras, Lie algebras and superalgebras, Leibniz algebras, quantum Lie algebras, and many others. We prove that if a finite group G acts on such an algebra A by automorphisms and anti-automorphisms and A satisfies an essential G-identity, then A satisfies an ordinary identity of degree bounded by a function that depends on the degree of the original identity and the order of G. We show in the case of ordinary Lie algebras that if L is a Lie algebra, a finite group G acts on L by automorphisms and anti-automorphisms, and the order of G is coprime to the characteristic of the field, then the existence of an identity on skew-symmetric elements implies the existence of an identity on the whole of L, with the same kind of dependence between the degrees of the identities. Finally, we generalize Amitsur's theorem on polynomial identities in associative algebras with involution to the case of alternative algebras with involution
Czech Academy of Sciences Publication Activity Database
Zuevsky, Alexander
2018-01-01
Roč. 16, č. 1 (2018), s. 1-8 ISSN 2391-5455 Institutional support: RVO:67985840 Keywords : Kac-Moody Lie algebras * cocycles Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 0.682, year: 2016 https://www.degruyter.com/view/j/math.2018.16.issue-1/math-2018-0002/math-2018-0002. xml
Czech Academy of Sciences Publication Activity Database
Zuevsky, Alexander
2018-01-01
Roč. 16, č. 1 (2018), s. 1-8 ISSN 2391-5455 Institutional support: RVO:67985840 Keywords : Kac-Moody Lie algebras * cocycles Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 0.682, year: 2016 https://www.degruyter.com/view/j/math.2018.16.issue-1/math-2018-0002/math-2018-0002.xml
Identities and derivations for Jacobian algebras
International Nuclear Information System (INIS)
Dzhumadil'daev, A.S.
2001-09-01
Constructions of n-Lie algebras by strong n-Lie-Poisson algebras are given. First cohomology groups of adjoint module of Jacobian algebras are calculated. Minimal identities of 3-Jacobian algebra are found. (author)
Generalizing the bms3 and 2D-conformal algebras by expanding the Virasoro algebra
Caroca, Ricardo; Concha, Patrick; Rodríguez, Evelyn; Salgado-Rebolledo, Patricio
2018-03-01
By means of the Lie algebra expansion method, the centrally extended conformal algebra in two dimensions and the bms3 algebra are obtained from the Virasoro algebra. We extend this result to construct new families of expanded Virasoro algebras that turn out to be infinite-dimensional lifts of the so-called Bk, Ck and Dk algebras recently introduced in the literature in the context of (super)gravity. We also show how some of these new infinite-dimensional symmetries can be obtained from expanded Kač-Moody algebras using modified Sugawara constructions. Applications in the context of three-dimensional gravity are briefly discussed.
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
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.
On higher-dimensional loop algebras, pseudodifferential operators and Fock space realizations
International Nuclear Information System (INIS)
Westerberg, A.
1997-01-01
We discuss a previously discovered extension of the infinite-dimensional Lie algebra map(M,g) which generalizes the Kac-Moody algebras in 1+1 dimensions and the Mickelsson-Faddeev algebras in 3+1 dimensions to manifolds M of general dimensions. Furthermore, we review the method of regularizing current algebras in higher dimensions using pseudodifferential operator (PSDO) symbol calculus. In particular, we discuss the issue of Lie algebra cohomology of PSDOs and its relation to the Schwinger terms arising in the quantization process. Finally, we apply this regularization method to the algebra with partial success, and discuss the remaining obstacles to the construction of a Fock space representation. (orig.)
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
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
Exceptional quantum subgroups for the rank two Lie algebras B2 and G2
Coquereaux, R.; Tahri, E.H.
2010-01-01
Exceptional modular invariants for the Lie algebras B2 (at levels 2,3,7,12) and G2 (at levels 3,4) can be obtained from conformal embeddings. We determine the associated alge bras of quantum symmetries and discover or recover, as a by-product, the graphs describing exceptional quantum subgroups of type B2 or G2 which encode their module structure over the associated fusion category. Global dimensions are given.
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
International Nuclear Information System (INIS)
Yan Qingyou; Zhang Yufeng; Wei Xiaopeng
2004-01-01
A new subalgebra G of the Lie algebra A 2 is first constructed. Then two loop algebra G-bar 1 , G-bar 2 are presented in terms of different definitions of gradations. Using G-bar 1 , G-bar 2 designs two isospectral problems, respectively. Again utilizing Tu-pattern obtains two types of various integrable Hamiltonian hierarchies of evolution equations. As reduction cases, the well-known Schroedinger equation and MKdV equation are obtained. At last, we turn the subalgebras G-bar 1 , G-bar 2 of the loop algebra A-bar 2 into equivalent subalgebras of the loop algebra A-bar 1 by making a suitable linear transformation so that the two types of 5-dimensional loop algebras are constructed. Two kinds of integrable couplings of the obtained hierarchies are showed. Specially, the integrable couplings of Schroedinger equation and MKdV equation are obtained, respectively
Quantum integrable systems related to lie algebras
International Nuclear Information System (INIS)
Olshanetsky, M.A.; Perelomov, A.M.
1983-01-01
Some quantum integrable finite-dimensional systems related to Lie algebras are considered. This review continues the previous review of the same authors (1981) devoted to the classical aspects of these systems. The dynamics of some of these systems is closely related to free motion in symmetric spaces. Using this connection with the theory of symmetric spaces some results such as the forms of spectra, wave functions, S-matrices, quantum integrals of motion are derived. In specific cases the considered systems describe the one-dimensional n-body systems interacting pairwise via potentials g 2 v(q) of the following 5 types: vsub(I)(q)=q - 2 , vsub(II)(q)=sinh - 2 q, vsub(III)(q)=sin - 2 q, vsub(IV)(q)=P(q), vsub(V)(q)=q - 2 +#betta# 2 q 2 . Here P(q) is the Weierstrass function, so that the first three cases are merely subcases on the fourth. The system characterized by the Toda nearest-neighbour potential exp(qsub(j)-qsub(j+1)) is moreover considered. This review presents from a general and universal point of view results obtained mainly over the past fifteen years. Besides, it contains some new results both of physical and mathematical interest. (orig.)
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
Kolev, Boris
2006-01-01
23 pages; International audience; This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. We investigate the special case where one of the structures is the canonical Lie-Poisson structure and the second one is constant. These structures called affine or modified Lie-Poisson structures are involved in the integrability of certain Euler equations that arise as models of shallow water waves.
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.
On Associative Conformal Algebras of Linear Growth
Retakh, Alexander
2000-01-01
Lie conformal algebras appear in the theory of vertex algebras. Their relation is similar to that of Lie algebras and their universal enveloping algebras. Associative conformal algebras play a role in conformal representation theory. We introduce the notions of conformal identity and unital associative conformal algebras and classify finitely generated simple unital associative conformal algebras of linear growth. These are precisely the complete algebras of conformal endomorphisms of finite ...
Contractions of Lie algebras and separation of variables. The n-dimensional sphere
International Nuclear Information System (INIS)
Izmest'ev, A.A.; Pogosyan, G.S.; Sisakyan, A.N.; Winternitz, P.
1998-01-01
Inonu-Wigner contractions from the rotation group O (n + 1) to the Euclidean group E (n) are used to relate the separation of variables in Laplace-Beltrami operators on n-dimensional spheres and Euclidean spaces. We consider all subgroup type coordinates corresponding to different chains of subgroups of O (n + 1) and E (n). In particular, the contractions relate the graphical formalism of 'trees' on spheres to the 'clusters' on Euclidean spaces (introduced in this article). The contractions are considered analytically on several levels: the vector fields realizing the Lie algebras, the complete sets of commuting operators characterizing separable coordinate systems, the coordinate systems themselves and the separated eigenfunctions
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)
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
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.
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
On d -Dimensional Lattice (co)sine n -Algebra
International Nuclear Information System (INIS)
Yao Shao-Kui; Zhang Chun-Hong; Zhao Wei-Zhong; Ding Lu; Liu Peng
2016-01-01
We present the (co)sine n-algebra which is indexed by the d-dimensional integer lattice. Due to the associative operators, this generalized (co)sine n-algebra is the higher order Lie algebra for the n even case. The particular cases are the d-dimensional lattice sine 3 and cosine 5-algebras with the special parameter values. We find that the corresponding d-dimensional lattice sine 3 and cosine 5-algebras are the Nambu 3-algebra and higher order Lie algebra, respectively. The limiting case of the d-dimensional lattice (co)sine n-algebra is also discussed. Moreover we construct the super sine n-algebra, which is the super higher order Lie algebra for the n even case. (paper)
On the classification of quantum W-algebras
International Nuclear Information System (INIS)
Bowcock, P.; Watts, G.T.M.
1992-01-01
In this paper we consider the structure of general quantum W-algebras. We introduce the notions of deformability, positive-definiteness, and reductivity of a W-algebra. We show that one can associate a reductive finite Lie algebra to each reductive W-algebra. The finite Lie algebra is also endowed with a preferred sl(2) subalgebra, which gives the conformal weights of the W-algebra. We extend this to cover W-algebras containing both bosonic and fermionic fields, and illustrate our ideas with the Poisson bracket algebras of generalised Drinfeld-Sokolov hamiltonian systems. We then discuss the possibilities of classifying deformable W-algebras which fall outside this class in the context of automorphisms of Lie algebras. In conclusion we list the cases in which the W-algebra has no weight-one fields, and further, those in which it has only one weight-two field. (orig.)
Lie-theoretic generating relations of two variable Laguerre polynomials
International Nuclear Information System (INIS)
Khan, Subuhi; Yasmin, Ghazala
2002-07-01
Generating relations involving two variable Lagneire polynonuals L n (x, y) are derived. The process involves the construction of a three dimensional Lie algebra isomorphic to special linear algebra sl(2) with the help of Weisner's method by giving suitable interpretations to the index n of the polynomials L n (x, y). (author)
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.
Quantization and representation theory of finite W algebras
International Nuclear Information System (INIS)
Boer, J. de; Tjin, T.
1993-01-01
In this paper we study the finitely generated algebras underlying W algebras. These so called 'finite W algebras' are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions are labeled by the inequivalent embeddings of sl 2 into the simple Lie algebra in question. For arbitrary embeddings a coordinate free formula for the reduced Poisson structure is derived. We also prove that any finite W algebra can be embedded into the Kirillov Poisson algebra of a (semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that generalized finite Toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finite W symmetry. In the second part we BRST quantize the finite W algebras. The BRST cohomoloy is calculated using a spectral sequence (which is different from the one used by Feigin and Frenkel). This allows us to quantize all finite W algebras in one stroke. Examples are given. In the last part of the paper we study the representation theory of finite W algebras. It is shown, using a quantum inversion of the generalized Miura transformation, that the representations of finite W algebras can be constructed from the representations of a certain Lie subalgebra of the original simple Lie algebra. As a byproduct of this we are able to construct the Fock realizations of arbitrary finite W algebras. (orig.)
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
Algebraic Methods to Design Signals
2015-08-27
to date on designing signals using algebraic and combinatorial methods. Mathematical tools from algebraic number theory, representation theory and... combinatorial objects in designing signals for communication purposes. Sequences and arrays with desirable autocorrelation properties have many...multiple access methods in mobile radio communication systems. We continue our mathematical framework based on group algebras, character theory
Lie-Hamilton systems on curved spaces: a geometrical approach
Herranz, Francisco J.; de Lucas, Javier; Tobolski, Mariusz
2017-12-01
A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional Lie algebra, a Vessiot-Guldberg Lie algebra, of Hamiltonian vector fields relative to a Poisson structure. Its general solution can be written as an autonomous function, the superposition rule, of a generic finite family of particular solutions and a set of constants. We pioneer the study of Lie-Hamilton systems on Riemannian spaces (sphere, Euclidean and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes) as well as on semi-Riemannian spaces (Newtonian spacetimes). Their corresponding constants of motion and superposition rules are obtained explicitly in a geometric way. This work extends the (graded) contraction of Lie algebras to a contraction procedure for Lie algebras of vector fields, Hamiltonian functions, and related symplectic structures, invariants, and superposition rules.
Non commutative geometry methods for group C{sup *}-algebras
Energy Technology Data Exchange (ETDEWEB)
Diep, Do Ngoc
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{sup *}-algebras: started in the elementary part, with one example of description of the structure of C{sup *}-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{sup *}-algebras were created and developed. (author). Refs.
Algebraic special functions and SO(3,2)
International Nuclear Information System (INIS)
Celeghini, E.; Olmo, M.A. del
2013-01-01
A ladder structure of operators is presented for the associated Legendre polynomials and the sphericas harmonics. In both cases these operators belong to the irreducible representation of the Lie algebra so(3,2) with quadratic Casimir equals to −5/4. As both are also bases of square-integrable functions, the universal enveloping algebra of so(3,2) is thus shown to be homomorphic to the space of linear operators acting on the L 2 functions defined on (−1,1)×Z and on the sphere S 2 , respectively. The presence of a ladder structure is suggested to be the general condition to obtain a Lie algebra representation defining in this way the “algebraic special functions” that are proposed to be the connection between Lie algebras and square-integrable functions so that the space of linear operators on the L 2 functions is homomorphic to the universal enveloping algebra. The passage to the group, by means of the exponential map, shows that the associated Legendre polynomials and the spherical harmonics support the corresponding unitary irreducible representation of the group SO(3,2). -- Highlights: •The algebraic ladder structure is constructed for the associated Legendre polynomials (ALP). •ALP and spherical harmonics support a unitary irreducible SO(3,2)-representation. •A ladder structure is the condition to get a Lie group representation defining “algebraic special functions”. •The “algebraic special functions” connect Lie algebras and L 2 functions
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 an extension of the Weil algebra
International Nuclear Information System (INIS)
Palev, Ch.
An extension of the Weil algebra Wsub(n), generated by an appropriate topology is considered. The topology is introduced in such a way that algebraic operations in Wsub(n) to be continuous. The algebraic operations in Wsub(n) are extended by a natural way to a complement, which is noted as an extended Weil algebra. It turns out that the last algebra contains isomorphically the Heisenberg group. By the same way an arbitrary enveloping algebra of a Lie group may be extended. The extended algebra will contain the initial Lie group. (S.P.)
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
3-Lie bialgebras (Lb,Cd and (Lb,Ce
Directory of Open Access Journals (Sweden)
Bai Ruipu
2016-05-01
Full Text Available Four dimensional $3$-Lie coalgebras with two-dimensional derived algebras, and four-dimensional $3$-Lie bialgebras of type $(L_b, C_c$ are classified. It is proved that there exist three classes of four dimensional $3$-Lie coalgebras with two-dimensional derived algebra which are $(L, C_{c_i}$, $i=1, 2, 3$ (Lemma 3.1, and ten classes of four dimensional $3$-Lie bialgebras of type $(L_b, C_c$ (Theorem 3.2.
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...
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..
Algebra, Geometry and Mathematical Physics Conference
Paal, Eugen; Silvestrov, Sergei; Stolin, Alexander
2014-01-01
This book collects the proceedings of the Algebra, Geometry and Mathematical Physics Conference, held at the University of Haute Alsace, France, October 2011. Organized in the four areas of algebra, geometry, dynamical symmetries and conservation laws and mathematical physics and applications, the book covers deformation theory and quantization; Hom-algebras and n-ary algebraic structures; Hopf algebra, integrable systems and related math structures; jet theory and Weil bundles; Lie theory and applications; non-commutative and Lie algebra and more. The papers explore the interplay between research in contemporary mathematics and physics concerned with generalizations of the main structures of Lie theory aimed at quantization, and discrete and non-commutative extensions of differential calculus and geometry, non-associative structures, actions of groups and semi-groups, non-commutative dynamics, non-commutative geometry and applications in physics and beyond. The book benefits a broad audience of researchers a...
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.)
Galilean contractions of W-algebras
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Jørgen Rasmussen
2017-09-01
Full Text Available Infinite-dimensional Galilean conformal algebras can be constructed by contracting pairs of symmetry algebras in conformal field theory, such as W-algebras. Known examples include contractions of pairs of the Virasoro algebra, its N=1 superconformal extension, or the W3 algebra. Here, we introduce a contraction prescription of the corresponding operator-product algebras, or equivalently, a prescription for contracting tensor products of vertex algebras. With this, we work out the Galilean conformal algebras arising from contractions of N=2 and N=4 superconformal algebras as well as of the W-algebras W(2,4, W(2,6, W4, and W5. The latter results provide evidence for the existence of a whole new class of W-algebras which we call Galilean W-algebras. We also apply the contraction prescription to affine Lie algebras and find that the ensuing Galilean affine algebras admit a Sugawara construction. The corresponding central charge is level-independent and given by twice the dimension of the underlying finite-dimensional Lie algebra. Finally, applications of our results to the characterisation of structure constants in W-algebras are proposed.
Tensor spaces and exterior algebra
Yokonuma, Takeo
1992-01-01
This book explains, as clearly as possible, tensors and such related topics as tensor products of vector spaces, tensor algebras, and exterior algebras. You will appreciate Yokonuma's lucid and methodical treatment of the subject. This book is useful in undergraduate and graduate courses in multilinear algebra. Tensor Spaces and Exterior Algebra begins with basic notions associated with tensors. To facilitate understanding of the definitions, Yokonuma often presents two or more different ways of describing one object. Next, the properties and applications of tensors are developed, including the classical definition of tensors and the description of relative tensors. Also discussed are the algebraic foundations of tensor calculus and applications of exterior algebra to determinants and to geometry. This book closes with an examination of algebraic systems with bilinear multiplication. In particular, Yokonuma discusses the theory of replicas of Chevalley and several properties of Lie algebras deduced from them.
International Nuclear Information System (INIS)
Yau, Donald
2011-01-01
We study a twisted generalization of Novikov algebras, called Hom-Novikov algebras, in which the two defining identities are twisted by a linear map. It is shown that Hom-Novikov algebras can be obtained from Novikov algebras by twisting along any algebra endomorphism. All algebra endomorphisms on complex Novikov algebras of dimensions 2 or 3 are computed, and their associated Hom-Novikov algebras are described explicitly. Another class of Hom-Novikov algebras is constructed from Hom-commutative algebras together with a derivation, generalizing a construction due to Dorfman and Gel'fand. Two other classes of Hom-Novikov algebras are constructed from Hom-Lie algebras together with a suitable linear endomorphism, generalizing a construction due to Bai and Meng.
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 ...
International Nuclear Information System (INIS)
Froehlich, J.
1977-01-01
Sufficient conditions on unbounded, symmetric operators A and B which imply that exp(itA)exp(isB)exp(-itA) satisfies the well known 'multiple commutator' formula are derived. This formula is then applied to prove new necessary and sufficient conditions for the integrability of representations of Lie algebras and canonical commutation relations and the commutativity of the spectral projections of two commuting, unbounded, self-adjoint operators. A classic theorem of Nelson's is obtained as a corollary. Our results are useful in relativistic quantum field theory. (orig.) [de
Controllability of linear vector fields on Lie groups
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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
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
Lie groups and grand unified theories
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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
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
Critical analysis of algebraic collective models
International Nuclear Information System (INIS)
Moshinsky, M.
1986-01-01
The author shall understand by algebraic collective models all those based on specific Lie algebras, whether the latter are suggested through simple shell model considerations like in the case of the Interacting Boson Approximation (IBA), or have a detailed microscopic foundation like the symplectic model. To analyze these models critically, it is convenient to take a simple conceptual example of them in which all steps can be implemented analytically or through elementary numerical analysis. In this note he takes as an example the symplectic model in a two dimensional space i.e. based on a sp(4,R) Lie algebra, and show how through its complete discussion we can get a clearer understanding of the structure of algebraic collective models of nuclei. In particular he discusses the association of Hamiltonians, related to maximal subalgebras of our basic Lie algebra, with specific types of spectra, and the connections between spectra and shapes
Quantum cluster algebra structures on quantum nilpotent algebras
Goodearl, K R
2017-01-01
All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to these problems, including the construction of quantum cluster algebra structures on quantum unipotent groups and quantum double Bruhat cells (the Berenstein-Zelevinsky conjecture), and treat these problems from a unified perspective. All such applications also establish equality between the constructed quantum cluster algebras and their upper counterparts.
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Kedem, Rinat
2008-01-01
Q-systems first appeared in the analysis of the Bethe equations for the XXX model and generalized Heisenberg spin chains (Kirillov and Reshetikhin 1987 Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Steklov. 160 211-21, 301). Such systems are known to exist for any simple Lie algebra and many other Kac-Moody algebras. We formulate the Q-system associated with any simple, simply-laced Lie algebras g in the language of cluster algebras (Fomin and Zelevinsky 2002 J. Am. Math. Soc. 15 497-529), and discuss the relation of the polynomiality property of the solutions of the Q-system in the initial variables, which follows from the representation-theoretical interpretation, to the Laurent phenomenon in cluster algebras (Fomin and Zelevinsky 2002 Adv. Appl. Math. 28 119-44)
Czech Academy of Sciences Publication Activity Database
Sati, H.; Schreiber, Urs
2017-01-01
Roč. 2017, č. 3 (2017), č. článku 87. ISSN 1126-6708 Institutional support: RVO:67985840 Keywords : Differential and Algebra ic Geometry * p-branes Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics http://link.springer.com/article/10.1007%2FJHEP03%282017%29087
Uncertainty relations and semi-groups in B-algebras
International Nuclear Information System (INIS)
Papaloucas, L.C.
1980-07-01
Starting from a B-algebra which satisfies the conditions of a structure theorem, we obtain directly a Lie algebra for which the Lie ring satisfies automatically the Heisenberg uncertainty relations. (author)
The central extensions of Kac-Moody-Malcev algebras
International Nuclear Information System (INIS)
Osipov, E.P.
1989-01-01
The authors introduce a class of infinite-dimensional Kac-Moody-Malcev algebras. The Kac-Moody-Malcev algebras are the generalization of Lie algebras of Kac-Moody type to the Malcev algebras. They demonstrate that the central extensions of Kac-Moody-Malcev algebras are given by the same cocycles as in the case of Lie algebras. It is given a construction of Virasoro algebra in terms of bilinear combinations of currents satisfying the Kac-Moody-Malcev commutation relations. Thus, it is given the generalization of the Sugawara Construction to the case of Kac-Moody-Malcev algebras. Analogues of Kac-Moody-Malcev algebras may be also introduced in the case of arbitrary Riemann surface
Goto, Masami; Abe, Osamu; Aoki, Shigeki; Hayashi, Naoto; Miyati, Tosiaki; Takao, Hidemasa; Iwatsubo, Takeshi; Yamashita, Fumio; Matsuda, Hiroshi; Mori, Harushi; Kunimatsu, Akira; Ino, Kenji; Yano, Keiichi; Ohtomo, Kuni
2013-07-01
This study aimed to investigate whether the effect of scanner for cortex volumetry with atlas-based method is reduced using Diffeomorphic Anatomical Registration Through Exponentiated Lie Algebra (DARTEL) normalization compared with standard normalization. Three-dimensional T1-weighted magnetic resonance images (3D-T1WIs) of 21 healthy subjects were obtained and evaluated for effect of scanner in cortex volumetry. 3D-T1WIs of the 21 subjects were obtained with five MRI systems. Imaging of each subject was performed on each of five different MRI scanners. We used the Voxel-Based Morphometry 8 tool implemented in Statistical Parametric Mapping 8 and WFU PickAtlas software (Talairach brain atlas theory). The following software default settings were used as bilateral region-of-interest labels: "Frontal Lobe," "Hippocampus," "Occipital Lobe," "Orbital Gyrus," "Parietal Lobe," "Putamen," and "Temporal Lobe." Effect of scanner for cortex volumetry using the atlas-based method was reduced with DARTEL normalization compared with standard normalization in Frontal Lobe, Occipital Lobe, Orbital Gyrus, Putamen, and Temporal Lobe; was the same in Hippocampus and Parietal Lobe; and showed no increase with DARTEL normalization for any region of interest (ROI). DARTEL normalization reduces the effect of scanner, which is a major problem in multicenter studies.
International Nuclear Information System (INIS)
Goto, Masami; Ino, Kenji; Yano, Keiichi; Abe, Osamu; Aoki, Shigeki; Hayashi, Naoto; Miyati, Tosiaki; Takao, Hidemasa; Mori, Harushi; Kunimatsu, Akira; Ohtomo, Kuni; Iwatsubo, Takeshi; Yamashita, Fumio; Matsuda, Hiroshi
2013-01-01
This study aimed to investigate whether the effect of scanner for cortex volumetry with atlas-based method is reduced using Diffeomorphic Anatomical Registration Through Exponentiated Lie Algebra (DARTEL) normalization compared with standard normalization. Three-dimensional T1-weighted magnetic resonance images (3D-T1WIs) of 21 healthy subjects were obtained and evaluated for effect of scanner in cortex volumetry. 3D-T1WIs of the 21 subjects were obtained with five MRI systems. Imaging of each subject was performed on each of five different MRI scanners. We used the Voxel-Based Morphometry 8 tool implemented in Statistical Parametric Mapping 8 and WFU PickAtlas software (Talairach brain atlas theory). The following software default settings were used as bilateral region-of-interest labels: ''Frontal Lobe,'' ''Hippocampus,'' ''Occipital Lobe,'' ''Orbital Gyrus,'' ''Parietal Lobe,'' ''Putamen,'' and ''Temporal Lobe.'' Effect of scanner for cortex volumetry using the atlas-based method was reduced with DARTEL normalization compared with standard normalization in Frontal Lobe, Occipital Lobe, Orbital Gyrus, Putamen, and Temporal Lobe; was the same in Hippocampus and Parietal Lobe; and showed no increase with DARTEL normalization for any region of interest (ROI). DARTEL normalization reduces the effect of scanner, which is a major problem in multicenter studies. (orig.)
Coherent states and classical limit of algebraic quantum models
International Nuclear Information System (INIS)
Scutaru, H.
1983-01-01
The algebraic models for collective motion in nuclear physics belong to a class of theories the basic observables of which generate selfadjoint representations of finite dimensional, real Lie algebras, or of the enveloping algebras of these Lie algebras. The simplest and most used for illustrations model of this kind is the Lipkin model, which is associated with the Lie algebra of the three dimensional rotations group, and which presents all characteristic features of an algebraic model. The Lipkin Hamiltonian is the image, of an element of the enveloping algebra of the algebra SO under a representation. In order to understand the structure of the algebraic models the author remarks that in both classical and quantum mechanics the dynamics is associated to a typical algebraic structure which we shall call a dynamical algebra. In this paper he shows how the constructions can be made in the case of the algebraic quantum systems. The construction of the symplectic manifold M can be made in this case using a quantum analog of the momentum map which he defines
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
The BRS algebra of a free differential algebra
International Nuclear Information System (INIS)
Boukraa, S.
1987-04-01
We construct in this work, the Weil and the universal BRS algebras of theories that can have as a gauge symmetry a free differential (Sullivan) algebra, the natural extension of Lie algebras allowing the definition of p-form gauge potentials (p>1). The finite gauge transformations of these potentials are deduced from the infinitesimal ones and the group structure is shown. The geometrical meaning of these p-form gauge potentials is given by the notion of a Quillen superconnection. (author). 19 refs
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-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
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)
Automorphism modular invariants of current algebras
International Nuclear Information System (INIS)
Gannon, T.; Walton, M.A.
1996-01-01
We consider those two-dimensional rational conformal field theories (RCFTs) whose chiral algebras, when maximally extended, are isomorphic to the current algebra formed from some untwisted affine Lie algebra at fixed level. In this case the partition function is specified by an automorphism of the fusion ring and corresponding symmetry of the Kac-Peterson modular matrices. We classify all such partition functions when the underlying finite-dimensional Lie algebra is simple. This gives all possible spectra for this class of RCFTs. While accomplishing this, we also find the primary fields with second smallest quantum dimension. (orig.). With 3 tabs
International Nuclear Information System (INIS)
Yu Zhang; Zhang Yufeng
2009-01-01
Three semi-direct sum Lie algebras are constructed, which is an efficient and new way to obtain discrete integrable couplings. As its applications, three discrete integrable couplings associated with the modified K dV lattice equation are worked out. The approach can be used to produce other discrete integrable couplings of the discrete hierarchies of soliton equations.
Universal R-matrix for quantized (super) algebras
International Nuclear Information System (INIS)
Khoroshkin, S.M.; Tolstoj, V.N.
1991-01-01
For quantum deformations of finite-dimensional contragredient Lie (super)algebras an explicit formula for the universal R-matrix is given. This formula generalizes the analogous formulae for quantized semisimple Lie algebras obtained by M. Rosso, A.N. Kirillov and N. Reshetikhin, Yas.S. Soibelman and S.Z. Levendorskii. Approach is based on careful analysis of quantized rank 1 and 2 (super)algebras, a combinatorial structure of the root systems and algebraic properties of q-exponential functions. Quantum Weyl group is not used. 19 refs.; 2 tabs
Non-Hermitian systems of Euclidean Lie algebraic type with real energy spectra
International Nuclear Information System (INIS)
Dey, Sanjib; Fring, Andreas; Mathanaranjan, Thilagarajah
2014-01-01
We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean–Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries exhibiting various types of qualitative behaviour. On the basis of explicitly computed non-perturbative Dyson maps we construct metric operators, isospectral Hermitian counterparts for which we solve the corresponding time-independent Schrödinger equation for specific choices of the coupling constants. In these cases general analytical expressions for the solutions are obtained in the form of Mathieu functions, which we analyze numerically to obtain the corresponding energy spectra. We identify regions in the parameter space for which the corresponding spectra are entirely real and also domains where the PT symmetry is spontaneously broken and sometimes also regained at exceptional points. In some cases it is shown explicitly how the threshold region from real to complex spectra is characterized by the breakdown of the Dyson maps or the metric operator. We establish the explicit relationship to models currently under investigation in the context of beam dynamics in optical lattices. -- Highlights: •Different PT-symmetries lead to qualitatively different systems. •Construction of non-perturbative Dyson maps and isospectral Hermitian counterparts. •Numerical discussion of the eigenvalue spectra for one of the E(2)-systems. •Established link to systems studied in the context of optical lattices. •Setup for the E(3)-algebra is provided
Representations of some quantum tori Lie subalgebras
International Nuclear Information System (INIS)
Jiang, Jingjing; Wang, Song
2013-01-01
In this paper, we define the q-analog Virasoro-like Lie subalgebras in x ∞ =a ∞ (b ∞ , c ∞ , d ∞ ). The embedding formulas into x ∞ are introduced. Irreducible highest weight representations of A(tilde sign) q , B(tilde sign) q , and C(tilde sign) q -series of the q-analog Virasoro-like Lie algebras in terms of vertex operators are constructed. We also construct the polynomial representations of the A(tilde sign) q , B(tilde sign) q , C(tilde sign) q , and D(tilde sign) q -series of the q-analog Virasoro-like Lie algebras.
An introduction of gauge field by the Lie-isotopic lifting of the Hilbert space
International Nuclear Information System (INIS)
Nishioka, M.
1984-01-01
It is introduced the gauge field by the Lie-isotopic lifting of the Hilbert space. Our method is different from other's in that the commutator between the isotropic element and the generators of the Lie algebra does not vanish in our case, but vanishes in other cases. Our method uses the Lie-isotopic lifting of the Hilbert space, but others do not use it
Formalization and Implementation of Algebraic Methods in Geometry
Directory of Open Access Journals (Sweden)
Filip Marić
2012-02-01
Full Text Available We describe our ongoing project of formalization of algebraic methods for geometry theorem proving (Wu's method and the Groebner bases method, their implementation and integration in educational tools. The project includes formal verification of the algebraic methods within Isabelle/HOL proof assistant and development of a new, open-source Java implementation of the algebraic methods. The project should fill-in some gaps still existing in this area (e.g., the lack of formal links between algebraic methods and synthetic geometry and the lack of self-contained implementations of algebraic methods suitable for integration with dynamic geometry tools and should enable new applications of theorem proving in education.
Polynomial deformations of oscillator algebras in quantum theories with internal symmetries
International Nuclear Information System (INIS)
Karassiov, V.P.
1992-01-01
This paper reports that for last years some new Lie-algebraic structures (quantum groups or algebras, W-algebras, Casimir algebras) have been introduced in different areas of modern physics. All these objects are non-linear generalizations (deformations) of usual (linear) Lie algebras which are generated by a set B = {T a } of their generators T a satisfying a commutation relations (CR) of the form [T a , T b ] = f ab ({T c }) where f ab (...) are some functions of the generators T c given by power series. From the mathematical viewpoint such objects called as nonlinear or deformed Lie algebras G d may be treated as universal algebras or algebraic systems G d = left-angle B; +, · , [,] right-angle generated by a basic set B and the usual operations of the addition (+) and the multiplication (·) together with the Lie product ([T a , T b ] = T a T b - T b T a )
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 ...
On the q-deformation of certain infinite dimensional Lie algebras
International Nuclear Information System (INIS)
El Kinani, E.H.; Zakkari, M.
1995-07-01
A representation of the q-deformed centreless Virasoro algebra in terms of the Gauss derivatives D x and D y on the quantum plane C q [x,y] is given. Moreover, we obtain the deformed version of the algebra of the area-preserving diffeomorphisms of the torus T 2 . In the end, the correspondence between Psd(q,p,r) and the a-bar ∞ algebra is pointed out. (author). 11 refs
Loop homotopy algebras in closed string field theory
International Nuclear Information System (INIS)
Markl, M.
2001-01-01
Barton Zwiebach (1993) constructed ''string products'' on the Hilbert space of a combined conformal field theory of matter and ghosts, satisfying the ''main identity''. It has been well known that the ''tree level'' of the theory gives an example of a strongly homotopy Lie algebra (though, as we will see later, this is not the whole truth). Strongly homotopy Lie algebras are now well-understood objects. On the one hand, strongly homotopy Lie algebra is given by a square zero coderivation on the cofree cocommutative connected coalgebra on the other hand, strongly homotopy Lie algebras are algebras over the cobar dual of the operad Com for commutative algebras. No such characterization of the structure of string products for arbitrary genera has been available, though there are two series of papers directly pointing towards the requisite characterization. As far as the characterization in terms of (co)derivations is concerned, we need the concept of higher order (co)derivations. For our characterization we need to understand the behavior of these higher (co)derivations on (co)free (co)algebras. The necessary machinery for the operadic approach is that of modular operads. We also indicate how to adapt the loop homotopy structure to the case of open string field theory. (orig.)
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
Algebraic reduction of the 't Hooft-Polyakov monopole to the Dirac monopole
International Nuclear Information System (INIS)
Landi, G.; Marmo, G.
1988-01-01
In the context of the algebraic description of gauge fields by means of extensions of Lie algebras considered in previous articles by the authors, we define the notion of reduction of an extension of Lie algebras. Given a connection we define the holonomy algebra and the holonomy sequence of the connection and we prove that it is always possible to reduce the extension we start with to the holonomy sequence of the connection. As an example we construct a 't Hooft-Polyakov-like extension of algebras and reduce it to the extension which describes the Dirac monopole as discussed in a previous paper by the authors. The supersymmetric version of all results is obtained by replacing ordinary Lie algebras with Lie superalgebras. (orig.)
Energy Technology Data Exchange (ETDEWEB)
Matone, Marco [Universita di Padova, Dipartimento di Fisica e Astronomia ' ' G. Galilei' ' , Padua (Italy); Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Padua (Italy)
2016-11-15
Recently it has been introduced an algorithm for the Baker-Campbell-Hausdorff (BCH) formula, which extends the Van-Brunt and Visser recent results, leading to new closed forms of BCH formula. More recently, it has been shown that there are 13 types of such commutator algebras. We show, by providing the explicit solutions, that these include the generators of the semisimple complex Lie algebras. More precisely, for any pair, X, Y of the Cartan-Weyl basis, we find W, linear combination of X, Y, such that exp(X) exp(Y) = exp(W). The derivation of such closed forms follows, in part, by using the above mentioned recent results. The complete derivation is provided by considering the structure of the root system. Furthermore, if X, Y, and Z are three generators of the Cartan-Weyl basis, we find, for a wide class of cases, W, a linear combination of X, Y and Z, such that exp(X) exp(Y) exp(Z) = exp(W). It turns out that the relevant commutator algebras are type 1c-i, type 4 and type 5. A key result concerns an iterative application of the algorithm leading to relevant extensions of the cases admitting closed forms of the BCH formula. Here we provide the main steps of such an iteration that will be developed in a forthcoming paper. (orig.)
International Nuclear Information System (INIS)
Matone, Marco
2016-01-01
Recently it has been introduced an algorithm for the Baker-Campbell-Hausdorff (BCH) formula, which extends the Van-Brunt and Visser recent results, leading to new closed forms of BCH formula. More recently, it has been shown that there are 13 types of such commutator algebras. We show, by providing the explicit solutions, that these include the generators of the semisimple complex Lie algebras. More precisely, for any pair, X, Y of the Cartan-Weyl basis, we find W, linear combination of X, Y, such that exp(X) exp(Y) = exp(W). The derivation of such closed forms follows, in part, by using the above mentioned recent results. The complete derivation is provided by considering the structure of the root system. Furthermore, if X, Y, and Z are three generators of the Cartan-Weyl basis, we find, for a wide class of cases, W, a linear combination of X, Y and Z, such that exp(X) exp(Y) exp(Z) = exp(W). It turns out that the relevant commutator algebras are type 1c-i, type 4 and type 5. A key result concerns an iterative application of the algorithm leading to relevant extensions of the cases admitting closed forms of the BCH formula. Here we provide the main steps of such an iteration that will be developed in a forthcoming paper. (orig.)
Differential geometry on Hopf algebras and quantum groups
International Nuclear Information System (INIS)
Watts, P.
1994-01-01
The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the aforementioned structures for quantum groups are determined
Kimura, Taro; Pestun, Vasily
2018-06-01
For a quiver with weighted arrows, we define gauge-theory K-theoretic W-algebra generalizing the definition of Shiraishi et al. and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented by Nekrasov is isomorphic to the definition of the W-algebra in the operator formalism as a commutant of screening charges in the free field representation. Besides, we allow arbitrary quiver and expect interesting applications to representation theory of generalized Borcherds-Kac-Moody Lie algebras, their quantum affinizations and associated W-algebras.
On Quantum Lie Nilpotency of Order 2
Directory of Open Access Journals (Sweden)
E. A. Kireeva
2016-01-01
Full Text Available The paper investigates the free algebras of varieties of associative algebras modulo identities of quantum Lie nilpotency of order 1 and 2. Let q be an invertible element of the ground field K (or of its extension. The element[x,y]q = xy-qyxof the free associative algebra is called a quantum commutator. We consider the algebras modulo identities [x,y]q = 0 (1and [[x,y]q ,z]q = 0. (2It is natural to consider the aforementioned algebras as the quantum analogs of commutative algebras and algebras of Lie nilpotency of order 2. The free algebras of the varieties of associative algebras modulo the identity of Lie nilpotency of order 2, that is the identity[[x,y] ,z] =0,where [x,y]=xy-yx is a Lie commutator, are of great interest in the theory of algebras with polynomial identities, since it was proved by A.V.Grishin for algebras over fields of characteristic 2, and V.V.Shchigolev for algebras over fields of characteristic p>2, that these algebras contain non-finitely generated T-spaces.We prove in the paper that the algebras modulo identities (1 and (2 are nilpotent in the usual sense and calculate precisely the nilpotency order of these algebras. More precisely, we prove that the free algebra of the variety of associative algebras modulo identity (1 is nilpotent of order 2 if q ≠ ± 1, and nilpotent of order 3 if q = - 1 and the characteristic of K is not equal to 2. It is also proved that the free algebra of the variety of associative algebras modulo identity (2 is nilpotent of order 3 if q3 ≠ 1, q ≠ ± 1, nilpotent of order 4 if q3 = 1, q ≠ 1, and nilpotent of
A program for computing cohomology of Lie superalgebras of vector fields
International Nuclear Information System (INIS)
Kornyak, V.V.
1998-01-01
An algorithm and its C implementation for computing the cohomology of Lie algebras and superalgebras is described. When elaborating the algorithm we paid primary attention to cohomology in trivial, adjoint and coadjoint modules for Lie algebras and superalgebras of the formal vector fields. These algebras have found many applications to modern supersymmetric models of theoretical and mathematical physics. As an example, we present 3- and 5-cocycles from the cohomology in the trivial module for the Poisson algebra Po (2), as found by computer
Contraction of graded su(2) algebra
International Nuclear Information System (INIS)
Patra, M.K.; Tripathy, K.C.
1989-01-01
The Inoenu-Wigner contraction scheme is extended to Lie superalgebras. The structure and representations of extended BRS algebra are obtained from contraction of the graded su(2) algebra. From cohomological consideration, we demonstrate that the graded su(2) algebra is the only superalgebra which, on contraction, yields the full BRS algebra. (orig.)
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
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).
International Nuclear Information System (INIS)
Chen Famin; Wu Yongshi
2010-01-01
We present a superspace formulation of the D=3, N=4, 5 superconformal Chern-Simons Matter theories, with matter supermultiplets valued in a symplectic 3-algebra. We first construct an N=1 superconformal action and then generalize a method used by Gaitto and Witten to enhance the supersymmetry from N=1 to N=5. By decomposing the N=5 supermultiplets and the symplectic 3-algebra properly and proposing a new superpotential term, we construct the N=4 superconformal Chern-Simons matter theories in terms of two sets of generators of a (quaternion) symplectic 3-algebra. The N=4 theories can also be derived by requiring that the supersymmetry transformations are closed on-shell. The relationship between the 3-algebras, Lie superalgebras, Lie algebras, and embedding tensors (proposed in [E. A. Bergshoeff, O. Hohm, D. Roest, H. Samtleben, and E. Sezgin, J. High Energy Phys. 09 (2008) 101.]) is also clarified. The general N=4, 5 superconformal Chern-Simons matter theories in terms of ordinary Lie algebras can be re-derived in our 3-algebra approach. All known N=4, 5 superconformal Chern-Simons matter theories can be recovered in the present superspace formulation for super-Lie algebra realization of symplectic 3-algebras.
Quantum cluster algebras and quantum nilpotent algebras
Goodearl, Kenneth R.; Yakimov, Milen T.
2014-01-01
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197
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)
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)
On the Fock space realizations of nonlinear algebras describing the high spin fields in AdS spaces
International Nuclear Information System (INIS)
Burdik, C.; Navratil, O.; Pashnev, A.
2002-01-01
The method of construction of Fock space realizations of Lie algebras is generalized for nonlinear algebras. We consider as an example the nonlinear algebra of constraints which describe the totally symmetric fields with higher spins in the AdS space-time
Computer methods in general relativity: algebraic computing
Araujo, M E; Skea, J E F; Koutras, A; Krasinski, A; Hobill, D; McLenaghan, R G; Christensen, S M
1993-01-01
Karlhede & MacCallum [1] gave a procedure for determining the Lie algebra of the isometry group of an arbitrary pseudo-Riemannian manifold, which they intended to im- plement using the symbolic manipulation package SHEEP but never did. We have recently ﬁnished making this procedure explicit by giving an algorithm suitable for implemen- tation on a computer [2]. Specifically, we have written an algorithm for determining the isometry group of a spacetime (in four dimensions), and partially implemented this algorithm using the symbolic manipulation package CLASSI, which is an extension of SHEEP.
Introduction to quantum algebras
International Nuclear Information System (INIS)
Kibler, M.R.
1992-09-01
The concept of a quantum algebra is made easy through the investigation of the prototype algebras u qp (2), su q (2) and u qp (1,1). The latter quantum algebras are introduced as deformations of the corresponding Lie algebras; this is achieved in a simple way by means of qp-bosons. The Hopf algebraic structure of u qp (2) is also discussed. The basic ingredients for the representation theory of u qp (2) are given. Finally, in connection with the quantum algebra u qp (2), the qp-analogues of the harmonic oscillator are discussed and of the (spherical and hyperbolical) angular momenta. (author) 50 refs
Quantum Heisenberg groups and Sklyanin algebras
International Nuclear Information System (INIS)
Andruskiewitsch, N.; Devoto, J.; Tiraboschi, A.
1993-05-01
We define new quantizations of the Heisenberg group by introducing new quantizations in the universal enveloping algebra of its Lie algebra. Matrix coefficients of the Stone-von Neumann representation are preserved by these new multiplications on the algebra of functions on the Heisenberg group. Some of the new quantizations provide also a new multiplication in the algebra of theta functions; we obtain in this way Sklyanin algebras. (author). 23 refs
Contraction-based classification of supersymmetric extensions of kinematical lie algebras
International Nuclear Information System (INIS)
Campoamor-Stursberg, R.; Rausch de Traubenberg, M.
2010-01-01
We study supersymmetric extensions of classical kinematical algebras from the point of view of contraction theory. It is shown that contracting the supersymmetric extension of the anti-de Sitter algebra leads to a hierarchy similar in structure to the classical Bacry-Levy-Leblond classification.
Algebraic monoids, group embeddings, and algebraic combinatorics
Li, Zhenheng; Steinberg, Benjamin; Wang, Qiang
2014-01-01
This book contains a collection of fifteen articles and is dedicated to the sixtieth birthdays of Lex Renner and Mohan Putcha, the pioneers of the field of algebraic monoids. Topics presented include: v structure and representation theory of reductive algebraic monoids v monoid schemes and applications of monoids v monoids related to Lie theory v equivariant embeddings of algebraic groups v constructions and properties of monoids from algebraic combinatorics v endomorphism monoids induced from vector bundles v Hodge–Newton decompositions of reductive monoids A portion of these articles are designed to serve as a self-contained introduction to these topics, while the remaining contributions are research articles containing previously unpublished results, which are sure to become very influential for future work. Among these, for example, the important recent work of Michel Brion and Lex Renner showing that the algebraic semigroups are strongly π-regular. Graduate students as well a...
Robust Algebraic Multilevel Methods and Algorithms
Kraus, Johannes
2009-01-01
This book deals with algorithms for the solution of linear systems of algebraic equations with large-scale sparse matrices, with a focus on problems that are obtained after discretization of partial differential equations using finite element methods. Provides a systematic presentation of the recent advances in robust algebraic multilevel methods. Can be used for advanced courses on the topic.
An analogue of Wagner's theorem for decompositions of matrix algebras
International Nuclear Information System (INIS)
Ivanov, D N
2004-01-01
Wagner's celebrated theorem states that a finite affine plane whose collineation group is transitive on lines is a translation plane. The notion of an orthogonal decomposition (OD) of a classically semisimple associative algebra introduced by the author allows one to draw an analogy between finite affine planes of order n and ODs of the matrix algebra M n (C) into a sum of subalgebras conjugate to the diagonal subalgebra. These ODs are called WP-decompositions and are equivalent to the well-known ODs of simple Lie algebras of type A n-1 into a sum of Cartan subalgebras. In this paper we give a detailed and improved proof of the analogue of Wagner's theorem for WP-decompositions of the matrix algebra of odd non-square order an outline of which was earlier published in a short note in 'Russian Math. Surveys' in 1994. In addition, in the framework of the theory of ODs of associative algebras, based on the method of idempotent bases, we obtain an elementary proof of the well-known Kostrikin-Tiep theorem on irreducible ODs of Lie algebras of type A n-1 in the case where n is a prime-power.
Classical versus Computer Algebra Methods in Elementary Geometry
Pech, Pavel
2005-01-01
Computer algebra methods based on results of commutative algebra like Groebner bases of ideals and elimination of variables make it possible to solve complex, elementary and non elementary problems of geometry, which are difficult to solve using a classical approach. Computer algebra methods permit the proof of geometric theorems, automatic…
Energy Technology Data Exchange (ETDEWEB)
Goto, Masami; Ino, Kenji; Yano, Keiichi [University of Tokyo Hospital, Department of Radiological Technology, Bunkyo-ku, Tokyo (Japan); Abe, Osamu [Nihon University School of Medicine, Department of Radiology, Itabashi-ku, Tokyo (Japan); Aoki, Shigeki [Juntendo University, Department of Radiology, Bunkyo-ku, Tokyo (Japan); Hayashi, Naoto [University of Tokyo Hospital, Department of Computational Diagnostic Radiology and Preventive Medicine, Bunkyo-ku, Tokyo (Japan); Miyati, Tosiaki [Kanazawa University, Graduate School of Medical Science, Kanazawa (Japan); Takao, Hidemasa; Mori, Harushi; Kunimatsu, Akira; Ohtomo, Kuni [University of Tokyo Hospital, Department of Radiology and Department of Computational Diagnostic Radiology and Preventive Medicine, Bunkyo-ku, Tokyo (Japan); Iwatsubo, Takeshi [University of Tokyo, Department of Neuropathology, Bunkyo-ku, Tokyo (Japan); Yamashita, Fumio [Iwate Medical University, Department of Radiology, Yahaba, Iwate (Japan); Matsuda, Hiroshi [Integrative Brain Imaging Center National Center of Neurology and Psychiatry, Department of Nuclear Medicine, Kodaira, Tokyo (Japan); Collaboration: Japanese Alzheimer' s Disease Neuroimaging Initiative
2013-07-15
This study aimed to investigate whether the effect of scanner for cortex volumetry with atlas-based method is reduced using Diffeomorphic Anatomical Registration Through Exponentiated Lie Algebra (DARTEL) normalization compared with standard normalization. Three-dimensional T1-weighted magnetic resonance images (3D-T1WIs) of 21 healthy subjects were obtained and evaluated for effect of scanner in cortex volumetry. 3D-T1WIs of the 21 subjects were obtained with five MRI systems. Imaging of each subject was performed on each of five different MRI scanners. We used the Voxel-Based Morphometry 8 tool implemented in Statistical Parametric Mapping 8 and WFU PickAtlas software (Talairach brain atlas theory). The following software default settings were used as bilateral region-of-interest labels: ''Frontal Lobe,'' ''Hippocampus,'' ''Occipital Lobe,'' ''Orbital Gyrus,'' ''Parietal Lobe,'' ''Putamen,'' and ''Temporal Lobe.'' Effect of scanner for cortex volumetry using the atlas-based method was reduced with DARTEL normalization compared with standard normalization in Frontal Lobe, Occipital Lobe, Orbital Gyrus, Putamen, and Temporal Lobe; was the same in Hippocampus and Parietal Lobe; and showed no increase with DARTEL normalization for any region of interest (ROI). DARTEL normalization reduces the effect of scanner, which is a major problem in multicenter studies. (orig.)
Shafarevich, Igor Rostislavovich
2005-01-01
This book is wholeheartedly recommended to every student or user of mathematics. Although the author modestly describes his book as 'merely an attempt to talk about' algebra, he succeeds in writing an extremely original and highly informative essay on algebra and its place in modern mathematics and science. From the fields, commutative rings and groups studied in every university math course, through Lie groups and algebras to cohomology and category theory, the author shows how the origins of each algebraic concept can be related to attempts to model phenomena in physics or in other branches
Supersymmetry in physics: an algebraic overview
International Nuclear Information System (INIS)
Ramond, P.
1983-01-01
In 1970, while attempting to generalize the Veneziano model (string model) to include fermions, I introduced a new algebraic structure which turned out to be a graded Lie algebra; it was used as a spectrum-generating algebra. This approach was soon after generalized to include interactions, yielding a complete model of fermions and boson (RNS model). In an unrelated work in the Soviet Union, it was shown how to generalize the Poincare group to include fermionic charges. However it was not until 1974 that an interacting field theory invariant under the Graded Poincare group in 3 + 1 dimensions was built (WZ model). Supersymmetric field theories turned out to have less divergent ultraviolet behavior than non-supersymmetric field theories. Gravity was generalized to include supersymmetry, to a theory called supergravity. By now many interacting local field theories exhibiting supersymmetry have been built and studied from 1 + 1 to 10 + 1 dimensions. Supersymmetric local field theories in less than 9 + 1 dimensions, can be understood as limits of multilocal (string) supersymmetric theories, in 9 + 1 dimensions. On the other hand, graded Lie algebras have been used in non-relativistic physics as approximate symmetries of Hamiltonians. The most striking such use so far helps comparing even and odd nuclei energy levels. It is believed that graded Lie algebras can be used whenever paired and unpaired fermions excitations can coexist. In this overview of a tremendously large field, I will only survey finite graded Lie algebras and their representations. For non-relativistic applications, all of GLA are potentially useful, while for relativistic applications, only these which include the Poincare group are to be considered
The algebra and geometry of SU(3) matrices
Mallesh, KS; Mukunda, N
1997-01-01
We give an elementary treatment of the defining representation and Lie algebra of the three-dimensional unitary unimodular group SU(3). The geometrical properties of the Lie algebra, which is an eight dimensional real Linear vector space, are developed in an SU(3) covariant manner. The f and d symbols of SU(3) lead to two ways of 'multiplying' two vectors to produce a third, and several useful geometric and algebraic identities are derived. The axis-angle parametrization of SU(3) is developed...
Brauer algebras of simply laced type
Cohen, A.M.; Frenk, B.J.; Wales, D.B.
2009-01-01
The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A n - 1 on n - 1 nodes. Here we describe an algebra
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.
Non-Hermitian systems of Euclidean Lie algebraic type with real energy spectra
Dey, Sanjib; Fring, Andreas; Mathanaranjan, Thilagarajah
2014-07-01
We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean-Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries exhibiting various types of qualitative behaviour. On the basis of explicitly computed non-perturbative Dyson maps we construct metric operators, isospectral Hermitian counterparts for which we solve the corresponding time-independent Schrödinger equation for specific choices of the coupling constants. In these cases general analytical expressions for the solutions are obtained in the form of Mathieu functions, which we analyze numerically to obtain the corresponding energy spectra. We identify regions in the parameter space for which the corresponding spectra are entirely real and also domains where the PT symmetry is spontaneously broken and sometimes also regained at exceptional points. In some cases it is shown explicitly how the threshold region from real to complex spectra is characterized by the breakdown of the Dyson maps or the metric operator. We establish the explicit relationship to models currently under investigation in the context of beam dynamics in optical lattices.
Novikov algebras with associative bilinear forms
Energy Technology Data Exchange (ETDEWEB)
Zhu Fuhai; Chen Zhiqi [School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071 (China)
2007-11-23
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. The goal of this paper is to study Novikov algebras with non-degenerate associative symmetric bilinear forms, which we call quadratic Novikov algebras. Based on the classification of solvable quadratic Lie algebras of dimension not greater than 4 and Novikov algebras in dimension 3, we show that quadratic Novikov algebras up to dimension 4 are commutative. Furthermore, we obtain the classification of transitive quadratic Novikov algebras in dimension 4. But we find that not every quadratic Novikov algebra is commutative and give a non-commutative quadratic Novikov algebra in dimension 6.
Semi-infinite Weil complex and the Virasoro algebra
International Nuclear Information System (INIS)
Feigin, B.; Frenkel, E.
1991-01-01
We define a semi-infinite analogue of the Weil algebra associated with an infinite-dimensional Lie algebra. It can be used for the definition of semi-infinite characteristic classes by analogy with the Chern-Weil construction. The second term of a spectral sequence of this Weil complex consists of the semi-infinite cohomology of the Lie algebra with coefficients in its 'adjoint semi-infinite symmetric powers'. We compute this cohomology for the Virasoro algebra. This is just the BRST cohomology of the bosonic βγ-system with the central charge 26. We give a complete description of the Fock representations of this bosonic system as modules over the Virasoro algebra, using Friedan-Martinec-Shenker bosonization. We derive a combinatorial identity from this result. (orig.)
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
Algebraic conformal field theory
International Nuclear Information System (INIS)
Fuchs, J.; Nationaal Inst. voor Kernfysica en Hoge-Energiefysica
1991-11-01
Many conformal field theory features are special versions of structures which are present in arbitrary 2-dimensional quantum field theories. So it makes sense to describe 2-dimensional conformal field theories in context of algebraic theory of superselection sectors. While most of the results of the algebraic theory are rather abstract, conformal field theories offer the possibility to work out many formulae explicitly. In particular, one can construct the full algebra A-bar of global observables and the endomorphisms of A-bar which represent the superselection sectors. Some explicit results are presented for the level 1 so(N) WZW theories; the algebra A-bar is found to be the enveloping algebra of a Lie algebra L-bar which is an extension of the chiral symmetry algebra of the WZW theory. (author). 21 refs., 6 figs
Lie algebras for the Dirac-Clifford ring
International Nuclear Information System (INIS)
Mignaco, J.A.; Linhares, C.A.
1992-01-01
It is shown in a general way that the Dirac-Clifford ring formed by the Dirac matrices and all their products, for all even and odd spacetime dimensions D, span the cumulation algebras SU(2 D/2 ) for even D and SU(2 (D- 1 )/2 ) + SU(2 (D-1)/2 ) for odd D. Some physical consequences of these results are discussed. (author)
L{sub ∞} algebras and field theory
Energy Technology Data Exchange (ETDEWEB)
Hohm, Olaf [Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY (United States); Zwiebach, Barton [Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA (United States)
2017-03-15
We review and develop the general properties of L{sub ∞} algebras focusing on the gauge structure of the associated field theories. Motivated by the L{sub ∞} 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{sub ∞} 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{sub ∞} 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{sub ∞} algebra for the interacting theory. The analysis suggests that L{sub ∞} algebras provide a classification of perturbative gauge invariant classical field theories. (copyright 2017 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
International Nuclear Information System (INIS)
Guenaydin, M.
1979-05-01
Quadratic Jordan formulation of quantum mechanics in terms of Jordan triple product is presented. This formulation extends to the case of octonionic quantum mechanics for which no Hilbert space formulation exists. Using ternary algebraic techniques we then five the constructions of the derivation, structure and Tits-Koecher (Moebius) algebras of Jordan superalgebras. (orig.) [de
Block (or Hamiltonian) Lie Symmetry of Dispersionless D-Type Drinfeld–Sokolov Hierarchy
International Nuclear Information System (INIS)
Li Chuan-Zhong; He Jing-Song; Su Yu-Cai
2014-01-01
In this paper, the dispersionless D-type Drinfeld–Sokolov hierarchy, i.e. a reduction of the dispersionless two-component BKP hierarchy, is studied. The additional symmetry flows of this hierarchy are presented. These flows form an infinite-dimensional Lie algebra of Block type as well as a Lie algebra of Hamiltonian type
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.
Algebra and topology for applications to physics
Rozhkov, S. S.
1987-01-01
The principal concepts of algebra and topology are examined with emphasis on applications to physics. In particular, attention is given to sets and mapping; topological spaces and continuous mapping; manifolds; and topological groups and Lie groups. The discussion also covers the tangential spaces of the differential manifolds, including Lie algebras, vector fields, and differential forms, properties of differential forms, mapping of tangential spaces, and integration of differential forms.
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
Methods of algebraic geometry in control theory
Falb, Peter
1999-01-01
"Control theory represents an attempt to codify, in mathematical terms, the principles and techniques used in the analysis and design of control systems. Algebraic geometry may, in an elementary way, be viewed as the study of the structure and properties of the solutions of systems of algebraic equations. The aim of this book is to provide access to the methods of algebraic geometry for engineers and applied scientists through the motivated context of control theory" .* The development which culminated with this volume began over twenty-five years ago with a series of lectures at the control group of the Lund Institute of Technology in Sweden. I have sought throughout to strive for clarity, often using constructive methods and giving several proofs of a particular result as well as many examples. The first volume dealt with the simplest control systems (i.e., single input, single output linear time-invariant systems) and with the simplest algebraic geometry (i.e., affine algebraic geometry). While this is qui...
The quantum poisson-Lie T-duality and mirror symmetry
International Nuclear Information System (INIS)
Parkhomenko, S.E.
1999-01-01
Poisson-Lie T-duality in quantum N=2 superconformal Wess-Zumino-Novikov-Witten models is considered. The Poisson-Lie T-duality transformation rules of the super-Kac-Moody algebra currents are found from the conjecture that, as in the classical case, the quantum Poisson-Lie T-duality transformation is given by an automorphism which interchanges the isotropic subalgebras of the underlying Manin triple in one of the chirality sectors of the model. It is shown that quantum Poisson-Lie T-duality acts on the N=2 super-Virasoro algebra generators of the quantum models as a mirror symmetry acts: in one of the chirality sectors it is a trivial transformation while in another chirality sector it changes the sign of the U(1) current and interchanges the spin-3/2 currents. A generalization of Poisson-Lie T-duality for the quantum Kazama-Suzuki models is proposed. It is shown that quantum Poisson-Lie T-duality acts in these models as a mirror symmetry also
Indian Academy of Sciences (India)
project of the Spanish Ministerio de Educación y Ciencia MTM2007-60333. References. [1] Calderón A J, On split Lie algebras with symmetric root systems, Proc. Indian. Acad. Sci (Math. Sci.) 118(2008) 351–356. [2] Calderón A J, On split Lie triple systems, Proc. Indian. Acad. Sci (Math. Sci.) 119(2009). 165–177.
An su(1, 1) algebraic approach for the relativistic Kepler-Coulomb problem
International Nuclear Information System (INIS)
Salazar-Ramirez, M; Granados, V D; MartInez, D; Mota, R D
2010-01-01
We apply the Schroedinger factorization method to the radial second-order equation for the relativistic Kepler-Coulomb problem. From these operators we construct two sets of one-variable radial operators which are realizations for the su(1, 1) Lie algebra. We use this algebraic structure to obtain the energy spectrum and the supersymmetric ground state for this system.
A note on the Akivis algebra of a smooth hyporeductive loop
International Nuclear Information System (INIS)
Issa, A.N.
2002-05-01
Using the fundamental tensors of a smooth loop and the differential geometric characterization of smooth hyporeductive loops, the Akivis operations of a local smooth hyporeductive loop are expressed through the two binary and the one ternary operations of the hyporeductive triple algebra (h.t.a.) associated with the given hyporeductive loop. Those Akivis operations are also given in terms of Lie brackets of a Lie algebra of vector fields with the hyporeductive decomposition which generalizes the reductive decomposition of Lie algebras. A nontrivial real two-dimensional h.t.a. is presented. (author)
Analytical Lie-algebraic solution of a 3D sound propagation problem in the ocean
Energy Technology Data Exchange (ETDEWEB)
Petrov, P.S., E-mail: petrov@poi.dvo.ru [Il' ichev Pacific Oceanological Institute, 43 Baltiyskaya str., Vladivostok, 690041 (Russian Federation); Prants, S.V., E-mail: prants@poi.dvo.ru [Il' ichev Pacific Oceanological Institute, 43 Baltiyskaya str., Vladivostok, 690041 (Russian Federation); Petrova, T.N., E-mail: petrova.tn@dvfu.ru [Far Eastern Federal University, 8 Sukhanova str., 690950, Vladivostok (Russian Federation)
2017-06-21
The problem of sound propagation in a shallow sea with variable bottom slope is considered. The sound pressure field produced by a time-harmonic point source in such inhomogeneous 3D waveguide is expressed in the form of a modal expansion. The expansion coefficients are computed using the adiabatic mode parabolic equation theory. The mode parabolic equations are solved explicitly, and the analytical expressions for the modal coefficients are obtained using a Lie-algebraic technique. - Highlights: • A group-theoretical approach is applied to a problem of sound propagation in a shallow sea with variable bottom slope. • An analytical solution of this problem is obtained in the form of modal expansion with analytical expressions of the coefficients. • Our result is the only analytical solution of the 3D sound propagation problem with no translational invariance. • This solution can be used for the validation of the numerical propagation models.
On the huge Lie superalgebra of pseudo superdifferential operators and super KP-hierarchies
International Nuclear Information System (INIS)
Sedra, M.B.
1995-08-01
Lie superalgebraic methods are used to establish a connection between the huge Lie superalgebra Ξ of super (pseudo) differential operators and various super KP-hierarchies. We show in particular that Ξ splits into 5 = 2 x 2 + 1 graded algebras expected to correspond to five classes of super KP-hierarchies generalizing the well-known Manin-Radul and Figueroa O'Farrill-Ramos supersymmetric KP-hierarchies. (author). 10 refs
String field theory. Algebraic structure, deformation properties and superstrings
International Nuclear Information System (INIS)
Muenster, Korbinian
2013-01-01
This thesis discusses several aspects of string field theory. The first issue is bosonic open-closed string field theory and its associated algebraic structure - the quantum open-closed homotopy algebra. We describe the quantum open-closed homotopy algebra in the framework of homotopy involutive Lie bialgebras, as a morphism from the loop homotopy Lie algebra of closed string to the involutive Lie bialgebra on the Hochschild complex of open strings. The formulation of the classical/quantum open-closed homotopy algebra in terms of a morphism from the closed string algebra to the open string Hochschild complex reveals deformation properties of closed strings on open string field theory. In particular, we show that inequivalent classical open string field theories are parametrized by closed string backgrounds up to gauge transformations. At the quantum level the correspondence is obstructed, but for other realizations such as the topological string, a non-trivial correspondence persists. Furthermore, we proof the decomposition theorem for the loop homotopy Lie algebra of closed string field theory, which implies uniqueness of closed string field theory on a fixed conformal background. Second, the construction of string field theory can be rephrased in terms of operads. In particular, we show that the formulation of string field theory splits into two parts: The first part is based solely on the moduli space of world sheets and ensures that the perturbative string amplitudes are recovered via Feynman rules. The second part requires a choice of background and determines the real string field theory vertices. Each of these parts can be described equivalently as a morphism between appropriate cyclic and modular operads, at the classical and quantum level respectively. The algebraic structure of string field theory is then encoded in the composition of these two morphisms. Finally, we outline the construction of type II superstring field theory. Specific features of the
Algebraic Properties of First Integrals for Scalar Linear Third-Order ODEs of Maximal Symmetry
Directory of Open Access Journals (Sweden)
K. S. Mahomed
2013-01-01
Full Text Available By use of the Lie symmetry group methods we analyze the relationship between the first integrals of the simplest linear third-order ordinary differential equations (ODEs and their point symmetries. It is well known that there are three classes of linear third-order ODEs for maximal cases of point symmetries which are 4, 5, and 7. The simplest scalar linear third-order equation has seven-point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equation y′′′=0 is unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals.
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
Multicore Performance of Block Algebraic Iterative Reconstruction Methods
DEFF Research Database (Denmark)
Sørensen, Hans Henrik B.; Hansen, Per Christian
2014-01-01
Algebraic iterative methods are routinely used for solving the ill-posed sparse linear systems arising in tomographic image reconstruction. Here we consider the algebraic reconstruction technique (ART) and the simultaneous iterative reconstruction techniques (SIRT), both of which rely on semiconv......Algebraic iterative methods are routinely used for solving the ill-posed sparse linear systems arising in tomographic image reconstruction. Here we consider the algebraic reconstruction technique (ART) and the simultaneous iterative reconstruction techniques (SIRT), both of which rely...... on semiconvergence. Block versions of these methods, based on a partitioning of the linear system, are able to combine the fast semiconvergence of ART with the better multicore properties of SIRT. These block methods separate into two classes: those that, in each iteration, access the blocks in a sequential manner...... a fixed relaxation parameter in each method, namely, the one that leads to the fastest semiconvergence. Computational results show that for multicore computers, the sequential approach is preferable....
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
International Nuclear Information System (INIS)
Kanakoglou, K; Daskaloyannis, C
2008-01-01
Parabosonic algebra in finite or infinite degrees of freedom is considered as a Z 2 -graded associative algebra, and is shown to be a Z 2 -graded (or super) Hopf algebra. The super-Hopf algebraic structure of the parabosonic algebra is established directly without appealing to its relation to the osp(1/2n) Lie superalgebraic structure. The notion of super-Hopf algebra is equivalently described as a Hopf algebra in the braided monoidal category CZ 2 M. The bosonization technique for switching a Hopf algebra in the braided monoidal category H M (where H is a quasitriangular Hopf algebra) into an ordinary Hopf algebra is reviewed. In this paper, we prove that for the parabosonic algebra P B , beyond the application of the bosonization technique to the original super-Hopf algebra, a bosonization-like construction is also achieved using two operators, related to the parabosonic total number operator. Both techniques switch the same super-Hopf algebra P B to an ordinary Hopf algebra, thus producing two different variants of P B , with an ordinary Hopf structure
Boyko, Vyacheslav M; Popovych, Roman O; Shapoval, Nataliya M
2013-01-01
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.
The dual algebra of the Poincare group on Fock space
International Nuclear Information System (INIS)
Klink, W.H.; Iowa Univ., Iowa City, IA
1989-01-01
The Lie algebra of operators commuting with the Poincare group on the Fock space appropriate for a massive spinless particle is constructed in terms of raising and lowering operators indexed by a Lorentz invariant function. From the assumption that the phase operator is an element of this Lie algebra, it is shown that the scattering operator can be written as a unitary representation operator of the group associated with the Lie algebra. A simple choice of the phase operator shows that the Lorentz invariant function can be interpreted as a basic scattering amplitude, in the sense that all multiparticle scattering amplitudes can be written in terms of this basic scattering amplitude. (orig.)
Algebraic study of chiral anomalies
Indian Academy of Sciences (India)
2012-06-14
Jun 14, 2012 ... They form a group G which acts on the (affine) space of ... The curvature F of A is defined by (notice that in this paper the bracket is defined ... This purely algebraic formulation easily extends to the consideration of the Lie algebra of vector .... namely the case of perturbatively renormalizable theories in four ...
International Nuclear Information System (INIS)
Campoamor-Stursberg, R
2008-01-01
We show that the Inoenue-Wigner contraction naturally associated to a reduction chain s implies s' of semisimple Lie algebras induces a decomposition of the Casimir operators into homogeneous polynomials, the terms of which can be used to obtain additional mutually commuting missing label operators for this reduction. The adjunction of these scalars that are no more invariants of the contraction allow to solve the missing label problem for those reductions where the contraction provides an insufficient number of labelling operators.
International Nuclear Information System (INIS)
Curado, E.M.F.; Hassouni, Y.; Rego-Monteiro, M.A.; Rodrigues, Ligia M.C.S.
2008-01-01
We discuss the role of generalized Heisenberg algebras (GHA) in obtaining an algebraic method to describe physical systems. The method consists in finding the GHA associated to a physical system and the relations between its generators and the physical observables. We choose as an example the infinite square-well potential for which we discuss the representations of the corresponding GHA. We suggest a way of constructing a physical realization of the generators of some GHA and apply it to the square-well potential. An expression for the position operator x in terms of the generators of the algebra is given and we compute its matrix elements
Superalgebras with Grassmann algebra-valued structure constants from superfields
International Nuclear Information System (INIS)
Azcarraga, J.A. de; Lukierski, J.
1987-05-01
We introduce generalized Lie algebras and superalgebras with generators and structure constants taking values in a Grassmann algebra. Such algebraic structures describe the equal time algebras in the superfield formalism. As an example we consider the equal time commutators and anticommutators among bilinears made out of the D=1 quantum superfields describing the supersymmetric harmonic oscillator. (author). 10 refs
Energy Technology Data Exchange (ETDEWEB)
Gaffney, J M
1975-01-01
A reappraisal of electromagnetic field theories is made and an account is given of the radiation gauge, Gupta-Bleuler and Fermi methods of quantitising the electromagnetic fields. The Weyl algebra of the vector potential is constructed and the Fermi method is then related to a certain representation of the algebra. The representation is specified by a generating functional for a state on the algebra. The Weyl algebra of the physical field is then constructed as a factor algebra. The Schroedinger representation of the algebra is then studied and it was found that the Fermi method is just a generalization of this representation to an infinite number of degrees of freedom. The Schroedinger representation of Fermi method is constructed.
The algebra and geometry of SU(3) matrices
International Nuclear Information System (INIS)
Mallesh, K.S.; Mukunda, N.
1997-01-01
We give an elementary treatment of the defining representation and Lie algebra of the three-dimensional unitary unimodular group SU(3). The geometrical properties of the Lie algebra, which is an eight dimensional real linear vector space, are developed in an SU(3) covariant manner. The f and d symbols of SU(3) lead to two ways of multiplying two vectors to produce a third, and several useful geometric and algebraic identities are derived. The axis-angle parametrization of SU(3) is developed as a generalization of that for SU(2), and the specifically new features are brought out. Application to the dynamics of three-level system is outlined. (author)
q-structure algebra of Uq(g-circumflex) from its adjoint action
International Nuclear Information System (INIS)
El Hassouni, A.; Hassouni, Y.; Zakkari, M.
1994-08-01
We prove that the adjoint action of the quantum affine Lie algebra U q (g-circumflex), where g is a simple finite dimensional Lie algebra, reproduces the q-commutation relationship of U q (g-circumflex) if and only if g is of type A n , n ≥ 1. (author). 4 refs
A master identity for homotopy Gerstenhaber algebras
International Nuclear Information System (INIS)
Akman, F.
2000-01-01
We produce a master identity {m}{m,m,..}=0 for a certain type of homotopy Gerstenhaber algebras, in particular suitable for the prototype, namely the Hochschild complex of an associative algebra. This algebraic master identity was inspired by the work of Getzler-Jones and Kimura-Voronov-Zuckerman in the context of topological conformal field theories. To this end, we introduce the notion of a ''partitioned multilinear map'' and explain the mechanics of composing such maps. In addition, many new examples of pre-Lie algebras and homotopy Gerstenhaber algebras are given. (orig.)
Trell, Erik; Edeagu, Samuel; Animalu, Alexander
2017-01-01
From a brief recapitulation of the foundational works of Marius Sophus Lie and Herrmann Günther Grassmann, and including missing African links, a rhapsodic survey is made of the straight line of extension and existence that runs as the very fibre of generation and creation throughout Nature's all utterances, which must therefore ultimately be the web of Reality itself of which the Arts and Sciences are interpreters on equal explorer terms. Assuming their direct approach, the straight line and its archaic and algebraic and artistic bearings and convolutions have been followed towards their inner reaches, which earlier resulted in a retrieval of the baryon and meson elementary particles and now equally straightforward the electron geodesics and the organic build of the periodic system of the elements.
Certain extensions of vertex operator algebras of affine type
International Nuclear Information System (INIS)
Li Haisheng
2001-01-01
We generalize Feigin and Miwa's construction of extended vertex operator (super)algebras A k (sl(2)) for other types of simple Lie algebras. For all the constructed extended vertex operator (super)algebras, irreducible modules are classified, complete reducibility of every module is proved and fusion rules are determined modulo the fusion rules for vertex operator algebras of affine type. (orig.)
Quantized Algebras of Functions on Homogeneous Spaces with Poisson Stabilizers
Neshveyev, Sergey; Tuset, Lars
2012-05-01
Let G be a simply connected semisimple compact Lie group with standard Poisson structure, K a closed Poisson-Lie subgroup, 0 topology on the spectrum of C( G q / K q ). Next we show that the family of C*-algebras C( G q / K q ), 0 < q ≤ 1, has a canonical structure of a continuous field of C*-algebras and provides a strict deformation quantization of the Poisson algebra {{C}[G/K]} . Finally, extending a result of Nagy, we show that C( G q / K q ) is canonically KK-equivalent to C( G/ K).
Affine Kac-Moody algebras and their representations
International Nuclear Information System (INIS)
Slansky, R.
1988-01-01
Highest weight representation theory of finite dimensional and affine Kac-Moody algebras is summarized from a unified point of view. Lattices of discrete additive quantum numbers and the presentation of Lie algebras on Cartan matrices are the central points of departure for the analysis. (author)
Quantum algebras and Poisson geometry in mathematical physics
Karasev, M V
2005-01-01
This collection presents new and interesting applications of Poisson geometry to some fundamental well-known problems in mathematical physics. The methods used by the authors include, in addition to advanced Poisson geometry, unexpected algebras with non-Lie commutation relations, nontrivial (quantum) Kählerian structures of hypergeometric type, dynamical systems theory, semiclassical asymptotics, etc.
Variational linear algebraic equations method
International Nuclear Information System (INIS)
Moiseiwitsch, B.L.
1982-01-01
A modification of the linear algebraic equations method is described which ensures a variational bound on the phaseshifts for potentials having a definite sign at all points. The method is illustrated by the elastic scattering of s-wave electrons by the static field of atomic hydrogen. (author)
Linear Algebraic Method for Non-Linear Map Analysis
International Nuclear Information System (INIS)
Yu, L.; Nash, B.
2009-01-01
We present a newly developed method to analyze some non-linear dynamics problems such as the Henon map using a matrix analysis method from linear algebra. Choosing the Henon map as an example, we analyze the spectral structure, the tune-amplitude dependence, the variation of tune and amplitude during the particle motion, etc., using the method of Jordan decomposition which is widely used in conventional linear algebra.
Some Applications of Algebraic System Solving
Roanes-Lozano, Eugenio
2011-01-01
Technology and, in particular, computer algebra systems, allows us to change both the way we teach mathematics and the mathematical curriculum. Curiously enough, unlike what happens with linear system solving, algebraic system solving is not widely known. The aim of this paper is to show that, although the theory lying behind the "exact…
Role of Lie algebra for confinement in non-abelian gauge field scheme
International Nuclear Information System (INIS)
Fukushima, K.; Sato, H.
2014-01-01
This article reports an explicit function form for confining classical Yang-Mills vector potentials and quantum fluctuations around the classical field. The classical vector potential, which is composed of a confining localized function and an unlocalized function, satisfies the classical Yang-Mills equation. The confining localized function contributes to the Wilson loop, while the unlocalized function makes no contribution to this loop. The confining linear potential between a heavy fermion and antifermion is due to (1) the Lie algebra and (2) the form of the confining localized function which has opposite signs at the positions of the particle and antiparticle along the Wilson loop in the time direction. Some classical confining parts of vector potentials also change sign on inversion of the coordinates of the axis perpendicular to the axis joining the two particles. The localized parts of the vector potentials are squeezed around the axis connecting the two particles, and the string tension of the confining linear potential is derived. Quantum fluctuations are formulated using a field expression in terms of local basis functions in real spacetime. The quantum path integral gives the Coulomb potential between the two particles in addition to the linear potential due to the classical fields
A generalization of the deformed algebra of quantum group SU(2)q for Hopf algebra
International Nuclear Information System (INIS)
Ludu, A.; Gupta, R.K.
1992-12-01
A generalization of the deformation of Lie algebra of SU(2) group is established for the Hopf algebra, by modifying the J 3 component in all of its defining commutators. The modification is carried out in terms of a polynomial f, of J 3 and the q-deformation parameter, which contains the known q-deformation functionals as its particular cases. (author). 20 refs
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
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
Quantum deformations of conformal algebras with mass-like deformation parameters
International Nuclear Information System (INIS)
Frydryszak, Andrzej; Lukierski, Jerzy; Mozrzymas, Marek; Minnaert, Pierre
1998-01-01
We recall the mathematical apparatus necessary for the quantum deformation of Lie algebras, namely the notions of coboundary Lie algebras, classical r-matrices, classical Yang-Baxter equations (CYBE), Froebenius algebras and parabolic subalgebras. Then we construct the quantum deformation of D=1, D=2 and D=3 conformal algebras, showing that this quantization introduce fundamental mass parameters. Finally we consider with more details the quantization of D=4 conformal algebra. We build three classes of sl(4,C) classical r-matrices, satisfying CYBE and depending respectively on 8, 10 and 12 generators of parabolic subalgebras. We show that only the 8-dimensional r-matrices allow to impose the D=4 conformal o(4,2)≅su(2,2) reality conditions. Weyl reflections and Dynkin diagram automorphisms for o(4,2) define the class of admissible bases for given classical r-matrices
Mathematical methods linear algebra normed spaces distributions integration
Korevaar, Jacob
1968-01-01
Mathematical Methods, Volume I: Linear Algebra, Normed Spaces, Distributions, Integration focuses on advanced mathematical tools used in applications and the basic concepts of algebra, normed spaces, integration, and distributions.The publication first offers information on algebraic theory of vector spaces and introduction to functional analysis. Discussions focus on linear transformations and functionals, rectangular matrices, systems of linear equations, eigenvalue problems, use of eigenvectors and generalized eigenvectors in the representation of linear operators, metric and normed vector
Satake diagrams of affine Kac-Moody algebras
Energy Technology Data Exchange (ETDEWEB)
Tripathy, L K [S B R Government Womens' College, Berhampur, Orissa 760 001 (India); Pati, K C [Department of Physics, Khallikote College, Berhampur, Orissa 760 001 (India)
2006-02-10
Satake diagrams of affine Kac-Moody algebras (untwisted and twisted) are obtained from their Dynkin diagrams. These diagrams give a classification of restricted root systems associated with these algebras. In the case of simple Lie algebras, these root systems and Satake diagrams correspond to symmetric spaces which have recently found many physical applications in quantum integrable systems, quantum transport problems, random matrix theories etc. We hope these types of root systems may have similar applications in theoretical physics in future and may correspond to symmetric spaces analogue of affine Kac-Moody algebras if they exist.
The kinematic algebras from the scattering equations
International Nuclear Information System (INIS)
Monteiro, Ricardo; O’Connell, Donal
2014-01-01
We study kinematic algebras associated to the recently proposed scattering equations, which arise in the description of the scattering of massless particles. In particular, we describe the role that these algebras play in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex which is associated to each solution of those equations. We also identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant
Noncommutative o*(N) and usp*(2N) algebras and the corresponding gauge field theories
International Nuclear Information System (INIS)
Bars, I.; Sheikh-Jabbari, M.M.; Vasiliev, M.A.
2001-03-01
The extension of the noncommutative u * (N) Lie algebra to noncommutative orthogonal and symplectic Lie algebras is studied. Using an anti-automorphism of the star-matrix algebra, we show that the u * (N) can consistently be restricted to o * (N) and usp * (N) algebras that have new mathematical structures. We give explicit fundamental matrix representations of these algebras, through which the formulation for the corresponding noncommutative gauge field theories are obtained. In addition, we present a D-brane configuration with an orientifold which realizes geometrically our algebraic construction, thus embedding the new noncommutative gauge theories in superstring theory in the presence of a constant background magnetic field. Some algebraic generalizations that may have applications in other areas of physics are also discussed. (author)
Characterizing ξ-Lie Multiplicative Isomorphisms on Von Neumann Algebras
Directory of Open Access Journals (Sweden)
Yamin Song
2014-01-01
Full Text Available Let ℳ and be von Neumann algebras without central summands of type I1. Assume that ξ∈ℂ with ξ≠1. In this paper, all maps Φ:ℳ→ satisfying ΦAB-ξBA=ΦAΦB-ξΦBΦ(A are characterized.
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.)
Group C∗-algebras without the completely bounded approximation property
DEFF Research Database (Denmark)
Haagerup, U.
2016-01-01
It is proved that: (1) The Fourier algebra A(G) of a simple Lie group G of real rank at least 2 with finite center does not have a multiplier bounded approximate unit. (2) The reduced C∗-algebra C∗ r of any lattice in a non-compact simple Lie group of real rank at least 2 with finite center does...... not have the completely bounded approximation property. Hence, the results obtained by de Canniere and the author for SOe (n, 1), n ≥ 2, and by Cowling for SU(n, 1) do not generalize to simple Lie groups of real rank at least 2. © 2016 Heldermann Verlag....
Representations of the algebra Uq'(son) related to quantum gravity
International Nuclear Information System (INIS)
Klimyk, A.U.
2002-01-01
The aim of this paper is to review our results on finite dimensional irreducible representations of the nonstandard q-deformation U q ' (so n ) of the universal enveloping algebra U(so(n)) of the Lie algebra so(n) which does not coincide with the Drinfeld-Jimbo quantum algebra U q (so n ).This algebra is related to algebras of observables in quantum gravity and to algebraic geometry.Irreducible finite dimensional representations of the algebra U q ' (so n ) for q not a root of unity and for q a root of unity are given
The κ-(AdS quantum algebra in (3+1 dimensions
Directory of Open Access Journals (Sweden)
Ángel Ballesteros
2017-03-01
Full Text Available The quantum duality principle is used to obtain explicitly the Poisson analogue of the κ-(AdS quantum algebra in (3+1 dimensions as the corresponding Poisson–Lie structure on the dual solvable Lie group. The construction is fully performed in a kinematical basis and deformed Casimir functions are also explicitly obtained. The cosmological constant Λ is included as a Poisson–Lie group contraction parameter, and the limit Λ→0 leads to the well-known κ-Poincaré algebra in the bicrossproduct basis. A twisted version with Drinfel'd double structure of this κ-(AdS deformation is sketched.
Commutative algebra constructive methods finite projective modules
Lombardi, Henri
2015-01-01
Translated from the popular French edition, this book offers a detailed introduction to various basic concepts, methods, principles, and results of commutative algebra. It takes a constructive viewpoint in commutative algebra and studies algorithmic approaches alongside several abstract classical theories. Indeed, it revisits these traditional topics with a new and simplifying manner, making the subject both accessible and innovative. The algorithmic aspects of such naturally abstract topics as Galois theory, Dedekind rings, Prüfer rings, finitely generated projective modules, dimension theory of commutative rings, and others in the current treatise, are all analysed in the spirit of the great developers of constructive algebra in the nineteenth century. This updated and revised edition contains over 350 well-arranged exercises, together with their helpful hints for solution. A basic knowledge of linear algebra, group theory, elementary number theory as well as the fundamentals of ring and module theory is r...
Jorgensen, Palle E T
1987-01-01
Historically, operator theory and representation theory both originated with the advent of quantum mechanics. The interplay between the subjects has been and still is active in a variety of areas.This volume focuses on representations of the universal enveloping algebra, covariant representations in general, and infinite-dimensional Lie algebras in particular. It also provides new applications of recent results on integrability of finite-dimensional Lie algebras. As a central theme, it is shown that a number of recent developments in operator algebras may be handled in a particularly e
Coset realization of unifying W-algebras
International Nuclear Information System (INIS)
Blumenhagen, R.; Huebel, R.
1994-06-01
We construct several quantum coset W-algebras, e.g. sl(2,R)/U(1) and sl(2,R)+sl(2,R)/sl(2,R), and argue that they are finitely nonfreely generated. Furthermore, we discuss in detail their role as unifying W-algebras of Casimir W-algebras. We show that it is possible to give coset realizations of various types of unifying W-algebras, e.g. the diagonal cosets based on the symplectic Lie algebras sp(2n) realize the unifying W-algebras which have previously been introduced as 'WD -n '. In addition, minimal models of WD -n are studied. The coset realizations provide a generalization of level-rank-duality of dual coset pairs. As further examples of finitely nonfreely generated quantum W-algebras we discuss orbifolding of W-algebras which on the quantum level has different properties than in the classical case. We demonstrate in some examples that the classical limit according to Bowcock and Watts of these nonfreely finitely generated quantum W-algebras probably yields infinitely nonfreely generated classical W-algebras. (orig.)
Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
Escobar-Ruiz, Mauricio A.; Kalnins, Ernest G.; Miller, Willar, Jr.; Subag, Eyal
2017-03-01
Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by Bôcher contractions of the conformal Lie algebra {so}(4,C) to itself. In this paper we give a precise definition of Bôcher contractions and show how they can be classified. They subsume well known contractions of {e}(2,C) and {so}(3,C) and have important physical and geometric meanings, such as the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. We also classify abstract nondegenerate quadratic algebras in terms of an invariant that we call a canonical form. We describe an algorithm for finding the canonical form of such algebras. We calculate explicitly all canonical forms arising from quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces. We further discuss contraction of quadratic algebras, focusing on those coming from superintegrable systems.
BRST operator for superconformal algebras with quadratic nonlinearity
International Nuclear Information System (INIS)
Khviengia, Z.; Sezgin, E.
1993-07-01
We construct the quantum BRST operators for a large class of superconformal and quasi-superconformal algebras with quadratic nonlinearity. The only free parameter in these algebras is the level of the (super) Kac-Moody sector. The nilpotency of the quantum BRST operator imposes a condition on the level. We find this condition for (quasi) superconformal algebras with a Kac-Moody sector based on a simple Lie algebra and for the Z 2 x Z 2 -graded superconformal algebras with a Kac-Moody sector based on the superalgebra osp(N modul 2M) or sl (N + 2 modul N). (author). 22 refs, 3 tabs
Relation of deformed nonlinear algebras with linear ones
International Nuclear Information System (INIS)
Nowicki, A; Tkachuk, V M
2014-01-01
The relation between nonlinear algebras and linear ones is established. For a one-dimensional nonlinear deformed Heisenberg algebra with two operators we find the function of deformation for which this nonlinear algebra can be transformed to a linear one with three operators. We also establish the relation between the Lie algebra of total angular momentum and corresponding nonlinear one. This relation gives a possibility to simplify and to solve the eigenvalue problem for the Hamiltonian in a nonlinear case using the reduction of this problem to the case of linear algebra. It is demonstrated in an example of a harmonic oscillator. (paper)
Degenerate representation from tensorial identities and quantum realisations of YBZF algebras
International Nuclear Information System (INIS)
Iosifescu, M.; Scutaru, H.
1987-06-01
The second- degree irreducible tensors in the enveloping algebra of the classical semisimple Lie algebras are determined and the irreducible representations on which these tensors vanish are derived.(authors)
Algebraic Methods in Plane Geometry
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 13; Issue 10. Algebraic Methods in ... General Article Volume 13 Issue 10 October 2008 pp 916-928 ... Keywords. Conics; family of curves; Pascal's theorem; homogeneous coordinates; Butterfly theorem; abelian group; associativity of addition; group law.
A new derivation of the highest-weight polynomial of a unitary lie algebra
International Nuclear Information System (INIS)
P Chau, Huu-Tai; P Van, Isacker
2000-01-01
A new method is presented to derive the expression of the highest-weight polynomial used to build the basis of an irreducible representation (IR) of the unitary algebra U(2J+1). After a brief reminder of Moshinsky's method to arrive at the set of equations defining the highest-weight polynomial of U(2J+1), an alternative derivation of the polynomial from these equations is presented. The method is less general than the one proposed by Moshinsky but has the advantage that the determinantal expression of the highest-weight polynomial is arrived at in a direct way using matrix inversions. (authors)
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.)
On the structure of transitively differential algebras
Post, Gerhard F.
1999-01-01
We study finite-dimensional Lie algebras of polynomial vector fields in $n$ variables that contain the vector fields ${\\partial}/{\\partial x_i} \\; (i=1,\\ldots, n)$ and $x_1{\\partial}/{\\partial x_1}+ \\dots + x_n{\\partial}/{\\partial x_n}$. We derive some general results on the structure of such Lie
Applications of an anti-symmetry loop algebra and its expanding forms
International Nuclear Information System (INIS)
Zhang Yufeng; Yan Qingyou
2004-01-01
Constructing an anti-symmetry subalgebra A-tilde 2 of loop algebra A-tilde 2 gives the well-known Jaulent-Miodek (JM) hierarchy, the JM equation and its new Lax pair. Further, the Darboux transformation of the JM equation is deduced by anstaz method. By making use of a high-order loop algebra and Tu scheme, an expanding integrable model of the JM hierarchy is obtained. A direct expansion A-macron 2 * of loop algebra A-tilde 2 by considering the definition of Lie algebra is presented, which is used to establish two isospectral problems. It follows that corresponding two new integrable systems are engendered, which possess bi-Hamiltonian structures, respectively. Furthermore, a scalar transformation is applied to turn the loop algebra A-bar 2 * into its equivalent subalgebra A-tilde 1 of loop algebra A-tilde 1 . With the help of A-tilde 1 , another new high-order loop algebra G-bar is constructed, which is used to obtain an expanding integrable model of one of two integrable systems presented
AIR Tools - A MATLAB package of algebraic iterative reconstruction methods
DEFF Research Database (Denmark)
Hansen, Per Christian; Saxild-Hansen, Maria
2012-01-01
We present a MATLAB package with implementations of several algebraic iterative reconstruction methods for discretizations of inverse problems. These so-called row action methods rely on semi-convergence for achieving the necessary regularization of the problem. Two classes of methods are impleme......We present a MATLAB package with implementations of several algebraic iterative reconstruction methods for discretizations of inverse problems. These so-called row action methods rely on semi-convergence for achieving the necessary regularization of the problem. Two classes of methods...... are implemented: Algebraic Reconstruction Techniques (ART) and Simultaneous Iterative Reconstruction Techniques (SIRT). In addition we provide a few simplified test problems from medical and seismic tomography. For each iterative method, a number of strategies are available for choosing the relaxation parameter...
International Nuclear Information System (INIS)
Barbarin, F.; Sorba, P.; Ragoucy, E.
1996-01-01
The property of some finite W algebras to be the commutant of a particular subalgebra of a simple Lie algebra G is used to construct realizations of G. When G ≅ so (4,2), unitary representations of the conformal and Poincare algebras are recognized in this approach, which can be compared to the usual induced representation technique. When G approx=(2, R), the anyonic parameter can be seen as the eigenvalue of a W generator in such W representations of G. The generalization of such properties to the affine case is also discussed in the conclusion, where an alternative of the Wakimoto construction for sl(2) k is briefly presented. (authors)
International Nuclear Information System (INIS)
Bina, B.; Guenaydin, M.
1997-01-01
We give a complete classification of the real forms of simple non-linear superconformal algebras (SCA) and quasi-superconformal algebras (QSCA) and present a unified realization of these algebras with simple symmetry groups. This classification is achieved by establishing a correspondence between simple non-linear QSCA's and SCA's and quaternionic and super-quaternionic symmetric spaces of simple Lie groups and Lie supergroups, respectively. The unified realization we present involves a dimension zero scalar field (dilaton), dimension-1 symmetry currents, and dimension-1/2 free bosons for QSCA's and dimension-1/2 free fermions for SCA's. The free bosons and fermions are associated with the quaternionic and super-quaternionic symmetric spaces of corresponding Lie groups and Lie supergroups, respectively. We conclude with a discussion of possible applications of our results. (orig.)
A general Euclidean connection for so(n,m) lie algebra and the algebraic approach to scattering
International Nuclear Information System (INIS)
Ionescu, R.A.
1994-11-01
We obtain a general Euclidean connection for so(n,m). This Euclidean connection allows an algebraic derivation of the S matrix and it reduces to the known one in suitable circumstances. (author). 8 refs
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 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
The algebraic locus of Feynman integrals
Kol, Barak
2016-01-01
In the Symmetries of Feynman Integrals (SFI) approach, a diagram's parameter space is foliated by orbits of a Lie group associated with the diagram. SFI is related to the important methods of Integrations By Parts and of Differential Equations. It is shown that sometimes there exist a locus in parameter space where the set of SFI differential equations degenerates into an algebraic equation, thereby enabling a solution in terms of integrals associated with degenerations of the diagram. This i...
Algebraic Verification Method for SEREs Properties via Groebner Bases Approaches
Directory of Open Access Journals (Sweden)
Ning Zhou
2013-01-01
Full Text Available This work presents an efficient solution using computer algebra system to perform linear temporal properties verification for synchronous digital systems. The method is essentially based on both Groebner bases approaches and symbolic simulation. A mechanism for constructing canonical polynomial set based symbolic representations for both circuit descriptions and assertions is studied. We then present a complete checking algorithm framework based on these algebraic representations by using Groebner bases. The computational experience result in this work shows that the algebraic approach is a quite competitive checking method and will be a useful supplement to the existent verification methods based on simulation.
Basic algebraic topology and its applications
Adhikari, Mahima Ranjan
2016-01-01
This book provides an accessible introduction to algebraic topology, a ﬁeld at the intersection of topology, geometry and algebra, together with its applications. Moreover, it covers several related topics that are in fact important in the overall scheme of algebraic topology. Comprising eighteen chapters and two appendices, the book integrates various concepts of algebraic topology, supported by examples, exercises, applications and historical notes. Primarily intended as a textbook, the book oﬀers a valuable resource for undergraduate, postgraduate and advanced mathematics students alike. Focusing more on the geometric than on algebraic aspects of the subject, as well as its natural development, the book conveys the basic language of modern algebraic topology by exploring homotopy, homology and cohomology theories, and examines a variety of spaces: spheres, projective spaces, classical groups and their quotient spaces, function spaces, polyhedra, topological groups, Lie groups and cell complexes, etc. T...
On the algebraic structure of differential calculus on quantum groups
International Nuclear Information System (INIS)
Rad'ko, O.V.; Vladimirov, A.A.
1997-01-01
Intrinsic Hopf algebra structure of the Woronowicz differential complex is shown to generate quite naturally a bicovariant algebra of four basic objects within a differential calculus on quantum groups - coordinate functions, differential forms, Lie derivatives, and inner derivatives - as the cross-product algebra of two mutually dual graded Hopf algebras. This construction, properly taking into account Hopf-algebraic properties of Woronowicz's bicovariant calculus, provides a direct proof of the Cartan identity and of many other useful relations. A detailed comparison with other approaches is also given
Geometric approach to the (BRS-) differential algebras of supersymmetric YM-theories
International Nuclear Information System (INIS)
Gieres, F.
1987-01-01
The (BRS-) differential algebra of susy YM-theories is defined in terms of superfields and forms on rigid U(N)-superspace. For d = 4 and N = 1.2 we show that it projects to the ''BRS-component field algebra in the WZ-gauge'' without any supergauge fixing. In this process the supergeometry is destroyed with the result that the final algebra becomes a prototype for a differential algebra which cannot be associated with an ordinary Lie algebra
The algebras of large N matrix mechanics
Energy Technology Data Exchange (ETDEWEB)
Halpern, M.B.; Schwartz, C.
1999-09-16
Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.
Quantum field theories on algebraic curves. I. Additive bosons
International Nuclear Information System (INIS)
Takhtajan, Leon A
2013-01-01
Using Serre's adelic interpretation of cohomology, we develop a 'differential and integral calculus' on an algebraic curve X over an algebraically closed field k of constants of characteristic zero, define algebraic analogues of additive multi-valued functions on X and prove the corresponding generalized residue theorem. Using the representation theory of the global Heisenberg algebra and lattice Lie algebra, we formulate quantum field theories of additive and charged bosons on an algebraic curve X. These theories are naturally connected with the algebraic de Rham theorem. We prove that an extension of global symmetries (Witten's additive Ward identities) from the k-vector space of rational functions on X to the vector space of additive multi-valued functions uniquely determines these quantum theories of additive and charged bosons.
Holonomies for connections with values in L_infty algebras
DEFF Research Database (Denmark)
Arias Abad, Camilo; Schaetz, Florian
2014-01-01
Given a flat connection $\\alpha$ on a manifold $M$ with values in a filtered $L_\\infty$-algebra $g$, we construct a morphism $hol^\\infty_\\alpha: C(M) B\\hat{U}_\\infty(g)$, generalizing the holonomies of flat connections with values in Lie algebras. The construction is based on Gugenheim's $A_\\inft...
International Nuclear Information System (INIS)
Unge, Rikard von
2002-01-01
We extend the path-integral formalism for Poisson-Lie T-duality to include the case of Drinfeld doubles which can be decomposed into bi-algebras in more than one way. We give the correct shift of the dilaton, correcting a mistake in the literature. We then use the fact that the six dimensional Drinfeld doubles have been classified to write down all possible conformal Poisson-Lie T-duals of three dimensional space times and we explicitly work out two duals to the constant dilaton and zero anti-symmetric tensor Bianchi type V space time and show that they satisfy the string equations of motion. This space-time was previously thought to have no duals because of the tracefulness of the structure constants. (author)
Fiber-wise linear Poisson structures related to W∗-algebras
Odzijewicz, Anatol; Jakimowicz, Grzegorz; Sliżewska, Aneta
2018-01-01
In the framework of Banach differential geometry we investigate the fiber-wise linear Poisson structures as well as the Lie groupoid and Lie algebroid structures which are defined in the canonical way by the structure of a W∗-algebra (von Neumann algebra) M. The main role in this theory is played by the complex Banach-Lie groupoid G(M) ⇉ L(M) of partially invertible elements of M over the lattice L(M) of orthogonal projections of M. The Atiyah sequence and the predual Atiyah sequence corresponding to this groupoid are investigated from the point of view of Banach Poisson geometry. In particular we show that the predual Atiyah sequence fits in a short exact sequence of complex Banach sub-Poisson V B-groupoids with G(M) ⇉ L(M) as the side groupoid.
Affine Lie algebraic origin of constrained KP hierarchies
International Nuclear Information System (INIS)
Aratyn, H.; Gomes, J.F.; Zimerman, A.H.
1994-07-01
It is presented an affine sl(n+1) algebraic construction of the basic constrained KP hierarchy. This hierarchy is analyzed using two approaches, namely linear matrix eigenvalue problem on hermitian symmetric space and constrained KP Lax formulation and we show that these approaches are equivalent. The model is recognized to be generalized non-linear Schroedinger (GNLS) hierarchy and it is used as a building block for a new class of constrained KP hierarchies. These constrained KP hierarchies are connected via similarity-Backlund transformations and interpolate between GNLS and multi-boson KP-Toda hierarchies. The construction uncovers origin of the Toda lattice structure behind the latter hierarchy. (author). 23 refs
SYNTHESIS METHODS OF ALGEBRAIC NORMAL FORM OF MANY-VALUED LOGIC FUNCTIONS
Directory of Open Access Journals (Sweden)
A. V. Sokolov
2016-01-01
Full Text Available The rapid development of methods of error-correcting coding, cryptography, and signal synthesis theory based on the principles of many-valued logic determines the need for a more detailed study of the forms of representation of functions of many-valued logic. In particular the algebraic normal form of Boolean functions, also known as Zhegalkin polynomial, that well describe many of the cryptographic properties of Boolean functions is widely used. In this article, we formalized the notion of algebraic normal form for many-valued logic functions. We developed a fast method of synthesis of algebraic normal form of 3-functions and 5-functions that work similarly to the Reed-Muller transform for Boolean functions: on the basis of recurrently synthesized transform matrices. We propose the hypothesis, which determines the rules of the synthesis of these matrices for the transformation from the truth table to the coefficients of the algebraic normal form and the inverse transform for any given number of variables of 3-functions or 5-functions. The article also introduces the definition of algebraic degree of nonlinearity of the functions of many-valued logic and the S-box, based on the principles of many-valued logic. Thus, the methods of synthesis of algebraic normal form of 3-functions applied to the known construction of recurrent synthesis of S-boxes of length N = 3k, whereby their algebraic degrees of nonlinearity are computed. The results could be the basis for further theoretical research and practical applications such as: the development of new cryptographic primitives, error-correcting codes, algorithms of data compression, signal structures, and algorithms of block and stream encryption, all based on the perspective principles of many-valued logic. In addition, the fast method of synthesis of algebraic normal form of many-valued logic functions is the basis for their software and hardware implementation.
Current algebras and many-body physics
International Nuclear Information System (INIS)
Albertin, U.K.
1989-01-01
Several applications of current algebras in many body physics are examined. The first is the interacting Bose gas in three dimensions. Theories for phonons, vortices and rotons are all described within the current algebra formalism. Next the one dimensional electron gas is examined within the approximation of linear dispersion so that relativistic current algebra techniques may be used. The relation with Thirring strings and compactified boson models is examined, and points of enhanced symmetry in the compactified boson models are shown to lie on phase transition lines for the electron gas. Finally, mathematical aspects of the current algebra are studied. The theory of induced representations of the diffeomorphism group are used to describe the Aharanov-Bohm effect, the thermodynamics of the Bose gas, and the Bose gas in the presence of vortex filaments
Homological methods, representation theory, and cluster algebras
Trepode, Sonia
2018-01-01
This text presents six mini-courses, all devoted to interactions between representation theory of algebras, homological algebra, and the new ever-expanding theory of cluster algebras. The interplay between the topics discussed in this text will continue to grow and this collection of courses stands as a partial testimony to this new development. The courses are useful for any mathematician who would like to learn more about this rapidly developing field; the primary aim is to engage graduate students and young researchers. Prerequisites include knowledge of some noncommutative algebra or homological algebra. Homological algebra has always been considered as one of the main tools in the study of finite-dimensional algebras. The strong relationship with cluster algebras is more recent and has quickly established itself as one of the important highlights of today’s mathematical landscape. This connection has been fruitful to both areas—representation theory provides a categorification of cluster algebras, wh...
c-fans and Newton polyhedra of algebraic varieties
International Nuclear Information System (INIS)
Kazarnovskii, B Ya
2003-01-01
To every algebraic subvariety of a complex torus there corresponds a Euclidean geometric object called a c-fan. This correspondence determines an intersection theory for algebraic varieties. c-fans form a graded commutative algebra with visually defined operations. The c-fans of algebraic varieties lie in the subring of rational c-fans. It seems that other subrings may be used to construct an intersection theory for other categories of analytic varieties. We discover a relation between an old problem in the theory of convex bodies (the so-called Minkowski problem) and the ring of c-fans. This enables us to define a correspondence that sends any algebraic curve to a convex polyhedron in the space of characters of the torus
The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders
International Nuclear Information System (INIS)
Gurau, Razvan
2012-01-01
Random tensor models for a generic complex tensor generalize matrix models in arbitrary dimensions and yield a theory of random geometries. They support a 1/N expansion dominated by graphs of spherical topology. Their Schwinger Dyson equations, generalizing the loop equations of matrix models, translate into constraints satisfied by the partition function. The constraints have been shown, in the large N limit, to close a Lie algebra indexed by colored rooted D-ary trees yielding a first generalization of the Virasoro algebra in arbitrary dimensions. In this paper we complete the Schwinger Dyson equations and the associated algebra at all orders in 1/N. The full algebra of constraints is indexed by D-colored graphs, and the leading order D-ary tree algebra is a Lie subalgebra of the full constraints algebra.
Villarreal, Rafael
2015-01-01
The book stresses the interplay between several areas of pure and applied mathematics, emphasizing the central role of monomial algebras. It unifies the classical results of commutative algebra with central results and notions from graph theory, combinatorics, linear algebra, integer programming, and combinatorial optimization. The book introduces various methods to study monomial algebras and their presentation ideals, including Stanley-Reisner rings, subrings and blowup algebra-emphasizing square free quadratics, hypergraph clutters, and effective computational methods.
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
W∞ and the Racah-Wigner algebra
International Nuclear Information System (INIS)
Pope, C.N.; Shen, X.; Romans, L.J.
1990-01-01
We examine the structure of a recently constructed W ∞ algebra, an extension of the Virasoro algebra that describes an infinite number of fields with all conformal spins 2,3..., with central terms for all spins. By examining its underlying SL(2,R) structure, we are able to exhibit its relation to the algebas of SL(2,R) tensor operators. Based upon this relationship, we generalise W ∞ to a one-parameter family of inequivalent Lie algebras W ∞ (μ), which for general μ requires the introduction of formally negative spins. Furthermore, we display a realisation of the W ∞ (μ) commutation relations in terms of an underlying associative product, which we denote with a lone star. This product structure shares many formal features with the Racah-Wigner algebra in angular-momentum theory. We also discuss the relation between W ∞ and the symplectic algebra on a cone, which can be viewed as a co-adjoint orbit of SL(2,R). (orig.)
Beltrami parametrization and gauging of Virasoro and w-infinity algebras
International Nuclear Information System (INIS)
Tatar, L.
1992-07-01
The gauging of the Virasoro and w-infinity algebras are discussed from the point of view of BRST symmetry. Both algebras are realised as ''Russian formulas'' for the curvatures built from the generators of the Lie algebras and the corresponding gauge fields. The generalized curvatures are used to determine the gauge invariant Lagrangians as well as the anomaly structures of the conformal two dimensional theory and the w-gravity. (author). 21 refs
INdAM Meeting on Homological and Computational Methods in Commutative Algebra
Gubeladze, Joseph; Römer, Tim
2017-01-01
This volume collects contributions by leading experts in the area of commutative algebra related to the INdAM meeting “Homological and Computational Methods in Commutative Algebra” held in Cortona (Italy) from May 30 to June 3, 2016 . The conference and this volume are dedicated to Winfried Bruns on the occasion of his 70th birthday. In particular, the topics of this book strongly reflect the variety of Winfried Bruns’ research interests and his great impact on commutative algebra as well as its applications to related fields. The authors discuss recent and relevant developments in algebraic geometry, commutative algebra, computational algebra, discrete geometry and homological algebra. The book offers a unique resource, both for young and more experienced researchers seeking comprehensive overviews and extensive bibliographic references.
On the geometry of Riemannian manifolds with a Lie structure at infinity
Directory of Open Access Journals (Sweden)
Bernd Ammann
2004-01-01
Full Text Available We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with a Lie structure at infinity to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.
The Weyl approach to the representation theory of reflection equation algebra
International Nuclear Information System (INIS)
Saponov, P A
2004-01-01
The present paper deals with the representation theory of reflection equation algebra, connected to a Hecke type R-matrix. Up to some reasonable additional conditions, the R-matrix is arbitrary (not necessary originating from quantum groups). We suggest a universal method for constructing finite dimensional irreducible representations in the framework of the Weyl approach well known in the representation theory of classical Lie groups and algebras. With this method a series of irreducible modules is constructed. The modules are parametrized by Young diagrams. The spectrum of central elements s k Tr q L k is calculated in the single-row and single-column representations. A rule for the decomposition of the tensor product of modules into a direct sum of irreducible components is also suggested
Quantum algebras in nuclear structure
International Nuclear Information System (INIS)
Bonatsos, D.; Daskaloyannis, C.
1995-01-01
Quantum algebras is a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction through the necessary mathematical tools (q-numbers, q-analysis, q-oscillators, q-algebras), the su q (2) rotator model and its extensions, the construction of deformed exactly soluble models (Interacting Boson Model, Moszkowski model), the use of deformed bosons in the description of pairing correlations, and the symmetries of the anisotropic quantum harmonic oscillator with rational ratios of frequencies, which underline the structure of superdeformed and hyperdeformed nuclei are discussed in some details. A brief description of similar applications to molecular structure and an outlook are also given. (author) 2 Tabs., 324 Refs
Algebraic structure of the Green's ansatz and its q-deformed analogue
International Nuclear Information System (INIS)
Palev, T.D.
1994-08-01
The algebraic structure of the Green's ansatz is analyzed in such a way that its generalization to the case of q-deformed para-Bose and para-Fermi operators is becoming evident. To this end the underlying Lie (super) algebraic properties of the parastatistics are essentially used. (author). 41 refs
Experimental and Theoretical Methods in Algebra, Geometry and Topology
Veys, Willem; Bridging Algebra, Geometry, and Topology
2014-01-01
Algebra, geometry and topology cover a variety of different, but intimately related research fields in modern mathematics. This book focuses on specific aspects of this interaction. The present volume contains refereed papers which were presented at the International Conference “Experimental and Theoretical Methods in Algebra, Geometry and Topology”, held in Eforie Nord (near Constanta), Romania, during 20-25 June 2013. The conference was devoted to the 60th anniversary of the distinguished Romanian mathematicians Alexandru Dimca and Ştefan Papadima. The selected papers consist of original research work and a survey paper. They are intended for a large audience, including researchers and graduate students interested in algebraic geometry, combinatorics, topology, hyperplane arrangements and commutative algebra. The papers are written by well-known experts from different fields of mathematics, affiliated to universities from all over the word, they cover a broad range of topics and explore the research f...
New topological invariants for non-abelian antisymmetric tensor fields from extended BRS algebra
International Nuclear Information System (INIS)
Boukraa, S.; Maillet, J.M.; Nijhoff, F.
1988-09-01
Extended non-linear BRS and Gauge transformations containing Lie algebra cocycles, and acting on non-abelian antisymmetric tensor fields are constructed in the context of free differential algebras. New topological invariants are given in this framework. 6 refs
Tang, Wanjie; Li, Bin; Huang, Xiaoqi; Jiang, Xiaoyu; Li, Fei; Wang, Lijuan; Chen, Taolin; Wang, Jinhui; Gong, Qiyong; Yang, Yanchun
2013-10-01
Few studies have used neuroimaging to characterize treatment-refractory obsessive-compulsive disorder (OCD). This study sought to explore gray matter structure in patients with treatment-refractory OCD and compare it with that of healthy controls. A total of 18 subjects with treatment-refractory OCD and 26 healthy volunteers were analyzed by MRI using a 3.0-T scanner and voxel-based morphometry (VBM). Diffeomorphic anatomical registration using exponentiated Lie algebra (DARTEL) was used to identify structural changes in gray matter associated with treatment-refractory OCD. A partial correlation model was used to analyze whether morphometric changes were associated with Yale-Brown Obsessive-Compulsive Scale scores and illness duration. Gray matter volume did not differ significantly between the two groups. Treatment-refractory OCD patients showed significantly lower gray matter density than healthy subjects in the left posterior cingulate cortex (PCC) and mediodorsal thalamus (MD) and significantly higher gray matter density in the left dorsal striatum (putamen). These changes did not correlate with symptom severity or illness duration. Our findings provide new evidence of deficits in gray matter density in treatment-refractory OCD patients. These patients may show characteristic density abnormalities in the left PCC, MD and dorsal striatum (putamen), which should be verified in longitudinal studies. © 2013. Published by Elsevier Inc. All rights reserved.
'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
Algorithmic and experimental methods in algebra, geometry, and number theory
Decker, Wolfram; Malle, Gunter
2017-01-01
This book presents state-of-the-art research and survey articles that highlight work done within the Priority Program SPP 1489 “Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory”, which was established and generously supported by the German Research Foundation (DFG) from 2010 to 2016. The goal of the program was to substantially advance algorithmic and experimental methods in the aforementioned disciplines, to combine the different methods where necessary, and to apply them to central questions in theory and practice. Of particular concern was the further development of freely available open source computer algebra systems and their interaction in order to create powerful new computational tools that transcend the boundaries of the individual disciplines involved. The book covers a broad range of topics addressing the design and theoretical foundations, implementation and the successful application of algebraic algorithms in order to solve mathematical research problems. It off...
From groups to categorial algebra introduction to protomodular and mal’tsev categories
Bourn, Dominique
2017-01-01
This book gives a thorough and entirely self-contained, in-depth introduction to a specific approach to group theory, in a large sense of that word. The focus lie on the relationships which a group may have with other groups, via “universal properties”, a view on that group “from the outside”. This method of categorical algebra, is actually not limited to the study of groups alone, but applies equally well to other similar categories of algebraic objects. By introducing protomodular categories and Mal’tsev categories, which form a larger class, the structural properties of the category Gp of groups, show how they emerge from four very basic observations about the algebraic litteral calculus and how, studied for themselves at the conceptual categorical level, they lead to the main striking features of the category Gp of groups. Hardly any previous knowledge of category theory is assumed, and just a little experience with standard algebraic structures such as groups and monoids. Examples and exercises...
On one method of realization of commutation relation algebra
International Nuclear Information System (INIS)
Sveshnikov, K.A.
1983-01-01
Method for constructing the commulation relation representations based on the purely algebraic construction of joined algebraic representation with specially selected composition law has been suggested9 Purely combinatorial construction realizing commulation relations representation has been obtained proceeding from formal equivalence of operatopr action on vector and adding a simbol to a sequences of symbols. The above method practically has the structure of calculating algorithm, which assigns some rule of ''word'' formation of an initial set of ''letters''. In other words, a computer language with definite relations between words (an analogy between quantum mechanics and computer linguistics has been applied)
Algebraic time-dependent variational approach to dynamical calculations
International Nuclear Information System (INIS)
Shi, S.; Rabitz, H.
1988-01-01
A set of time-dependent basis states is obtained with a group of unitary transformations generated by a Lie algebra. Applying the time-dependent variational principle to the trial function subspace constructed from the linear combination of the time-dependent basis states gives rise to a set of ''classical'' equations of motion for the group parameters and the expansion coefficients from which the time evolution of the system state can be determined. The formulation is developed for a general Lie algebra as well as for the commonly encountered algebra containing homogeneous polynominal products of the coordinate Q and momentum P operators (or equivalently the boson creation a/sup dagger/ and annihilation a operators) of order 0, 1, and 2. Explicit expressions for the transition amplitudes are derived by virtue of the cannonical transformation properties of the unitary transformation. The applicability of the present formalism in a variety of problems is implied by two illustrative examples: (a) a parametric amplifier; (b) the collinear collision of an atom with a Morse oscillator
On the algebraic realization of SU(4) symmetry
International Nuclear Information System (INIS)
Asatryan, G.M.; Zaslavsky, A.N.
1976-01-01
A possibility of nonlinear realization of the symmetry with linearization on the SU(4)xYxC group is discussed. Algebraic properties of SU(4) are restored from the Weinberg condition: amplitudes of goldstone scattering on particles should have a reasonable (as in the Regge theory) asymptotic behaviour. In this case the breaking appears to be minimal. Large values of psi meson masses lead to high-lying charmed trajectories in the SU(4) algebraic realization
Use of regularized algebraic methods in tomographic reconstruction
International Nuclear Information System (INIS)
Koulibaly, P.M.; Darcourt, J.; Blanc-Ferraud, L.; Migneco, O.; Barlaud, M.
1997-01-01
The algebraic methods are used in emission tomography to facilitate the compensation of attenuation and of Compton scattering. We have tested on a phantom the use of a regularization (a priori introduction of information), as well as the taking into account of spatial resolution variation with the depth (SRVD). Hence, we have compared the performances of the two methods by back-projection filtering (BPF) and of the two algebraic methods (AM) in terms of FWHM (by means of a point source), of the reduction of background noise (σ/m) on the homogeneous part of Jaszczak's phantom and of reconstruction speed (time unit = BPF). The BPF methods make use of a grade filter (maximal resolution, no noise treatment), single or associated with a Hann's low-pass (f c = 0.4), as well as of an attenuation correction. The AM which embody attenuation and scattering corrections are, on one side, the OS EM (Ordered Subsets, partitioning and rearranging of the projection matrix; Expectation Maximization) without regularization or SRVD correction, and, on the other side, the OS MAP EM (Maximum a posteriori), regularized and embodying the SRVD correction. A table is given containing for each used method (grade, Hann, OS EM and OS MAP EM) the values of FWHM, σ/m and time, respectively. One can observe that the OS MAP EM algebraic method allows ameliorating both the resolution, by taking into account the SRVD in the reconstruction process and noise treatment by regularization. In addition, due to the OS technique the reconstruction times are acceptable
Commutative algebra with a view toward algebraic geometry
Eisenbud, David
1995-01-01
Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algeb...
Universal enveloping algebras of Toda field theories and the light-cone asymmetry parameter
International Nuclear Information System (INIS)
Itoyama, H.; Moxhay, P.
1990-01-01
The generators of the universal enveloping algebras in Toda field theories associated with Lie algebras are constructed. These form spectrum-generating algebras of the system which survive the constraints acting on the larger current algebra structure. It is found that the same generators fail to be a symmetry in the case of affine Toda field theory despite their close relationship with Mandelstam's soliton operators. We introduce the light-cone asymmetry parameter; its significance and utility are demonstrated. (orig.)
On actions of algebraic groups | Omokaro | Journal of the Nigerian ...
African Journals Online (AJOL)
In [6] we discussed the Lie Algebra associated with an algebraic group G. In this work, we employ morphical action of G to obtain a necessary and sufficient condition for a finite dimensional subspace F of K[X] to be stable under all translations where K[X] denotes the set of polynomials in the variables x1,x2, …, xn.
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.
The Hamiltonian of the quantum trigonometric Calogero-Sutherland model in the exceptional algebra E8
International Nuclear Information System (INIS)
Fernandez Nunez, J; Garcia Fuertes, W; Perelomov, A M
2009-01-01
We express the Hamiltonian of the quantum trigonometric Calogero-Sutherland model for the Lie algebra E 8 and coupling constant κ by using the fundamental irreducible characters of the algebra as dynamical independent variables
An introduction to abstract algebra
Robinson, Derek JS
2003-01-01
This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and in the information and physical sciences. In addition to introducing the main concepts of modern algebra, the book contains numerous applications, which are intended to illustrate the concepts and to convince the reader of the utility and relevance of algebra today. In particular applications to Polya coloring theory, latin squares, Steiner systems and error correcting codes are described. Another feature of the book is that group theory and ring theory are carried further than is often done at this level. There is ample material here for a two semester course in abstract algebra. The importance of proof is stressed and rigorous proofs of almost all results are given. But care has been taken to lead the reader through the proofs by gentle stages. There are nearly 400 problems, of varying degrees of difficulty, to test the reader''s skill and progress. The book should be suitable for students ...
H. S. group: its algebra and its Galilei limit
Energy Technology Data Exchange (ETDEWEB)
De Ritis, R [Naples Univ. (Italy). Istituto di Fisica; Franchini, L [Dipartimento di Matematica dell' Universita della Calabria, Cosenza; Platania, G [Osservatorio Astronomico di Capodimonte, Naples (Italy)
1976-08-11
The infinitesimal generators of the invariance group suitable for the study of Newtonian cosmology are calculated. They form an infinite-dimensional Lie algebra, which is also studied in some particular limits.
Noncommutative Chern-Connes characters of some noncompact quantum algebras
International Nuclear Information System (INIS)
Do Ngoc Diep; Kuku, Aderemi O.
2001-09-01
We prove in this paper that the periodic cyclic homology of the quantized algebras of functions on coadjoint orbits of connected and simply connected Lie group, are isomorphic to the periodic cyclic homology of the quantized algebras of functions on coadjoint orbits of compact maximal subgroups, without localization. Some noncompact quantum groups and algebras were constructed and their irreducible representations were classified in recent works of Do Ngoc Diep and Nguyen Viet Hai [DH1]-[DH2] by using deformation quantization. In this paper we compute their K-groups, periodic cyclic homology groups and their Chern characters. (author)
Quantum algebras in phenomenological description of particle properties
International Nuclear Information System (INIS)
Gavrilik, A.M.
2001-01-01
Quantum and q-deformed algebras find their application not only in mathematical physics and field theoretical context, but also in phenomenology of particle properties. We describe (i) the use of quantum algebras U q (su n ) corresponding to Lie algebras of the groups SU n , taken for flavor symmetries of hadrons, in deriving new high-accuracy hadron mass sum rules, and (ii) the use of (multimode) q-oscillator algebras along with q-Bose gas picture in modelling the properties of the intercept λ of two-pion (two-kaon) correlations in heavy-ion collisions, as λ shows sizable observed deviation from the expected Bose-Einstein type behavior. The deformation parameter q is in case (i) argued and in case (ii) conjectured to be connected with the Cabibbo angle θ c
Infinite-parametric extension of the conformal algebra in D>2 space-time dimension
International Nuclear Information System (INIS)
Fradkin, E.S.; Linetsky, V.Ya.
1990-09-01
On the basis of the analytic continuations of semisimple Lie algebras discovered recently by us we construct manifestly quasiconformal infinite-dimensional algebras AC(so(4,1)) and PAC(so(3,2)) extending the conformal algebras in three-dimensional Euclidean and Minkowski space-time like the Virasoro algebra extends so(2,1). Their higher spin generalizations are also constructed. A counterpart of the central extension for D>2 and possible applications in exactly solvable conformal quantum field models in D>2 are discussed. (author). 31 refs, 2 figs
Singh, S; Modi, S; Bagga, D; Kaur, P; Shankar, L R; Khushu, S
2013-03-01
The present study aimed to investigate whether brain morphological differences exist between adult hypothyroid subjects and age-matched controls using voxel-based morphometry (VBM) with diffeomorphic anatomic registration via an exponentiated lie algebra algorithm (DARTEL) approach. High-resolution structural magnetic resonance images were taken in ten healthy controls and ten hypothyroid subjects. The analysis was conducted using statistical parametric mapping. The VBM study revealed a reduction in grey matter volume in the left postcentral gyrus and cerebellum of hypothyroid subjects compared to controls. A significant reduction in white matter volume was also found in the cerebellum, right inferior and middle frontal gyrus, right precentral gyrus, right inferior occipital gyrus and right temporal gyrus of hypothyroid patients compared to healthy controls. Moreover, no meaningful cluster for greater grey or white matter volume was obtained in hypothyroid subjects compared to controls. Our study is the first VBM study of hypothyroidism in an adult population and suggests that, compared to controls, this disorder is associated with differences in brain morphology in areas corresponding to known functional deficits in attention, language, motor speed, visuospatial processing and memory in hypothyroidism. © 2012 British Society for Neuroendocrinology.
Linear algebraic methods applied to intensity modulated radiation therapy.
Crooks, S M; Xing, L
2001-10-01
Methods of linear algebra are applied to the choice of beam weights for intensity modulated radiation therapy (IMRT). It is shown that the physical interpretation of the beam weights, target homogeneity and ratios of deposited energy can be given in terms of matrix equations and quadratic forms. The methodology of fitting using linear algebra as applied to IMRT is examined. Results are compared with IMRT plans that had been prepared using a commercially available IMRT treatment planning system and previously delivered to cancer patients.
Algebraic methods in statistical mechanics and quantum field theory
Emch, Dr Gérard G
2009-01-01
This systematic algebraic approach concerns problems involving a large number of degrees of freedom. It extends the traditional formalism of quantum mechanics, and it eliminates conceptual and mathematical difficulties common to the development of statistical mechanics and quantum field theory. Further, the approach is linked to research in applied and pure mathematics, offering a reflection of the interplay between formulation of physical motivations and self-contained descriptions of the mathematical methods.The four-part treatment begins with a survey of algebraic approaches to certain phys
International Nuclear Information System (INIS)
Hudetz, T.
1989-01-01
As a 'by-product' of the Connes-Narnhofer-Thirring theory of dynamical entropy for (originally non-Abelian) nuclear C * -algebras, the well-known variational principle for topological entropy is eqivalently reformulated in purly algebraically defined terms for (separable) Abelian C * -algebras. This 'algebraic variational principle' should not only nicely illustrate the 'feed-back' of methods developed for quantum dynamical systems to the classical theory, but it could also be proved directly by 'algebraic' methods and could thus further simplify the original proof of the variational principle (at least 'in principle'). 23 refs. (Author)
Super-Galilean conformal algebra in AdS/CFT
International Nuclear Information System (INIS)
Sakaguchi, Makoto
2010-01-01
Galilean conformal algebra (GCA) is an Inoenue-Wigner (IW) contraction of a conformal algebra, while Newton-Hooke string algebra is an IW contraction of an Anti-de Sitter (AdS) algebra, which is the isometry of an AdS space. It is shown that the GCA is a boundary realization of the Newton-Hooke string algebra in the bulk AdS. The string lies along the direction transverse to the boundary, and the worldsheet is AdS 2 . The one-dimensional conformal symmetry so(2,1) and rotational symmetry so(d) contained in the GCA are realized as the symmetry on the AdS 2 string worldsheet and rotational symmetry in the space transverse to the AdS 2 in AdS d+2 , respectively. It follows from this correspondence that 32 supersymmetric GCAs can be derived as IW contractions of superconformal algebras, psu(2,2|4), osp(8|4), and osp(8*|4). We also derive less supersymmetric GCAs from su(2,2|2), osp(4|4), osp(2|4), and osp(8*|2).
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. .
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.
Free Modal Algebras Revisited: The Step-by-Step Method
Bezhanishvili, N.; Ghilardi, Silvio; Jibladze, Mamuka
2012-01-01
We review the step-by-step method of constructing finitely generated free modal algebras. First we discuss the global step-by-step method, which works well for rank one modal logics. Next we refine the global step-by-step method to obtain the local step-by-step method, which is applicable beyond
Dynamical theory of subconstituents based on ternary algebras
International Nuclear Information System (INIS)
Bars, I.; Guenaydin, M.
1980-01-01
We propose a dynamical theory of possible fundamental constituents of matter. Our scheme is based on (super) ternary algebras which are building blocks of Lie (super) algebras. Elementary fields, called ''ternons,'' are associated with the elements of a (super) ternary algebra. Effective gauge bosons, ''quarks,'' and ''leptons'' are constructed as composite fields from ternons. We propose two- and four-dimensional (super) ternon theories whose structures are closely related to CP/sub N/ and Yang-Mills theories and their supersymmetric extensions. We conjecture that at large distances (low energies) the ternon theories dynamically produce effective gauge theories and thus may be capable of explaining the present particle-physics phenomenology. Such a scenario is valid in two dimensions
AIR Tools II: algebraic iterative reconstruction methods, improved implementation
DEFF Research Database (Denmark)
Hansen, Per Christian; Jørgensen, Jakob Sauer
2017-01-01
with algebraic iterative methods and their convergence properties. The present software is a much expanded and improved version of the package AIR Tools from 2012, based on a new modular design. In addition to improved performance and memory use, we provide more flexible iterative methods, a column-action method...
Coherent population transfer in multilevel systems with magnetic sublevels. II. Algebraic analysis
International Nuclear Information System (INIS)
Martin, J.; Shore, B.W.; Bergmann, K.
1995-01-01
We extend previous theoretical work on coherent population transfer by stimulated Raman adiabatic passage for states involving nonzero angular momentum. The pump and Stokes fields are either copropagating or counterpropagating with the corresponding linearly polarized electric-field vectors lying in a common plane with the magnetic-field direction. Zeeman splitting lifts the magnetic sublevel degeneracy. We present an algebraic analysis of dressed-state properties to explain the behavior noted in numerical studies. In particular, we discuss conditions which are likely to lead to a failure of complete population transfer. The applied strategy, based on simple methods of linear algebra, will also be successful for other types of discrete multilevel systems, provided the rotating-wave and adiabatic approximation are valid
Coherent population transfer in multilevel systems with magnetic sublevels. II. Algebraic analysis
Martin, J.; Shore, B. W.; Bergmann, K.
1995-07-01
We extend previous theoretical work on coherent population transfer by stimulated Raman adiabatic passage for states involving nonzero angular momentum. The pump and Stokes fields are either copropagating or counterpropagating with the corresponding linearly polarized electric-field vectors lying in a common plane with the magnetic-field direction. Zeeman splitting lifts the magnetic sublevel degeneracy. We present an algebraic analysis of dressed-state properties to explain the behavior noted in numerical studies. In particular, we discuss conditions which are likely to lead to a failure of complete population transfer. The applied strategy, based on simple methods of linear algebra, will also be successful for other types of discrete multilevel systems, provided the rotating-wave and adiabatic approximation are valid.
The hidden symmetries and their algebraic structure of the static axially symmetric SDYM fields
International Nuclear Information System (INIS)
Hao Sanru
1993-01-01
A new explicit transformation about the static axially symmetric self-dual Yang-Mills (SDYM) fields is presented. The theory has proved that the new transformation is a symmetric one. For the two kinds of the Lie algebraic generators of the Lie group SL (N. R) /SO (N), the corresponding transformations are given. By making use of the Yang-Baxter equality and their square brackets, the loop and conformal algebraic structures of the symmetric transformations for the basic fields have been obtained. All the results obtained can be directly generalized to the other models
Exact solution of some linear matrix equations using algebraic methods
Djaferis, T. E.; Mitter, S. K.
1977-01-01
A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.
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
Some nonunitary, indecomposable representations of the Euclidean algebra e(3)
International Nuclear Information System (INIS)
Douglas, Andrew; De Guise, Hubert
2010-01-01
The Euclidean group E(3) is the noncompact, semidirect product group E(3)≅R 3 x SO(3). It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra e(3) is the complexification of the Lie algebra of E(3). We construct three distinct families of finite-dimensional, nonunitary representations of e(3) and show that each representation is indecomposable. The representations of the first family are explicitly realized as subspaces of the polynomial ring F[X,Y,Z] with the action of e(3) given by differential operators. The other families are constructed via duals and tensor products of the representations within the first family. We describe subrepresentations, quotients and duals of these indecomposable representations.
Jurco, B.; Schraml, S.; Schupp, P.; Wess, J.
2000-11-01
An enveloping algebra-valued gauge field is constructed, its components are functions of the Lie algebra-valued gauge field and can be constructed with the Seiberg-Witten map. This allows the formulation of a dynamics for a finite number of gauge field components on non-commutative spaces.
Parity proofs of the Kochen–Specker theorem based on the Lie algebra E8
International Nuclear Information System (INIS)
Waegell, Mordecai; Aravind, P K
2015-01-01
The 240 root vectors of the Lie algebra E8 lead to a system of 120 rays in a real eight-dimensional Hilbert space that contains a large number of parity proofs of the Kochen–Specker (KS) theorem. After introducing the rays in a triacontagonal representation due to Coxeter, we present their KS diagram in the form of a ‘basis table’ showing all 2025 bases (i.e., sets of eight mutually orthogonal rays) formed by the rays. Only a few of the bases are actually listed, but simple rules are given, based on the symmetries of E8, for obtaining all the other bases from the ones shown. The basis table is an object of great interest because all the parity proofs of E8 can be exhibited as subsets of it. We show how the triacontagonal representation of E8 facilitates the identification of substructures that are more easily searched for their parity proofs. We have found hundreds of different types of parity proofs, ranging from 9 bases (or contexts) at the low end to 35 bases at the high end, and involving projectors of various ranks and multiplicities. After giving an overview of the proofs we found, we present a few concrete examples of the proofs that illustrate both their generic features as well as some of their more unusual properties. In particular, we present a proof involving 34 rays and 9 bases that appears to provide the most compact proof of the KS theorem found to date in eight-dimensions. (paper)
International Nuclear Information System (INIS)
Jurco, B.; Schraml, S.; Wess, J.; Schupp, P.
2000-01-01
An enveloping algebra-valued gauge field is constructed, its components are functions of the Lie algebra-valued gauge field and can be constructed with the Seiberg-Witten map. This allows the formulation of a dynamics for a finite number of gauge field components on non-commutative spaces. (orig.)
Energy Technology Data Exchange (ETDEWEB)
Jurco, B. [Max-Planck-Institut fuer Mathematik, Bonn (Germany); Schraml, S.; Wess, J. [Max-Planck-Institut fuer Physik, Foehringer Ring 6, 80805 Muenchen (Germany); Sektion Physik, Universitaet Muenchen, Theresienstrasse 37, 80333 Muenchen (Germany); Schupp, P. [Sektion Physik, Universitaet Muenchen, Theresienstrasse 37, 80333 Muenchen (Germany)
2000-11-01
An enveloping algebra-valued gauge field is constructed, its components are functions of the Lie algebra-valued gauge field and can be constructed with the Seiberg-Witten map. This allows the formulation of a dynamics for a finite number of gauge field components on non-commutative spaces. (orig.)
Interactions Between Representation Ttheory, Algebraic Topology and Commutative Algebra
Pitsch, Wolfgang; Zarzuela, Santiago
2016-01-01
This book includes 33 expanded abstracts of selected talks given at the two workshops "Homological Bonds Between Commutative Algebra and Representation Theory" and "Brave New Algebra: Opening Perspectives," and the conference "Opening Perspectives in Algebra, Representations, and Topology," held at the Centre de Recerca Matemàtica (CRM) in Barcelona between January and June 2015. These activities were part of the one-semester intensive research program "Interactions Between Representation Theory, Algebraic Topology and Commutative Algebra (IRTATCA)." Most of the abstracts present preliminary versions of not-yet published results and cover a large number of topics (including commutative and non commutative algebra, algebraic topology, singularity theory, triangulated categories, representation theory) overlapping with homological methods. This comprehensive book is a valuable resource for the community of researchers interested in homological algebra in a broad sense, and those curious to learn the latest dev...
An Improved Algebraic Method for Transit Signal Priority Scheme and Its Impact on Traffic Emission
Ji, Yanjie; Hu, Bo; Han, Jing; Tang, Dounan
2014-01-01
Transit signal priority has a positive effect on improving traffic congestion and reducing transit delay and also has an influence on traffic emission. In this paper, an optimal transit signal priority scheme based on an improved algebraic method was developed and its impact on vehicle emission was evaluated as well. The improved algebraic method was proposed on the basis of classical algebraic method and has improvements in three aspects. First, the calculation rules of split loss are more r...
Another Representation for the Maximal Lie Algebra of sl(n+2,ℝ in Terms of Operators
Directory of Open Access Journals (Sweden)
Tooba Feroze
2009-01-01
of the special linear group of order (n+2, over the real numbers, sl(n+2,ℝ. In this paper, we provide an alternate representation of the symmetry algebra by simple relabelling of indices. This provides one more proof of the result that the symmetry algebra of (ya″=0 is sl(n+2,ℝ.
Minimal deformation of the commutative algebra and the linear group GL(n)
International Nuclear Information System (INIS)
Zupnik, B.M.
1993-01-01
We consider the relations of generalized commutativity in the algebra of formal series M q (x i ), which conserve a tensor I q -graduation and depend on parameters q(i,k). We choose the I q -invariant version of differential calculus on M q . A new construction of the symmetrized tensor product for M q -type algebras and the corresponding definition of minimally deformed linear group QGL(n) and Lie algebra qgl(n) are proposed. We study the connection of QGL(n) and qgl(n) with the special matrix algebra Mat(n, Q) containing matrices with noncommutative elements. A definition of the deformed determinant in the algebra Mat(n, Q) is given. The exponential parametrization in the algebra Mat(n, Q) is considered on the basis of Campbell-Hausdorf formula
Non-asymptotic fractional order differentiators via an algebraic parametric method
Liu, Dayan; Gibaru, O.; Perruquetti, Wilfrid
2012-01-01
Recently, Mboup, Join and Fliess [27], [28] introduced non-asymptotic integer order differentiators by using an algebraic parametric estimation method [7], [8]. In this paper, in order to obtain non-asymptotic fractional order differentiators we apply this algebraic parametric method to truncated expansions of fractional Taylor series based on the Jumarie's modified Riemann-Liouville derivative [14]. Exact and simple formulae for these differentiators are given where a sliding integration window of a noisy signal involving Jacobi polynomials is used without complex mathematical deduction. The efficiency and the stability with respect to corrupting noises of the proposed fractional order differentiators are shown in numerical simulations. © 2012 IEEE.
Non-asymptotic fractional order differentiators via an algebraic parametric method
Liu, Dayan
2012-08-01
Recently, Mboup, Join and Fliess [27], [28] introduced non-asymptotic integer order differentiators by using an algebraic parametric estimation method [7], [8]. In this paper, in order to obtain non-asymptotic fractional order differentiators we apply this algebraic parametric method to truncated expansions of fractional Taylor series based on the Jumarie\\'s modified Riemann-Liouville derivative [14]. Exact and simple formulae for these differentiators are given where a sliding integration window of a noisy signal involving Jacobi polynomials is used without complex mathematical deduction. The efficiency and the stability with respect to corrupting noises of the proposed fractional order differentiators are shown in numerical simulations. © 2012 IEEE.
Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group
International Nuclear Information System (INIS)
Majid, S.
1993-01-01
Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups U q (g). They have the same FRT generators l ± but a matrix braided-coproduct ΔL=LxL, where L=l + Sl - , and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices BM 1 (2); it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum double D(U q (sl 2 )) (also known as the 'quantum Lorentz group') is the semidirect product as an algebra of two copies of U q (sl 2 ), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-limits as doubles of the Lie algebras of Poisson Lie groups. (orig.)
A Multi-component Matrix Loop Algebra and Its Application
International Nuclear Information System (INIS)
Dong Huanhe; Zhang Ning
2005-01-01
A set of multi-component matrix Lie algebra is constructed. It follows that a type of new loop algebra A M-1 is presented. An isospectral problem is established. Integrable multi-component hierarchy is obtained by Tu pattern, which possesses tri-Hamiltonian structures. Furthermore, it can be reduced to the well-known AKNS hierarchy and BPT hierarchy. Therefore, the major result of this paper can be regarded as a unified expression integrable model of the AKNS hierarchy and the BPT hierarchy.
Algebraic theory of locally nilpotent derivations
Freudenburg, Gene
2017-01-01
This book explores the theory and application of locally nilpotent derivations, a subject motivated by questions in affine algebraic geometry and having fundamental connections to areas such as commutative algebra, representation theory, Lie algebras and differential equations. The author provides a unified treatment of the subject, beginning with 16 First Principles on which the theory is based. These are used to establish classical results, such as Rentschler's Theorem for the plane and the Cancellation Theorem for Curves. More recent results, such as Makar-Limanov's theorem for locally nilpotent derivations of polynomial rings, are also discussed. Topics of special interest include progress in classifying additive actions on three-dimensional affine space, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem. A lot of new material is included in this expanded second edition, such as canonical factoriza...
Representations of the q-deformed algebras Uq (so2,1) and Uq (so3,1)
International Nuclear Information System (INIS)
Gavrilik, O.M.; Klimyk, A.U.
1993-01-01
Representations of algebra U q (so 2 ,1) are studied. This algebra is a q-deformation of the universal enveloping algebra U(so 2 ,1) of the Lie algebra of the group SO 0 (2,1) and differs from the quantum algebra U q (SU 1 ,1). Classifications of irreducible representations and of infinitesimally irreducible representations of U q (SU 1 ,1). The sets of irreducible representations and of infinitesimally unitary irreducible representations of the algebra U q (so 3 ,1) are given. We also consider representations of U q (so n ,1) which are of class 1 with respect to subalgebra U q (so n ). (author). 22 refs
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
Topics in noncommutative algebra the theorem of Campbell, Baker, Hausdorff and Dynkin
Bonfiglioli, Andrea
2012-01-01
Motivated by the importance of the Campbell, Baker, Hausdorff, Dynkin Theorem in many different branches of Mathematics and Physics (Lie group-Lie algebra theory, linear PDEs, Quantum and Statistical Mechanics, Numerical Analysis, Theoretical Physics, Control Theory, sub-Riemannian Geometry), this monograph is intended to: 1) fully enable readers (graduates or specialists, mathematicians, physicists or applied scientists, acquainted with Algebra or not) to understand and apply the statements and numerous corollaries of the main result; 2) provide a wide spectrum of proofs from the modern literature, comparing different techniques and furnishing a unifying point of view and notation; 3) provide a thorough historical background of the results, together with unknown facts about the effective early contributions by Schur, Poincaré, Pascal, Campbell, Baker, Hausdorff and Dynkin; 4) give an outlook on the applications, especially in Differential Geometry (Lie group theory) and Analysis (PDEs of subelliptic type); ...
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
BPS algebras, genus zero and the heterotic Monster
Paquette, Natalie M.; Persson, Daniel; Volpato, Roberto
2017-10-01
In this note, we expand on some technical issues raised in (Paquette et al 2016 Commun. Number Theory Phys. 10 433-526) by the authors, as well as providing a friendly introduction to and summary of our previous work. We construct a set of heterotic string compactifications to 0 + 1 dimensions intimately related to the Monstrous moonshine module of Frenkel, Lepowsky, and Meurman (and orbifolds thereof). Using this model, we review our physical interpretation of the genus zero property of Monstrous moonshine. Furthermore, we show that the space of (second-quantized) BPS-states forms a module over the Monstrous Lie algebras mg —some of the first and most prominent examples of Generalized Kac-Moody algebras—constructed by Borcherds and Carnahan. In particular, we clarify the structure of the module present in the second-quantized string theory. We also sketch a proof of our methods in the language of vertex operator algebras, for the interested mathematician.
Lie Symmetry Analysis of the Inhomogeneous Toda Lattice Equation via Semi-Discrete Exterior Calculus
International Nuclear Information System (INIS)
Liu Jiang; Wang Deng-Shan; Yin Yan-Bin
2017-01-01
In this work, the Lie point symmetries of the inhomogeneous Toda lattice equation are obtained by semi-discrete exterior calculus, which is a semi-discrete version of Harrison and Estabrook’s geometric approach. A four-dimensional Lie algebra and its one-, two- and three-dimensional subalgebras are given. Two similarity reductions of the inhomogeneous Toda lattice equation are obtained by using the symmetry vectors. (paper)
Z2×Z2 generalizations of 𝒩 =2 super Schrödinger algebras and their representations
Aizawa, N.; Segar, J.
2017-11-01
We generalize the real and chiral N =2 super Schrödinger algebras to Z2×Z2-graded Lie superalgebras. This is done by D-module presentation, and as a consequence, the D-module presentations of Z2×Z2-graded superalgebras are identical to the ones of super Schrödinger algebras. We then generalize the calculus over the Grassmann number to Z2×Z2 setting. Using it and the standard technique of Lie theory, we obtain a vector field realization of Z2×Z2-graded superalgebras. A vector field realization of the Z2×Z2 generalization of N =1 super Schrödinger algebra is also presented.
The bubble algebra: structure of a two-colour Temperley-Lieb Algebra
International Nuclear Information System (INIS)
Grimm, Uwe; Martin, Paul P
2003-01-01
We define new diagram algebras providing a sequence of multiparameter generalizations of the Temperley-Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional statistical mechanics. These algebras give a rigorous foundation to the various 'multi-colour algebras' of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to a wide variety of such algebras occurring in statistical mechanics. We demonstrate how these algebras may be used to solve the Yang-Baxter equations
K-theory and periodic cyclic homology of some noncompact quantum algebras
International Nuclear Information System (INIS)
Do Ngoc Diep; Kuku, Aderemi O.
2003-07-01
We prove in this paper that the periodic cyclic homology of the quantized algebras of functions on coadjoint orbits of connected and simply connected Lie group, are isomorphic to the periodic cyclic homology of the quantized algebras of functions on coadjoint orbits of compact maximal subgroups, without localization. Some noncompact quantum groups and algebras were constructed and their irreducible representations were classified in recent works of Do Ngoc Diep and Nguyen Viet Hai [DH1]-[DH2] and Do Due Hanh [DD] by using deformation quantization. In this paper we compute their K-groups, periodic cyclic homology groups and their Chern characters. (author)
Miyanishi, Masayoshi
2000-01-01
Open algebraic surfaces are a synonym for algebraic surfaces that are not necessarily complete. An open algebraic surface is understood as a Zariski open set of a projective algebraic surface. There is a long history of research on projective algebraic surfaces, and there exists a beautiful Enriques-Kodaira classification of such surfaces. The research accumulated by Ramanujan, Abhyankar, Moh, and Nagata and others has established a classification theory of open algebraic surfaces comparable to the Enriques-Kodaira theory. This research provides powerful methods to study the geometry and topology of open algebraic surfaces. The theory of open algebraic surfaces is applicable not only to algebraic geometry, but also to other fields, such as commutative algebra, invariant theory, and singularities. This book contains a comprehensive account of the theory of open algebraic surfaces, as well as several applications, in particular to the study of affine surfaces. Prerequisite to understanding the text is a basic b...
Study on time of flight property of electron optical systems by differential algebraic method
International Nuclear Information System (INIS)
Cheng Min; Tang Tiantong; Yao Zhenhua
2002-01-01
Differential algebraic method is a powerful and promising technique in computer numerical analysis. When applied to nonlinear dynamics systems, the arbitrary high-order transfer properties of the systems can be computed directly with high precision. In this paper, the principle of differential algebra is applied to study on the time of flight (TOF) property of electron optical systems and their arbitrary order TOF transfer properties can be numerically calculated out. As an example, TOF transfer properties of a uniform magnetic sector field analyzer have been studied by differential algebraic method. Relative errors of the first-order and second-order TOF transfer coefficients of the magnetic sector field analyzer are of the order 10 -11 or smaller compared with the analytic solutions. It is proved that differential algebraic TOF method is of high accuracy and very helpful for high-order TOF transfer property analysis of electron optical systems. (author)
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....
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.
Equations of motion for a spectrum-generating algebra: Lipkin-Meshkov-Glick model
International Nuclear Information System (INIS)
Rosensteel, G; Rowe, D J; Ho, S Y
2008-01-01
For a spectrum-generating Lie algebra, a generalized equations-of-motion scheme determines numerical values of excitation energies and algebra matrix elements. In the approach to the infinite particle number limit or, more generally, whenever the dimension of the quantum state space is very large, the equations-of-motion method may achieve results that are impractical to obtain by diagonalization of the Hamiltonian matrix. To test the method's effectiveness, we apply it to the well-known Lipkin-Meshkov-Glick (LMG) model to find its low-energy spectrum and associated generator matrix elements in the eigenenergy basis. When the dimension of the LMG representation space is 10 6 , computation time on a notebook computer is a few minutes. For a large particle number in the LMG model, the low-energy spectrum makes a quantum phase transition from a nondegenerate harmonic vibrator to a twofold degenerate harmonic oscillator. The equations-of-motion method computes critical exponents at the transition point
Orbifold Riemann surfaces: Teichmueller spaces and algebras of geodesic functions
Energy Technology Data Exchange (ETDEWEB)
Mazzocco, Marta [Loughborough University, Loughborough (United Kingdom); Chekhov, Leonid O [Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center), Moscow (Russian Federation)
2009-12-31
A fat graph description is given for Teichmueller spaces of Riemann surfaces with holes and with Z{sub 2}- and Z{sub 3}-orbifold points (conical singularities) in the Poincare uniformization. The corresponding mapping class group transformations are presented, geodesic functions are constructed, and the Poisson structure is introduced. The resulting Poisson algebras are then quantized. In the particular cases of surfaces with n Z{sub 2}-orbifold points and with one and two holes, the respective algebras A{sub n} and D{sub n} of geodesic functions (classical and quantum) are obtained. The infinite-dimensional Poisson algebra D{sub n}, which is the semiclassical limit of the twisted q-Yangian algebra Y'{sub q}(o{sub n}) for the orthogonal Lie algebra o{sub n}, is associated with the algebra of geodesic functions on an annulus with n Z{sub 2}-orbifold points, and the braid group action on this algebra is found. From this result the braid group actions are constructed on the finite-dimensional reductions of this algebra: the p-level reduction and the algebra D{sub n}. The central elements for these reductions are found. Also, the algebra D{sub n} is interpreted as the Poisson algebra of monodromy data of a Frobenius manifold in the vicinity of a non-semisimple point. Bibliography: 36 titles.
Regular algebraic monoids | Oduwale | Journal of the Nigerian ...
African Journals Online (AJOL)
... in terms of their unit groups, and related toroidal data. That is what we accomplish here. Keywords: Unipotent radical, . toroidal data, normal, irreducible, regular, zero element, maximal torus, Zariski closure, Lie algebra. Journal of the Nigerian Association of Mathematical Physics, Volume 19 (November, 2011), pp 37 – 40 ...
On the economical solution method for a system of linear algebraic equations
Directory of Open Access Journals (Sweden)
Jan Awrejcewicz
2004-01-01
Full Text Available The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in ℝ3 is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error of O(hx12+hx22. The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.
Givant, Steven
2017-01-01
This monograph details several different methods for constructing simple relation algebras, many of which are new with this book. By drawing these seemingly different methods together, all are shown to be aspects of one general approach, for which several applications are given. These tools for constructing and analyzing relation algebras are of particular interest to mathematicians working in logic, algebraic logic, or universal algebra, but will also appeal to philosophers and theoretical computer scientists working in fields that use mathematics. The book is written with a broad audience in mind and features a careful, pedagogical approach; an appendix contains the requisite background material in relation algebras. Over 400 exercises provide ample opportunities to engage with the material, making this a monograph equally appropriate for use in a special topics course or for independent study. Readers interested in pursuing an extended background study of relation algebras will find a comprehensive treatme...
Fermion current algebras and Schwinger terms in (3+1)-dimensions
International Nuclear Information System (INIS)
Langmann, E.
1994-01-01
We discuss the restricted linear group in infinite dimensions modeled by the Schatten class of rank 2p=4 which contains the (3+1)-dimensional analogs of the loop groups and is closely related to Yang-Mills theory with fermions in (3+1)-dimensions. We give an alternative to the construction of the ''highest weight'' representation of this group found by Mickelsson and Rajeev. Our approach is close to quantum field theory, with the elements of this group regarded as Bogoliubov transformations for fermions in an external Yang-Mills field. Though these cannot be unitarily implemented in the physically relevant representation of the fermion field algebra, we argue that they can be implemented by sesquilinear forms, and that there is a (regularized) product of forms providing an appropriate group structure. On the Lie algebra level, this gives an explicit, non-perturbative construction of fermion current algebras in (3+1) space-time dimensions which explicitly shows that the ''wave function renormalization'' required for a consistent definition of the currents and their Lie bracket naturally leads to the Schwinger term identical with the Mickelsson-Rajeev cocycle. Though the explicit form of the Schwinger term is given only for the case p=2, our arguments apply also to the restricted linear groups modeled by Schatten classes of rank 2p=6, 8, .. corresponding to current algebras in (d+1)-dimensions, d=5, 7, .. (orig.)
Quantum affine algebras and deformations of the virasoro and W-algebras
International Nuclear Information System (INIS)
Frenkel, E.; Reshetikhin, N.
1996-01-01
Using the Wakimoto realization of quantum affine algebras we define new Poisson algebras, which are q-deformations of the classical W-algebras. We also define their free field realizations, i.e. homomorphisms into some Heisenberg-Poisson algebras. The formulas for these homomorphisms coincide with formulas for spectra of transfer-matrices in the corresponding quantum integrable models derived by the Bethe-Ansatz method. (orig.)
International Nuclear Information System (INIS)
Ludu, A.; Greiner, M.
1995-09-01
A non-linear associative algebra is realized in terms of translation and dilation operators, and a wavelet structure generating algebra is obtained. We show that this algebra is a q-deformation of the Fourier series generating algebra, and reduces to this for certain value of the deformation parameter. This algebra is also homeomorphic with the q-deformed su q (2) algebra and some of its extensions. Through this algebraic approach new methods for obtaining the wavelets are introduced. (author). 20 refs
An Algebraic Method of Synchronous Pulsewidth Modulation for Converters for Adjustable Speed Drives
DEFF Research Database (Denmark)
Oleschuk, Valentin; Blaabjerg, Frede
2002-01-01
This paper describes the basic peculiarities of a new method of feedforward synchronous pulsewidth modulation (PWM) of the output voltage of converters, based on one-stage closed-form strategy of PWM with pure algebraic control dependencies. It is applied to voltage source inverters with a contin......This paper describes the basic peculiarities of a new method of feedforward synchronous pulsewidth modulation (PWM) of the output voltage of converters, based on one-stage closed-form strategy of PWM with pure algebraic control dependencies. It is applied to voltage source inverters...... with a continuous scheme of conventional voltage space vector modulation and with two basic variants of symmetrical discontinuous PWM. Simulations give the behaviour of the proposed method and show the advantage of algebraic synchronous PWM compared with the typical asynchronous, for low indices of the frequency...
Applied algebra codes, ciphers and discrete algorithms
Hardy, Darel W; Walker, Carol L
2009-01-01
This book attempts to show the power of algebra in a relatively simple setting.-Mathematical Reviews, 2010… The book supports learning by doing. In each section we can find many examples which clarify the mathematics introduced in the section and each section is followed by a series of exercises of which approximately half are solved in the end of the book. Additional the book comes with a CD-ROM containing an interactive version of the book powered by the computer algebra system Scientific Notebook. … the mathematics in the book are developed as needed and the focus of the book lies clearly o
2nd EACA International School on Computer Algebra and its Applications
Gimenez, Philippe; Sáenz-de-Cabezón, Eduardo
2017-01-01
Featuring up-to-date coverage of three topics lying at the intersection of combinatorics and commutative algebra, namely Koszul algebras, primary decompositions and subdivision operations in simplicial complexes, this book has its focus on computations. "Computations and combinatorics in commutative algebra" has been written by experts in both theoretical and computational aspects of these three subjects and is aimed at a broad audience, from experienced researchers who want to have an easy but deep review of the topics covered to postgraduate students who need a quick introduction to the techniques. The computational treatment of the material, including plenty of examples and code, will be useful for a wide range of professionals interested in the connections between commutative algebra and combinatorics.
Algebraic definition of topological W gravity
International Nuclear Information System (INIS)
Hosono, S.
1992-01-01
In this paper, the authors propose a definition of the topological W gravity using some properties of the principal three-dimensional subalgebra of a simple Lie algebra due to Kostant. In the authors' definition, structures of the two-dimensional topological gravity are naturally embedded in the extended theories. In accordance with the definition, the authors will present some explicit calculations for the W 3 gravity
Algebras of functions on compact quantum groups, Schubert cells and quantum tori
International Nuclear Information System (INIS)
Levendorskij, S.; Soibelman, Ya.
1991-01-01
The structure of Poisson Lie groups on a simple compact group are parametrized by pairs (a, u), where aelement ofR, uelement ofΛ 2 f R , and f R is a real Cartan subalgebra of complexification of Lie algebra of the group in question. In the present article the description of the symplectic leaves for all pairs (a, u) is given. Also, the corresponding quantized algebras of functions are constructed and their irreducible representations are described. In the course of investigation Schubert cells and quantum tori appear. At the end of the article the quantum analog of the Weyl group is constructed and some of its applications, among them the formula for the universal R-matrix, are given. (orig.)
Institute of Scientific and Technical Information of China (English)
Daxing Wu; Ying Zhao; Jian Liao; Huifang Yin; Wei Wang
2011-01-01
Voxel-based morphometry-diffeomorphic anatomical registration using exponentiated lie algebra analysis was used to investigate the structural characteristics of white matter in young males with antisocial personality disorder (APD) and healthy controls without APD. The results revealed that APD subjects, relative to healthy subjects, exhibited increased white matter volume in the bilateral prefrontal lobe, right insula, precentral gyrus, bilateral superior temporal gyrus, right postcentral gyrus, right inferior parietal lobule, right precuneus, right middle occipital lobe, right parahippocampal gyrus and bilateral cingulate, and decreased volume in the middle temporal cortex and right cerebellum. The white matter volume in the medial frontal gyrus was significantly correlated with antisocial type scores on the Personality Diagnostic Questionnaire in APD subjects. These experimental findings indicate that white matter abnormalities in several brain areas may contribute to antisocial behaviors in APD subjects.