DEFF Research Database (Denmark)
Ara, Pere; Dykema, Kenneth J.; Rørdam, Mikael
2013-01-01
The proofs of Theorem 2.2 of K. J. Dykema and M. Rørdam, Purely infinite simple C∗-algebras arising from free product constructions}, Canad. J. Math. 50 (1998), 323--341 and of Theorem 3.1 of K. J. Dykema, Purely infinite simple C∗-algebras arising from free product constructions, II, Math. Scand...
Pure homology of algebraic varieties
Weber, Andrzej
2003-01-01
We show that for a complete complex algebraic variety the pure component of homology coincides with the image of intersection homology. Therefore pure homology is topologically invariant. To obtain slightly more general results we introduce "image homology" for noncomplete varieties.
Pure spinors, free differential algebras, and the supermembrane
International Nuclear Information System (INIS)
Fre, Pietro; Grassi, Pietro Antonio
2007-01-01
The Lagrangian formalism for the supermembrane in any 11d supergravity background is constructed in the pure spinor framework. Our gauge-fixed action is manifestly BRST, supersymmetric, and 3d Lorentz invariant. The relation between the Free Differential Algebras (FDA) underlying 11d supergravity and the BRST symmetry of the membrane action is exploited. The 'gauge-fixing' has a natural interpretation as the variation of the Chevalley cohomology class needed for the extension of 11d super-Poincare superalgebra to M-theory FDA. We study the solution of the pure spinor constraints in full detail
Pure spinors, free differential algebras, and the supermembrane
Energy Technology Data Exchange (ETDEWEB)
Fre, Pietro [Dipartimento di Fisica Teorica, Universita di Torino, and INFN-Sezione di Torino, Via P. Giuria 1, I-10125 Turin (Italy); Grassi, Pietro Antonio [Centro Studi e Ricerche E. Fermi, Compendio Viminale, I-00184 Rome (Italy) and DISTA, Universita del Piemonte Orientale, and INFN-Sezione di Torino, Via Bellini 25/G, Alessandria 15100 (Italy) and CERN, Theory Unit, CH-1211 Geneva 23 (Switzerland)]. E-mail: pgrassi@insti.physics.sunysb.edu
2007-02-12
The Lagrangian formalism for the supermembrane in any 11d supergravity background is constructed in the pure spinor framework. Our gauge-fixed action is manifestly BRST, supersymmetric, and 3d Lorentz invariant. The relation between the Free Differential Algebras (FDA) underlying 11d supergravity and the BRST symmetry of the membrane action is exploited. The 'gauge-fixing' has a natural interpretation as the variation of the Chevalley cohomology class needed for the extension of 11d super-Poincare superalgebra to M-theory FDA. We study the solution of the pure spinor constraints in full detail.
Purely infinite C*-algebras arising from crossed products
DEFF Research Database (Denmark)
Rørdam, Mikael; Sierakowski, Adam
2012-01-01
We study conditions that will ensure that a crossed product of a C-algebra by a discrete exact group is purely innite (simple or non-simple). We are particularly interested in the case of a discrete nonamenable exact group acting on a commutative C-algebra, where our sucient conditions can...
Constructing Meanings and Utilities within Algebraic Tasks
Ainley, Janet; Bills, Liz; Wilson, Kirsty
2004-01-01
The Purposeful Algebraic Activity project aims to explore the potential of spreadsheets in the introduction to algebra and algebraic thinking. We discuss two sub-themes within the project: tracing the development of pupils' construction of meaning for variable from arithmetic-based activity, through use of spreadsheets, and into formal algebra,…
Pure Jauch-Piron states on von Neumann algebras
International Nuclear Information System (INIS)
Hamhalter, J.
1993-01-01
We study Jauch-Piron states and two-valued measures on von Neumann algebra. We prove as the main result that, under some set-theoretical assumption, a pure state of a von Neumann algebra A not containing a central abelian portion is Jauch-Piron if and only if it is σ-additive. Moreover, we show that this result holds for type I factor indenpendently on the set-theoretical axiomatics. As a consequence we obtain a lucid characterization of pure Jauch-Piron states on von Neumann algebras acting on a Hilbert space with real-nonmeasurable dimension (this can be viewed as a generalization of the paper). We also characterize the von Neumann algebras whose logic of projections is Jauch-Piron. Finally, we prove that every two-valued measure on the projection logic of A, where A contains no type I 2 central portion, has to be concentrated at an abelian direct summand of A. (orig.)
Automatic Construction of Finite Algebras
Institute of Scientific and Technical Information of China (English)
张健
1995-01-01
This paper deals with model generation for equational theories,i.e.,automatically generating (finite)models of a given set of (logical) equations.Our method of finite model generation and a tool for automatic construction of finite algebras is described.Some examples are given to show the applications of our program.We argue that,the combination of model generators and theorem provers enables us to get a better understanding of logical theories.A brief comparison betwween our tool and other similar tools is also presented.
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...
A purely infinite AH-algebra and an application to AF-embeddability
Rordam, Mikael
2002-01-01
We show that there exists a purely infinite AH-algebra. The AH-algebra arises as an inductive limit of C*-algebras of the form C_0([0,1),M_k) and it absorbs the Cuntz algebra O_\\infty tensorially. Thus one can reach an O_\\infty-absorbing C*-algebra as an inductive limit of the finite and elementary C*-algebras C_0([0,1),M_k). As an application we give a new proof of a recent theorem of Ozawa that the cone over any separable exact C*-algebra is AF-embeddable, and we exhibit a concrete AF-algeb...
Fermionic constructions of exceptional Kac-Moody algebras
International Nuclear Information System (INIS)
Schwimmer, A.
1985-01-01
The author discusses the fermionic representations of SO(2n) Kac Moody algebras. He describes construction of the E/sub 8/ algebra in terms of free fermionic operators, and generalises procedures for the basic representations of the Kac-Moody algebras appearing in Freudenthal's magic square
Non-freely generated W-algebras and construction of N=2 super W-algebras
International Nuclear Information System (INIS)
Blumenhagen, R.
1994-07-01
Firstly, we investigate the origin of the bosonic W-algebras W(2, 3, 4, 5), W(2, 4, 6) and W(2, 4, 6) found earlier by direct construction. We present a coset construction for all three examples leading to a new type of finitely, non-freely generated quantum W-algebras, which we call unifying W-algebras. Secondly, we develop a manifest covariant formalism to construct N = 2 super W-algebras explicitly on a computer. Applying this algorithm enables us to construct the first four examples of N = 2 super W-algebras with two generators and the N = 2 super W 4 algebra involving three generators. The representation theory of the former ones shows that all examples could be divided into four classes, the largest one containing the N = 2 special type of spectral flow algebras. Besides the W-algebra of the CP(3) Kazama-Suzuki coset model, the latter example with three generators discloses a second solution which could also be explained as a unifying W-algebra for the CP(n) models. (orig.)
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.)
Construction of the K=8 fractional superconformal algebras
International Nuclear Information System (INIS)
Argyres, P.C.; Grochocinski, J.M.; Tye, S.H.H.
1993-01-01
We construct the K=8 fractional superconformal algebras. There are two such extended Virasoro algebras, one of which was constructed earlier, involving a fractional spin (equivalently, conformal dimension) 6/5 current. The new algebra involves two additional fractional spin currents with spin 13/5. Both algebras are non-local and satisfy non-abelian braiding relations. The construction of the algebras uses the ismorphism between the Z 8 parafermion theory and the tensor product of two tricritical Ising models. For the special value of the central charge c=52/55, corresponding to the eighth member of the unitary minimal series, the 13/5 currents of the new algebra decouple, while two spin 23/5 currents (level-2 current algebra descendants of the 13/5 currents) emerge. In addition, it is shown that the K=8 algebra involving the spin 13/5 currents at central charge c=12/5 is the appropriate algebra for the construction of the K=8 (four-dimensional) fractional superstring. (orig.)
Construction Example for Algebra System Using Harmony Search Algorithm
Directory of Open Access Journals (Sweden)
FangAn Deng
2015-01-01
Full Text Available The construction example of algebra system is to verify the existence of a complex algebra system, and it is a NP-hard problem. In this paper, to solve this kind of problems, firstly, a mathematical optimization model for construction example of algebra system is established. Secondly, an improved harmony search algorithm based on NGHS algorithm (INGHS is proposed to find as more solutions as possible for the optimization model; in the proposed INGHS algorithm, to achieve the balance between exploration power and exploitation power in the search process, a global best strategy and parameters dynamic adjustment method are present. Finally, nine construction examples of algebra system are used to evaluate the optimization model and performance of INGHS. The experimental results show that the proposed algorithm has strong performance for solving complex construction example problems of algebra system.
Construction and decoding of a class of algebraic geometry codes
DEFF Research Database (Denmark)
Justesen, Jørn; Larsen, Knud J.; Jensen, Helge Elbrønd
1989-01-01
A class of codes derived from algebraic plane curves is constructed. The concepts and results from algebraic geometry that were used are explained in detail; no further knowledge of algebraic geometry is needed. Parameters, generator and parity-check matrices are given. The main result is a decod...... is a decoding algorithm which turns out to be a generalization of the Peterson algorithm for decoding BCH decoder codes......A class of codes derived from algebraic plane curves is constructed. The concepts and results from algebraic geometry that were used are explained in detail; no further knowledge of algebraic geometry is needed. Parameters, generator and parity-check matrices are given. The main result...
Twisted vertex algebras, bicharacter construction and boson-fermion correspondences
International Nuclear Information System (INIS)
Anguelova, Iana I.
2013-01-01
The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new boson-fermion correspondence of type D-A. Further, we define a new concept of twisted vertex algebra of order N, which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing classes of examples of twisted vertex algebras, as well as for deriving formulas for the operator product expansions, analytic continuations, and normal ordered products. By using the underlying Hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for two important groups of examples. We show that the correspondences of types B, C, and D-A are isomorphisms of twisted vertex algebras
Fermionic construction of vertex operators for twisted affine algebras
International Nuclear Information System (INIS)
Frappat, L.; Sorba, P.; Sciarrino, A.
1988-03-01
We construct vertex operator representations of the twisted affine algebras in terms of fermionic (or parafermionic in some cases) elementary fields. The folding method applied to the extended Dynkin diagrams of the affine algebras allows us to determine explicitly these fermionic fields as vertex operators
Constructing canonical bases of quantized enveloping algebras
Graaf, W.A. de
2001-01-01
An algorithm for computing the elements of a given weight of the canonical basis of a quantized enveloping algebra is described. Subsequently, a similar algorithm is presented for computing the canonical basis of a finite-dimensional module.
An invitation to general algebra and universal constructions
Bergman, George M
2015-01-01
Rich in examples and intuitive discussions, this book presents General Algebra using the unifying viewpoint of categories and functors. Starting with a survey, in non-category-theoretic terms, of many familiar and not-so-familiar constructions in algebra (plus two from topology for perspective), the reader is guided to an understanding and appreciation of the general concepts and tools unifying these constructions. Topics include: set theory, lattices, category theory, the formulation of universal constructions in category-theoretic terms, varieties of algebras, and adjunctions. A large number of exercises, from the routine to the challenging, interspersed through the text, develop the reader's grasp of the material, exhibit applications of the general theory to diverse areas of algebra, and in some cases point to outstanding open questions. Graduate students and researchers wishing to gain fluency in important mathematical constructions will welcome this carefully motivated book.
Normal pure states of the von Nuemann algebra of bounded operators as Kaehler manifold
International Nuclear Information System (INIS)
Cirelli, R.; Lanzavecchia, P.; Mania, A.
1983-01-01
The projective space of a complex Hilbert space H is considered both as a Kaehler manifold and as the set of pure states of the von Neumann algebra B(H). A link is given between these two structures. Special attention is devoted to topology, orientation and automorphisms of the structures and Wigner's theorem. (author)
Constructive Learning in Undergraduate Linear Algebra
Chandler, Farrah Jackson; Taylor, Dewey T.
2008-01-01
In this article we describe a project that we used in our undergraduate linear algebra courses to help our students successfully master fundamental concepts and definitions and generate interest in the course. We describe our philosophy and discuss the projects overall success.
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)
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
Supersymmetric construction of exactly solvable potentials and nonlinear algebras
International Nuclear Information System (INIS)
Junker, G.; Roy, P.
1998-01-01
Using algebraic tools of supersymmetric quantum mechanics we construct classes of conditionally exactly solvable potentials being the supersymmetric partners of the linear or radial harmonic oscillator. With the help of the raising and lowering operators of these harmonic oscillators and the SUSY operators we construct ladder operators for these new conditionally solvable systems. It is found that these ladder operators together with the Hamilton operator form a nonlinear algebra which is of quadratic and cubic type for the SUSY partners of the linear and radial harmonic oscillator
On the algebra of local unitary invariants of pure and mixed quantum states
International Nuclear Information System (INIS)
Vrana, Peter
2011-01-01
We study the structure of the inverse limit of the graded algebras of local unitary invariant polynomials using its Hilbert series. For k subsystems, we show that the inverse limit is a free algebra and the number of algebraically independent generators with homogenous degree 2m equals the number of conjugacy classes of index m subgroups in a free group on k - 1 generators. Similarly, we show that the inverse limit in the case of k-partite mixed state invariants is free and the number of algebraically independent generators with homogenous degree m equals the number of conjugacy classes of index m subgroups in a free group on k generators. The two statements are shown to be equivalent. To illustrate the equivalence, using the representation theory of the unitary groups, we obtain all invariants in the m = 2 graded parts and express them in a simple form both in the case of mixed and pure states. The transformation between the two forms is also derived. Analogous invariants of higher degree are also introduced.
Anionic construction of the SLq,s(2) algebra
International Nuclear Information System (INIS)
Matheus-Valle, J.L.; Monteiro, M.R.
1993-01-01
Considering anionic oscillators in a two-dimensional lattice, the quantum semi-group sl (q,s ) (2) is realized by means of a generalized Schwinger construction. It is found that the parameter q of the algebra is connected to the statistical parameter, whereas the s parameter is related to a s-deformed oscillator introduced at each point of the lattice. (author)
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
An algorithm to construct the basic algebra of a skew group algebra
Horobeţ, E.
2016-01-01
We give an algorithm for the computation of the basic algebra Morita equivalent to a skew group algebra of a path algebra by obtaining formulas for the number of vertices and arrows of the new quiver Qb. We apply this algorithm to compute the basic algebra corresponding to all simple quaternion
An algebraic method for constructing stable and consistent autoregressive filters
International Nuclear Information System (INIS)
Harlim, John; Hong, Hoon; Robbins, Jacob L.
2015-01-01
In this paper, we introduce an algebraic method to construct stable and consistent univariate autoregressive (AR) models of low order for filtering and predicting nonlinear turbulent signals with memory depth. By stable, we refer to the classical stability condition for the AR model. By consistent, we refer to the classical consistency constraints of Adams–Bashforth methods of order-two. One attractive feature of this algebraic method is that the model parameters can be obtained without directly knowing any training data set as opposed to many standard, regression-based parameterization methods. It takes only long-time average statistics as inputs. The proposed method provides a discretization time step interval which guarantees the existence of stable and consistent AR model and simultaneously produces the parameters for the AR models. In our numerical examples with two chaotic time series with different characteristics of decaying time scales, we find that the proposed AR models produce significantly more accurate short-term predictive skill and comparable filtering skill relative to the linear regression-based AR models. These encouraging results are robust across wide ranges of discretization times, observation times, and observation noise variances. Finally, we also find that the proposed model produces an improved short-time prediction relative to the linear regression-based AR-models in forecasting a data set that characterizes the variability of the Madden–Julian Oscillation, a dominant tropical atmospheric wave pattern
Twisting products in Hopf algebras and the construction of the quantum double
International Nuclear Information System (INIS)
Ferrer Santos, W.R.
1992-04-01
Let H be a finite dimensional Hopf algebra and B an (H, H*)-comodule algebra. The purpose of this note is to present a construction in which the product of B is twisted by the given actions. The constructions of the smash product and of the Quantum Double appear as special cases. (author). 7 refs
Tabak, John
2004-01-01
Looking closely at algebra, its historical development, and its many useful applications, Algebra examines in detail the question of why this type of math is so important that it arose in different cultures at different times. The book also discusses the relationship between algebra and geometry, shows the progress of thought throughout the centuries, and offers biographical data on the key figures. Concise and comprehensive text accompanied by many illustrations presents the ideas and historical development of algebra, showcasing the relevance and evolution of this branch of mathematics.
Flanders, Harley
1975-01-01
Algebra presents the essentials of algebra with some applications. The emphasis is on practical skills, problem solving, and computational techniques. Topics covered range from equations and inequalities to functions and graphs, polynomial and rational functions, and exponentials and logarithms. Trigonometric functions and complex numbers are also considered, together with exponentials and logarithms.Comprised of eight chapters, this book begins with a discussion on the fundamentals of algebra, each topic explained, illustrated, and accompanied by an ample set of exercises. The proper use of a
Algebraic diagrammatic construction formalism with three-body interactions
Raimondi, Francesco; Barbieri, Carlo
2018-05-01
Background: Self-consistent Green's function theory has recently been extended to the basic formalism needed to account for three-body interactions [Carbone, Cipollone, Barbieri, Rios, and Polls, Phys. Rev. C 88, 054326 (2013), 10.1103/PhysRevC.88.054326]. The contribution of three-nucleon forces has so far been included in ab initio calculations on nuclear matter and finite nuclei only as averaged two-nucleon forces. Purpose: We derive the working equations for all possible two- and three-nucleon terms that enter the expansion of the self-energy up to the third order, thus including the interaction-irreducible (i.e., not averaged) diagrams with three-nucleon forces that have been previously neglected. Methods: We employ the algebraic diagrammatic construction up to the third order as an organization scheme for generating a nonperturbative self-energy, in which ring (particle-hole) and ladder (particle-particle) diagrams are resummed to all orders. Results: We derive expressions of the static and dynamic self-energy up to the third order, by taking into account the set of diagrams required when either the skeleton or nonskeleton expansions of the single-particle propagator are assumed. A hierarchy of importance among different diagrams is revealed, and a particular emphasis is given to a third-order diagram [see Fig. 2(c)] that is expected to play a significant role among those featuring an interaction-irreducible three-nucleon force. Conclusion: A consistent formalism to resum at infinite order correlations induced by three-nucleon forces in the self-consistent Green's function theory is now available and ready to be implemented in the many-body solvers.
Sepanski, Mark R
2010-01-01
Mark Sepanski's Algebra is a readable introduction to the delightful world of modern algebra. Beginning with concrete examples from the study of integers and modular arithmetic, the text steadily familiarizes the reader with greater levels of abstraction as it moves through the study of groups, rings, and fields. The book is equipped with over 750 exercises suitable for many levels of student ability. There are standard problems, as well as challenging exercises, that introduce students to topics not normally covered in a first course. Difficult problems are broken into manageable subproblems
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.)
Sugawara construction and the q-deformation of Virasoro (super)algebra
Energy Technology Data Exchange (ETDEWEB)
Chaichian, M. (Theory Div., CERN, Geneva (Switzerland)); Presnajder, P. (Dept. of Theoretical Physics, Comenius Univ., Bratislava (Czechoslovakia))
1992-02-27
The q-deformed Virasoro algebra is obtained using the bosonic annihilation and creation operators of the q-deformed infinite Heisenberg algebra H({infinity}){sub q}, which has the Hopf structure. The generators of the q-deformed Virasoro algebra are expressed as a Sugawara construction in terms of normal ordered binomials in these annihilation and creation operators and become double indexed as the reminiscence of a degeneracy removal. The obtained q-deformed Virasoro algebra with central extension reduces to the standard one in the non-deformed limit and in special representations (but not in general) possesses a simple (cocommutative) Hopf structure (not related to the one in H({infinity}){sub q}). The fermionic annihilation and creation operators corresponding to the q-deformed infinite Heisenberg superalgebra s-H({infinity}){sub q} necessary for a similar construction of the q-deformed Virasoro superalgebra are presented. (orig.).
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.
An explicit construction of Wakimoto realizations of current algebras
International Nuclear Information System (INIS)
Boer, J. de; Feher, L.
1996-05-01
It is known from a work of Feigin and Frenkel that a Wakimoto type, generalized free field realization of the current algebra g k can be associated with each parabolic subalgebra P = (g 0 + g + ) of the Lie algebra g, where in the standard case g 0 is the Cartan and P is the Borel subalgebra. In this letter we obtain an explicit formula for the Wakimoto realization in the general case. Using Hamiltonian reduction of the WZNW model, we first derive a Poisson bracket realization of the g-valued current in terms of simplistic bosons belonging to g + and a current belonging to g 0 . We then quantize the formula by determining the correct normal ordering. We also show that the affine-Sugawara stress-energy tensor takes the expected quadratic form in the constituents. (author)
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.
Linear Algebra and Smarandache Linear Algebra
Vasantha, Kandasamy
2003-01-01
The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and vector spaces over finite p...
Pure infiniteness and ideal structure of C*-algebras associated to Fell bundles
DEFF Research Database (Denmark)
Kwasniewski, Bartosz; Szymanski, Wojciech
2017-01-01
are introduced and investigated by themselves and in relation to the partial dynamical system dual to B. Several criteria of pure infiniteness of C^*_r(B) are given. It is shown that they generalize and unify corresponding results obtained in the context of crossed products, by the following duos: Laca...
Algebraic construction of interacting higher spin field theories
International Nuclear Information System (INIS)
Fougere, F.
1991-10-01
We develop a general framework which we believe may provide some insights into the structure of interacting 'high spin' field theories. A finite or infinite set of classical spin fields is described by means of a field defined on an enlarged spacetime manifold. The free action and its gauge symmetries are gathered into a nilpotent differential operator on this manifold. In particular, the choice of Grassmann-valued extra coordinates leads to theories involving only a finite set of fields, the possible contents (spin multiplicities, degree of reducibility, etc.) of which are classified according to the representations of a unitary algebra. The interacting theory is characterized by a functional of the field on the enlarged manifold. We show that there is among these functionals a natural graded Lie algebra structure allowing one to rewrite the gauge invariance condition of the action in a concise form which is a nonlinear generalization of the nilpotency condition of the free theory. We obtain the general solution of this 'classical master equation' , which can be built recurrently starting form the cubic vertex, and we study its symmetries. Our formalism lends itself to a systematic introduction of additional conditions, such as locality, polynomiality, etc. We write down the general form of the solutions exhibiting a scale invariance. The case of a spin 1 field yields, as a unique solution, Yang-Mills theory. In view of quantization, we show that the solution of the classical master equation straightforwardly provides a solution of the (quantum) Batalin-Vilkoviski master equation. One may then obtain a gauge fixed action in the usual way
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.
Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic space
International Nuclear Information System (INIS)
Binz, Ernst; Honegger, Reinhard; Rieckers, Alfred
2004-01-01
A systematic approach to the C*-Weyl algebra W(E,σ) over a possibly degenerate pre-symplectic form σ on a real vector space E of possibly infinite dimension is elaborated in an almost self-contained manner. The construction is based on the theory of Kolmogorov decompositions for σ-positive-definite functions on involutive semigroups and their associated projective unitary group representations. The σ-positive-definite functions provide also the C*-norm of W(E,σ), the latter being shown to be *-isomorphic to the twisted group C*-algebra of the discrete vector group E. The connections to related constructions are indicated. The treatment of the fundamental symmetries is outlined for arbitrary pre-symplectic σ. The construction method is especially applied to the trivial symplectic form σ=0, leading to the commutative Weyl algebra over E, which is shown to be isomorphic to the C*-algebra of the almost periodic continuous function on the topological dual E τ ' of E with respect to an arbitrary locally convex Hausdorff topology τ on E. It is demonstrated that the almost periodic compactification aE τ ' of E τ ' is independent of the chosen locally convex τ on E, and that aE τ ' is continuously group isomorphic to the character group E of E. Applications of the results to the procedures of strict and continuous deformation quantizations are mentioned in the outlook
The algebraic construction of the scale-invariant asymtotic theory
International Nuclear Information System (INIS)
Gatto, R.; Sartori, G.
1975-01-01
The procedure proposed in the preceding paper to construct the asymptotic scale-invariant theory is applied to massive free fields. The contracted fields (of the asymptotic theory) are calculated in terms of the original fields by two different procedures. The contracted charges are calculated and their general relation to the original charges is verified. The problem of defining a vacuum state for the contracted fields and charges is solved. The relation to the problem of non-equivalent representations of the commutator relations is pointed out
General construction of the Virasoro algebra from superalgebras
International Nuclear Information System (INIS)
Frappat, L.; Ragoucy, E.; Thuillier, F.
1991-02-01
The Master equations leading to a generalization of the Sugawara construction for superalgebras are given. The OSp(1|2) k case is worked out in detail. In particular, a new continuous one parameter solution is found at level k=4, which is related by K-conjugation to the continuous solution of SU(2) 4 . This new solution for OSp(1|2) 4 corresponds to the sum of a theory of a free boson (c=1) and of a model of a non-unitary minimal series with c=-14/11. (author) 13 refs., 2 tabs
Einstein algebras and general relativity
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Heller, M.
1992-01-01
A purely algebraic structure called an Einstein algebra is defined in such a way that every spacetime satisfying Einstein's equations is an Einstein algebra but not vice versa. The Gelfand representation of Einstein algebras is defined, and two of its subrepresentations are discussed. One of them is equivalent to the global formulation of the standard theory of general relativity; the other one leads to a more general theory of gravitation which, in particular, includes so-called regular singularities. In order to include other types of singularities one must change to sheaves of Einstein algebras. They are defined and briefly discussed. As a test of the proposed method, the sheaf of Einstein algebras corresponding to the space-time of a straight cosmic string with quasiregular singularity is constructed. 22 refs
The Yoneda algebra of a K2 algebra need not be another K2 algebra
Cassidy, T.; Phan, C.; Shelton, B.
2010-01-01
The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. K2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a K2 algebra would be another K2 algebra. We show that this is not necessarily the case by constructing a monomial K2 algebra for which the corresponding Yoneda algebra is not K2.
Generalized WDVV equations for F4 pure N=2 Super-Yang-Mills theory
Hoevenaars, L.K.; Kersten, P.H.M.; Martini, Ruud
2001-01-01
An associative algebra of holomorphic differential forms is constructed associated with pure N=2 super-Yang–Mills theory for the Lie algebra F4. Existence and associativity of this algebra, combined with the general arguments in the work of Marshakov, Mironov and Morozov, proves that the
An Algebraic Construction of the First Integrals of the Stationary KdV Hierarchy
Matsushima, Masatomo; Ohmiya, Mayumi
2009-09-01
The stationary KdV hierarchy is constructed using a kind of recursion operator called Λ-operator. The notion of the maximal solution of the n-th stationary KdV equation is introduced. Using this maximal solution, a specific differential polynomial with the auxiliary spectral parameter called the spectral M-function is constructed as the quadratic form of the fundamental system of the eigenvalue problem for the 2-nd order linear ordinary differential equation which is related to the linearizing operator of the hierarchy. By calculating a perfect square condition of the quadratic form by an elementary algebraic method, the complete set of first integrals of this hierarchy is constructed.
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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.
Construction of Algebraic and Difference Equations with a Prescribed Solution Space
Directory of Open Access Journals (Sweden)
Moysis Lazaros
2017-03-01
Full Text Available This paper studies the solution space of systems of algebraic and difference equations, given as auto-regressive (AR representations A(σβ(k = 0, where σ denotes the shift forward operator and A(σ is a regular polynomial matrix. The solution space of such systems consists of forward and backward propagating solutions, over a finite time horizon. This solution space can be constructed from knowledge of the finite and infinite elementary divisor structure of A(σ. This work deals with the inverse problem of constructing a family of polynomial matrices A(σ such that the system A(σβ(k = 0 satisfies some given forward and backward behavior. Initially, the connection between the backward behavior of an AR representation and the forward behavior of its dual system is showcased. This result is used to construct a system satisfying a certain backward behavior. By combining this result with the method provided by Gohberg et al. (2009 for constructing a system with a forward behavior, an algorithm is proposed for computing a system satisfying the prescribed forward and backward behavior.
Quantum W-algebras and elliptic algebras
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Feigin, B.; Kyoto Univ.; Frenkel, E.
1996-01-01
We define a quantum W-algebra associated to sl N as an associative algebra depending on two parameters. For special values of the parameters, this algebra becomes the ordinary W-algebra of sl N , or the q-deformed classical W-algebra of sl N . We construct free field realizations of the quantum W-algebras and the screening currents. We also point out some interesting elliptic structures arising in these algebras. In particular, we show that the screening currents satisfy elliptic analogues of the Drinfeld relations in U q (n). (orig.)
DDF construction and D-brane boundary states in pure spinor formalism
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Mukhopadhyay, Partha
2006-01-01
Open string boundary conditions for non-BPS D-branes in type II string theories discussed in hep-th/0505157 give rise to two sectors with integer (R sector) and half-integer (NS sector) modes for the combined fermionic matter and bosonic ghost variables in pure spinor formalism. Exploiting the manifest supersymmetry of the formalism we explicitly construct the DDF (Del Giudice, Di Vecchia, Fubini) states in both the sectors which are in one-to-one correspondence with the states in light-cone Green-Schwarz formalism. We also give a proof of validity of this construction. A similar construction in the closed string sector enables us to define a physical Hilbert space in pure spinor formalism which is used to project the covariant boundary states of both the BPS and non-BPS instantonic D-branes. These projected boundary states take exactly the same form as those found in light-cone Green-Schwarz formalism and are suitable for computing the cylinder diagram with manifest open-closed duality
Pure spinor integration from the collating formula
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Grassi, P.A.; Sommovigo, L.
2011-01-01
We use the technique developed by Becchi and Imbimbo to construct a well-defined BRST-invariant path integral formulation of pure spinor amplitudes. The space of pure spinors can be viewed from the algebraic geometry point of view as a collection of open sets where the constraints can be solved and a set of free and independent variables can be defined. On the intersections of those open sets, the functional measure jumps and one has to add boundary terms to construct a well-defined path integral. The result is the definition of the pure spinor integration measure constructed in terms of differential forms on each single patch.
On one method of realization of commutation relation algebra
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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)
Identities and derivations for Jacobian algebras
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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)
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Takao, Masaru
1989-01-01
We review W-algebras which are generated by stress tensor and primary fields. Associativity plays an important role in determining the extended algebra and further implies the algebras to exist for special values of central charges. Explicitly constructing the algebras including primary fields of spin less than 4, we investigate the closure structure of the Jacobi identity of the extended algebras. (author)
Representations of quantum bicrossproduct algebras
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Arratia, Oscar; Olmo, Mariano A del
2002-01-01
We present a method to construct induced representations of quantum algebras which have a bicrossproduct structure. We apply this procedure to some quantum kinematical algebras in (1+1) dimensions with this kind of structure: null-plane quantum Poincare algebra, non-standard quantum Galilei algebra and quantum κ-Galilei algebra
Knippenberg, Stefan; Gieseking, Rebecca L.; Rehn, Dirk R.; Mukhopadhyay, Sukrit; Dreuw, Andreas; Bredas, Jean-Luc
2016-01-01
-of-merit for all-optical switching. For a series of model polymethines, we compare several algebraic diagrammatic construction (ADC) schemes for the polarization propagator with approximate second-order coupled cluster (CC2) theory, the widely used INDO
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
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
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
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
We give an equivalent construction of the infinitesimal time translation operator for partial differential evolution equation in the algebraic dynamics algorithm proposed by Shun-Jin Wang and his students. Our construction involves only simple partial differentials and avoids the derivative terms of δ function which appear in the course of computation by means of Wang-Zhang operator. We prove Wang’s equivalent theorem which says that our construction and Wang-Zhang’s are equivalent. We use our construction to deal with several typical equations such as nonlinear advection equation, Burgers equation, nonlinear Schrodinger equation, KdV equation and sine-Gordon equation, and obtain at least second order approximate solutions to them. These equations include the cases of real and complex field variables and the cases of the first and the second order time derivatives.
Profinite algebras and affine boundedness
Schneider, Friedrich Martin; Zumbrägel, Jens
2015-01-01
We prove a characterization of profinite algebras, i.e., topological algebras that are isomorphic to a projective limit of finite discrete algebras. In general profiniteness concerns both the topological and algebraic characteristics of a topological algebra, whereas for topological groups, rings, semigroups, and distributive lattices, profiniteness turns out to be a purely topological property as it is is equivalent to the underlying topological space being a Stone space. Condensing the core...
Ma, Tingting; Brown, Irving A.; Kulm, Gerald; Davis, Trina J.; Lewis, Chance W.; Allen, G. Donald
2016-01-01
From the perspectives of Graduate Research Assistants (GRAs), this study examines the design and implementation of a simulated teaching environment in "Second Life" (SL) for prospective teachers to teach algebra for diverse learners. Drawing upon the Learning-for-Use framework, the analyses provide evidence on the development of student…
From Rota-Baxter algebras to pre-Lie algebras
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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
Axis Problem of Rough 3-Valued Algebras
Institute of Scientific and Technical Information of China (English)
Jianhua Dai; Weidong Chen; Yunhe Pan
2006-01-01
The collection of all the rough sets of an approximation space has been given several algebraic interpretations, including Stone algebras, regular double Stone algebras, semi-simple Nelson algebras, pre-rough algebras and 3-valued Lukasiewicz algebras. A 3-valued Lukasiewicz algebra is a Stone algebra, a regular double Stone algebra, a semi-simple Nelson algebra, a pre-rough algebra. Thus, we call the algebra constructed by the collection of rough sets of an approximation space a rough 3-valued Lukasiewicz algebra. In this paper,the rough 3-valued Lukasiewicz algebras, which are a special kind of 3-valued Lukasiewicz algebras, are studied. Whether the rough 3-valued Lukasiewicz algebra is a axled 3-valued Lukasiewicz algebra is examined.
Categories and Commutative Algebra
Salmon, P
2011-01-01
L. Badescu: Sur certaines singularites des varietes algebriques.- D.A. Buchsbaum: Homological and commutative algebra.- S. Greco: Anelli Henseliani.- C. Lair: Morphismes et structures algebriques.- B.A. Mitchell: Introduction to category theory and homological algebra.- R. Rivet: Anneaux de series formelles et anneaux henseliens.- P. Salmon: Applicazioni della K-teoria all'algebra commutativa.- M. Tierney: Axiomatic sheaf theory: some constructions and applications.- C.B. Winters: An elementary lecture on algebraic spaces.
Introduction to relation algebras relation algebras
Givant, Steven
2017-01-01
The first volume of a pair that charts relation algebras from novice to expert level, this text offers a comprehensive grounding for readers new to the topic. Upon completing this introduction, mathematics students may delve into areas of active research by progressing to the second volume, Advanced Topics in Relation Algebras; computer scientists, philosophers, and beyond will be equipped to apply these tools in their own field. The careful presentation establishes first the arithmetic of relation algebras, providing ample motivation and examples, then proceeds primarily on the basis of algebraic constructions: subalgebras, homomorphisms, quotient algebras, and direct products. Each chapter ends with a historical section and a substantial number of exercises. The only formal prerequisite is a background in abstract algebra and some mathematical maturity, though the reader will also benefit from familiarity with Boolean algebra and naïve set theory. The measured pace and outstanding clarity are particularly ...
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Feigin, B.L.; Semikhatov, A.M.
2004-01-01
We construct W-algebra generalizations of the sl-circumflex(2) algebra-W algebras W n (2) generated by two currents E and F with the highest pole of order n in their OPE. The n=3 term in this series is the Bershadsky-Polyakov W 3 (2) algebra. We define these algebras as a centralizer (commutant) of the Uqs-bar (n vertical bar 1) quantum supergroup and explicitly find the generators in a factored, 'Miura-like' form. Another construction of the W n (2) algebras is in terms of the coset sl-circumflex(n vertical bar 1)/sl-circumflex(n). The relation between the two constructions involves the 'duality' (k+n-1)(k'+n-1)=1 between levels k and k' of two sl-circumflex(n) algebras
Edwards, Harold M
1995-01-01
In his new undergraduate textbook, Harold M Edwards proposes a radically new and thoroughly algorithmic approach to linear algebra Originally inspired by the constructive philosophy of mathematics championed in the 19th century by Leopold Kronecker, the approach is well suited to students in the computer-dominated late 20th century Each proof is an algorithm described in English that can be translated into the computer language the class is using and put to work solving problems and generating new examples, making the study of linear algebra a truly interactive experience Designed for a one-semester course, this text adopts an algorithmic approach to linear algebra giving the student many examples to work through and copious exercises to test their skills and extend their knowledge of the subject Students at all levels will find much interactive instruction in this text while teachers will find stimulating examples and methods of approach to the subject
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
Algebraic method for constructing singular steady solitary waves: a case study
Clamond, Didier; Dutykh, Denys; Galligo, André
2016-07-01
This article describes the use of algebraic methods in a phase plane analysis of ordinary differential equations. The method is illustrated by the study of capillary-gravity steady surface waves propagating in shallow water. We consider the (fully nonlinear, weakly dispersive) Serre-Green-Naghdi equation with surface tension, because it provides a tractable model that, at the same time, is not too simple, so interest in the method can be emphasized. In particular, we analyse a special class of solutions, the solitary waves, which play an important role in many fields of physics. In capillary-gravity regime, there are two kinds of localized infinitely smooth travelling wave solutions-solitary waves of elevation and of depression. However, if we allow the solitary waves to have an angular point, then the `zoology' of solutions becomes much richer, and the main goal of this study is to provide a complete classification of such singular localized solutions using the methods of the effective algebraic geometry.
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).
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Krivonos, S.O.; Sorin, A.S.
1994-06-01
We show that the Zamolodchikov's and Polyakov-Bershadsky nonlinear algebras W 3 and W (2) 3 can be embedded as subalgebras into some linear algebras with finite set of currents. Using these linear algebras we find new field realizations of W (2) 3 and W 3 which could be a starting point for constructing new versions of W-string theories. We also reveal a number of hidden relationships between W 3 and W (2) 3 . We conjecture that similar linear algebras can exist for other W-algebra as well. (author). 10 refs
Algebraic partial Boolean algebras
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Smith, Derek
2003-01-01
Partial Boolean algebras, first studied by Kochen and Specker in the 1960s, provide the structure for Bell-Kochen-Specker theorems which deny the existence of non-contextual hidden variable theories. In this paper, we study partial Boolean algebras which are 'algebraic' in the sense that their elements have coordinates in an algebraic number field. Several of these algebras have been discussed recently in a debate on the validity of Bell-Kochen-Specker theorems in the context of finite precision measurements. The main result of this paper is that every algebraic finitely-generated partial Boolean algebra B(T) is finite when the underlying space H is three-dimensional, answering a question of Kochen and showing that Conway and Kochen's infinite algebraic partial Boolean algebra has minimum dimension. This result contrasts the existence of an infinite (non-algebraic) B(T) generated by eight elements in an abstract orthomodular lattice of height 3. We then initiate a study of higher-dimensional algebraic partial Boolean algebras. First, we describe a restriction on the determinants of the elements of B(T) that are generated by a given set T. We then show that when the generating set T consists of the rays spanning the minimal vectors in a real irreducible root lattice, B(T) is infinite just if that root lattice has an A 5 sublattice. Finally, we characterize the rays of B(T) when T consists of the rays spanning the minimal vectors of the root lattice E 8
Lectures on the theory of pure motives
Murre, Jacob P; A, Chris
2013-01-01
The theory of motives was created by Grothendieck in the 1960s as he searched for a universal cohomology theory for algebraic varieties. The theory of pure motives is well established as far as the construction is concerned. Pure motives are expected to have a number of additional properties predicted by Grothendieck's standard conjectures, but these conjectures remain wide open. The theory for mixed motives is still incomplete. This book deals primarily with the theory of pure motives. The exposition begins with the fundamentals: Grothendieck's construction of the category of pure motives and examples. Next, the standard conjectures and the famous theorem of Jannsen on the category of the numerical motives are discussed. Following this, the important theory of finite dimensionality is covered. The concept of Chow-K�nneth decomposition is introduced, with discussion of the known results and the related conjectures, in particular the conjectures of Bloch-Beilinson type. We finish with a chapter on relative m...
Conference on Algebraic Geometry for Coding Theory and Cryptography
Lauter, Kristin; Walker, Judy
2017-01-01
Covering topics in algebraic geometry, coding theory, and cryptography, this volume presents interdisciplinary group research completed for the February 2016 conference at the Institute for Pure and Applied Mathematics (IPAM) in cooperation with the Association for Women in Mathematics (AWM). The conference gathered research communities across disciplines to share ideas and problems in their fields and formed small research groups made up of graduate students, postdoctoral researchers, junior faculty, and group leaders who designed and led the projects. Peer reviewed and revised, each of this volume's five papers achieves the conference’s goal of using algebraic geometry to address a problem in either coding theory or cryptography. Proposed variants of the McEliece cryptosystem based on different constructions of codes, constructions of locally recoverable codes from algebraic curves and surfaces, and algebraic approaches to the multicast network coding problem are only some of the topics covered in this vo...
Fractional supersymmetry and infinite dimensional lie algebras
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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
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
Hurwitz Algebras and the Octonion Algebra
Burdik, Čestmir; Catto, Sultan
2018-02-01
We explore some consequences of a theory of internal symmetries for elementary particles constructed on exceptional quantum mechanical spaces based on Jordan algebra formulation that admit exceptional groups as gauge groups.
Projector bases and algebraic spinors
International Nuclear Information System (INIS)
Bergdolt, G.
1988-01-01
In the case of complex Clifford algebras a basis is constructed whose elements satisfy projector relations. The relations are sufficient conditions for the elements to span minimal ideals and hence to define algebraic spinors
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...
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.)
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.
Construction and characterization of a pure protein hydrogel for drug delivery application.
Xu, Xu; Xu, ZhaoKang; Yang, XiaoFeng; He, YanHao; Lin, Rong
2017-02-01
Injectable hydrogels have a variety of applications, including regenerative medicine, tissue engineering and controlled drug delivery. In this paper, we reported on a pure protein hydrogel based on tetrameric recombinant proteins for the potential drug delivery application. This protein hydrogel was formed instantly by simply mixing two recombinant proteins (ULD-TIP1 and ULD-GGGWRESAI) through the specific protein-peptide interaction. The protein hydrogel was characterized by rheology and scanning electron microscopy (SEM). In vitro cytotoxicity test indicated that the developed protein hydrogel had no apparent cytotoxicity against L-929 cells and HCEC cells after 48h incubation. The formed protein hydrogels was gradually degraded after incubation in phosphate buffered solution (PBS, pH=7.4) for a period of 144h study, as indicated by in vitro degradation test. Encapsulation of model drug (sodium diclofenac; DIC) were achieved by simple mixing of drugs with hydrogelator and the entrapped drugs was almost completely released from hydrogels within 24h via a diffusion manner. As a conclusion, the simple and mild preparation procedure and good biocompatibility of protein hydrogel would render its good promising candidate for drug delivery applications. Copyright © 2016 Elsevier B.V. All rights reserved.
Hitt, Fernando; Morasse, Christian
2009-01-01
Introduction: In this document we stress the importance of developing in children a structure for advanced numerical-algebraic thinking that can provide an element of control when solving mathematical situations. We analyze pupils' conceptions that induce errors in algebra due to a lack of control in connection with their numerical thinking. We…
Polynomials in algebraic analysis
Multarzyński, Piotr
2012-01-01
The concept of polynomials in the sense of algebraic analysis, for a single right invertible linear operator, was introduced and studied originally by D. Przeworska-Rolewicz \\cite{DPR}. One of the elegant results corresponding with that notion is a purely algebraic version of the Taylor formula, being a generalization of its usual counterpart, well known for functions of one variable. In quantum calculus there are some specific discrete derivations analyzed, which are right invertible linear ...
Lefschetz, Solomon
2005-01-01
An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.
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Garcia, R.L.
1983-11-01
The Grassmann algebra is presented briefly. Exponential and logarithm of matrices functions, whose elements belong to this algebra, are studied with the help of the SCHOONSCHIP and REDUCE 2 algebraic manipulators. (Author) [pt
Seo, Young Joo; Kim, Young Hee
2016-01-01
In this paper we construct some real algebras by using elementary functions, and discuss some relations between several axioms and its related conditions for such functions. We obtain some conditions for real-valued functions to be a (edge) d -algebra.
Schneider, Hans
1989-01-01
Linear algebra is one of the central disciplines in mathematics. A student of pure mathematics must know linear algebra if he is to continue with modern algebra or functional analysis. Much of the mathematics now taught to engineers and physicists requires it.This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related t
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.
Mester, Dávid; Nagy, Péter R.; Kállay, Mihály
2018-03-01
A reduced-cost implementation of the second-order algebraic-diagrammatic construction [ADC(2)] method is presented. We introduce approximations by restricting virtual natural orbitals and natural auxiliary functions, which results, on average, in more than an order of magnitude speedup compared to conventional, density-fitting ADC(2) algorithms. The present scheme is the successor of our previous approach [D. Mester, P. R. Nagy, and M. Kállay, J. Chem. Phys. 146, 194102 (2017)], which has been successfully applied to obtain singlet excitation energies with the linear-response second-order coupled-cluster singles and doubles model. Here we report further methodological improvements and the extension of the method to compute singlet and triplet ADC(2) excitation energies and transition moments. The various approximations are carefully benchmarked, and conservative truncation thresholds are selected which guarantee errors much smaller than the intrinsic error of the ADC(2) method. Using the canonical values as reference, we find that the mean absolute error for both singlet and triplet ADC(2) excitation energies is 0.02 eV, while that for oscillator strengths is 0.001 a.u. The rigorous cutoff parameters together with the significantly reduced operation count and storage requirements allow us to obtain accurate ADC(2) excitation energies and transition properties using triple-ζ basis sets for systems of up to one hundred atoms.
The Unitality of Quantum B-algebras
Han, Shengwei; Xu, Xiaoting; Qin, Feng
2018-02-01
Quantum B-algebras as a generalization of quantales were introduced by Rump and Yang, which cover the majority of implicational algebras and provide a unified semantic for a wide class of substructural logics. Unital quantum B-algebras play an important role in the classification of implicational algebras. The main purpose of this paper is to construct unital quantum B-algebras from non-unital quantum B-algebras.
Spin-4 extended conformal algebras
International Nuclear Information System (INIS)
Kakas, A.C.
1988-01-01
We construct spin-4 extended conformal algebras using the second hamiltonian structure of the KdV hierarchy. In the presence of a U(1) current a family of spin-4 algebras exists but the additional requirement that the spin-1 and spin-4 currents commute fixes the algebra uniquely. (orig.)
Plumpton, C
1968-01-01
Sixth Form Pure Mathematics, Volume 1, Second Edition, is the first of a series of volumes on Pure Mathematics and Theoretical Mechanics for Sixth Form students whose aim is entrance into British and Commonwealth Universities or Technical Colleges. A knowledge of Pure Mathematics up to G.C.E. O-level is assumed and the subject is developed by a concentric treatment in which each new topic is used to illustrate ideas already treated. The major topics of Algebra, Calculus, Coordinate Geometry, and Trigonometry are developed together. This volume covers most of the Pure Mathematics required for t
Vertex algebras and algebraic curves
Frenkel, Edward
2004-01-01
Vertex algebras are algebraic objects that encapsulate the concept of operator product expansion from two-dimensional conformal field theory. Vertex algebras are fast becoming ubiquitous in many areas of modern mathematics, with applications to representation theory, algebraic geometry, the theory of finite groups, modular functions, topology, integrable systems, and combinatorics. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship with the geometry of algebraic curves. The notion of a vertex algebra is introduced in a coordinate-independent way, so that vertex operators become well defined on arbitrary smooth algebraic curves, possibly equipped with additional data, such as a vector bundle. Vertex algebras then appear as the algebraic objects encoding the geometric structure of various moduli spaces associated with algebraic curves. Therefore they may be used to give a geometric interpretation of various questions of representation theory. The book co...
African Journals Online (AJOL)
Tadesse
In this paper we introduce the concept of implicative algebras which is an equivalent definition of lattice implication algebra of Xu (1993) and further we prove that it is a regular Autometrized. Algebra. Further we remark that the binary operation → on lattice implicative algebra can never be associative. Key words: Implicative ...
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 ...
Algebraic K-theory and algebraic topology
Energy Technology Data Exchange (ETDEWEB)
Berrick, A J [Department of Mathematics, National University of Singapore (Singapore)
2003-09-15
This contribution treats the various topological constructions of Algebraic K-theory together with the underlying homotopy theory. Topics covered include the plus construction together with its various ramifications and applications, Topological Hochschild and Cyclic Homology as well as K-theory of the ring of integers.
Ruberti, M; Yun, R; Gokhberg, K; Kopelke, S; Cederbaum, L S; Tarantelli, F; Averbukh, V
2014-05-14
Here, we extend the L2 ab initio method for molecular photoionization cross-sections introduced in Gokhberg et al. [J. Chem. Phys. 130, 064104 (2009)] and benchmarked in Ruberti et al. [J. Chem. Phys. 139, 144107 (2013)] to the calculation of total photoionization cross-sections of molecules in electronically excited states. The method is based on the ab initio description of molecular electronic states within the many-electron Green's function approach, known as algebraic diagrammatic construction (ADC), and on the application of Stieltjes-Chebyshev moment theory to Lanczos pseudospectra of the ADC electronic Hamiltonian. The intermediate state representation of the dipole operator in the ADC basis is used to compute the transition moments between the excited states of the molecule. We compare the results obtained using different levels of the many-body theory, i.e., ADC(1), ADC(2), and ADC(2)x for the first two excited states of CO, N2, and H2O both at the ground state and the excited state equilibrium or saddle point geometries. We find that the single excitation ADC(1) method is not adequate even at the qualitative level and that the inclusion of double electronic excitations for description of excited state photoionization is essential. Moreover, we show that the use of the extended ADC(2)x method leads to a substantial systematic difference from the strictly second-order ADC(2). Our calculations demonstrate that a theoretical modelling of photoionization of excited states requires an intrinsically double excitation theory with respect to the ground state and cannot be achieved by the standard single excitation methods with the ground state as a reference.
Introduction to vertex algebras, Borcherds algebras and the Monster Lie algebras
International Nuclear Information System (INIS)
Gebert, R.W.
1993-09-01
The theory of vertex algebras constitutes a mathematically rigorous axiomatic formulation of the algebraic origins of conformal field theory. In this context Borcherds algebras arise as certain ''physical'' subspaces of vertex algebras. The aim of this review is to give a pedagogical introduction into this rapidly-developing area of mathematics. Based on the machinery of formal calculus we present the axiomatic definition of vertex algebras. We discuss the connection with conformal field theory by deriving important implications of these axioms. In particular, many explicit calculations are presented to stress the eminent role of the Jacobi identity axiom for vertex algebras. As a class of concrete examples the vertex algebras associated with even lattices are constructed and it is shown in detail how affine Lie algebras and the fake Monster Lie algebra naturally appear. This leads us to the abstract definition of Borcherds algebras as generalized Kac-Moody algebras and their basic properties. Finally, the results about the simplest generic Borcherds algebras are analysed from the point of view of symmetry in quantum theory and the construction of the Monster Lie algebra is sketched. (orig.)
The relation between quantum W algebras and Lie algebras
International Nuclear Information System (INIS)
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.)
Constructive algebraic integration theory
Spitters, B.A.W.
2006-01-01
For a long time people have been trying to develop probability theory starting from ‘finite’ events rather than collections of infinite events. In this way one can find natural replacements for measurable sets and integrable functions, but measurable functions seemed to be more difficult to find. We
Polishchuk, Alexander
2005-01-01
Quadratic algebras, i.e., algebras defined by quadratic relations, often occur in various areas of mathematics. One of the main problems in the study of these (and similarly defined) algebras is how to control their size. A central notion in solving this problem is the notion of a Koszul algebra, which was introduced in 1970 by S. Priddy and then appeared in many areas of mathematics, such as algebraic geometry, representation theory, noncommutative geometry, K-theory, number theory, and noncommutative linear algebra. The book offers a coherent exposition of the theory of quadratic and Koszul algebras, including various definitions of Koszulness, duality theory, Poincar�-Birkhoff-Witt-type theorems for Koszul algebras, and the Koszul deformation principle. In the concluding chapter of the book, they explain a surprising connection between Koszul algebras and one-dependent discrete-time stochastic processes.
Quantum deformed su(mvertical stroke n) algebra and superconformal algebra on quantum superspace
International Nuclear Information System (INIS)
Kobayashi, Tatsuo
1993-01-01
We study a deformed su(mvertical stroke n) algebra on a quantum superspace. Some interesting aspects of the deformed algebra are shown. As an application of the deformed algebra we construct a deformed superconformal algebra. From the deformed su(1vertical stroke 4) algebra, we derive deformed Lorentz, translation of Minkowski space, iso(2,2) and its supersymmetric algebras as closed subalgebras with consistent automorphisms. (orig.)
International Nuclear Information System (INIS)
Marquette, Ian
2013-01-01
We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently
International Nuclear Information System (INIS)
Ogievetsky, O.; Schmidke, W.B.; Wess, J.; Muenchen Univ.; Zumino, B.; Lawrence Berkeley Lab., CA
1992-01-01
The q-differential calculus for the q-Minkowski space is developed. The algebra of the q-derivatives with the q-Lorentz generators is found giving the q-deformation of the Poincare algebra. The reality structure of the q-Poincare algebra is given. The reality structure of the q-differentials is also found. The real Laplaacian is constructed. Finally the comultiplication, counit and antipode for the q-Poincare algebra are obtained making it a Hopf algebra. (orig.)
Goodstein, R L
2007-01-01
This elementary treatment by a distinguished mathematician employs Boolean algebra as a simple medium for introducing important concepts of modern algebra. Numerous examples appear throughout the text, plus full solutions.
Operadic formulation of topological vertex algebras and gerstenhaber or Batalin-Vilkovisky algebras
International Nuclear Information System (INIS)
Huang Yizhi
1994-01-01
We give the operadic formulation of (weak, strong) topological vertex algebras, which are variants of topological vertex operator algebras studied recently by Lian and Zuckerman. As an application, we obtain a conceptual and geometric construction of the Batalin-Vilkovisky algebraic structure (or the Gerstenhaber algebra structure) on the cohomology of a topological vertex algebra (or of a weak topological vertex algebra) by combining this operadic formulation with a theorem of Getzler (or of Cohen) which formulates Batalin-Vilkovisky algebras (or Gerstenhaber algebras) in terms of the homology of the framed little disk operad (or of the little disk operad). (orig.)
International Nuclear Information System (INIS)
Waldron, A.K.; Joshi, G.C.
1992-01-01
By considering representation theory for non-associative algebras the fundamental adjoint representations of the octonion algebra is constructed. It is then shown how these representations by associative matrices allow a consistent octonionic gauge theory to be realized. It was found that non-associativity implies the existence of new terms in the transformation laws of fields and the kinetic term of an octonionic Lagrangian. 13 refs
Currents on Grassmann algebras
International Nuclear Information System (INIS)
Coquereaux, R.; Ragoucy, E.
1993-09-01
Currents are defined on a Grassmann algebra Gr(N) with N generators as distributions on its exterior algebra (using the symmetric wedge product). The currents are interpreted in terms of Z 2 -graded Hochschild cohomology and closed currents in terms of cyclic cocycles (they are particular multilinear forms on Gr(N)). An explicit construction of the vector space of closed currents of degree p on Gr(N) is given by using Berezin integration. (authors). 10 refs
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...
Galilean contractions of W-algebras
Directory of Open Access Journals (Sweden)
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.
Jordan algebras versus C*- algebras
International Nuclear Information System (INIS)
Stormer, E.
1976-01-01
The axiomatic formulation of quantum mechanics and the problem of whether the observables form self-adjoint operators on a Hilbert space, are discussed. The relation between C*- algebras and Jordan algebras is studied using spectral theory. (P.D.)
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.
String field representation of the Virasoro algebra
Energy Technology Data Exchange (ETDEWEB)
Mertes, Nicholas [Institute of Physics AS CR,Na Slovance 2, Prague 8 (Czech Republic); Department of Physics, University of Miami,Coral Gables, FL (United States); Schnabl, Martin [Institute of Physics AS CR,Na Slovance 2, Prague 8 (Czech Republic)
2016-12-29
We construct a representation of the zero central charge Virasoro algebra using string fields in Witten’s open bosonic string field theory. This construction is used to explore extensions of the KBc algebra and find novel algebraic solutions of open string field theory.
Partially ordered algebraic systems
Fuchs, Laszlo
2011-01-01
Originally published in an important series of books on pure and applied mathematics, this monograph by a distinguished mathematician explores a high-level area in algebra. It constitutes the first systematic summary of research concerning partially ordered groups, semigroups, rings, and fields. The self-contained treatment features numerous problems, complete proofs, a detailed bibliography, and indexes. It presumes some knowledge of abstract algebra, providing necessary background and references where appropriate. This inexpensive edition of a hard-to-find systematic survey will fill a gap i
Hopf algebras in noncommutative geometry
International Nuclear Information System (INIS)
Varilly, Joseph C.
2001-10-01
We give an introductory survey to the use of Hopf algebras in several problems of non- commutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We show its relation to the algebra of transverse differential operators introduced by Connes and Moscovici in order to compute a local index formula in cyclic cohomology, and to the several Hopf algebras defined by Connes and Kreimer to simplify the combinatorics of perturbative renormalization. We explain how characteristic classes for a Hopf module algebra can be obtained from the cyclic cohomology of the Hopf algebra which acts on it. Finally, we discuss the theory of non- commutative spherical manifolds and show how they arise as homogeneous spaces of certain compact quantum groups. (author)
International Nuclear Information System (INIS)
Nikityuk, N.M.
1989-01-01
The results of investigations of using the algebraic coding theory for the creation of parallel encoders, majority coincidence schemes and coordinate processors for the first and second trigger levels are described. Concrete examples of calculation and structure of special-purpose processor using the table arithmetic method are given for multiplicity t ≤ 5. The question of using parallel and sequential syndrome coding methods for the registration of events with clusters is discussed. 30 refs.; 10 figs
Colored Quantum Algebra and Its Bethe State
International Nuclear Information System (INIS)
Wang Jin-Zheng; Jia Xiao-Yu; Wang Shi-Kun
2014-01-01
We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation. (general)
Extended Kac-Moody algebras and applications
International Nuclear Information System (INIS)
Ragoucy, E.; Sorba, P.
1991-04-01
The notion of a Kac-Moody algebra defined on the S 1 circle is extended to super Kac-Moody algebras defined on MxG N , M being a smooth closed compact manifold of dimension greater than one, and G N the Grassman algebra with N generators. All the central extensions of these algebras are computed. Then, for each such algebra the derivation algebra constructed from the MxG N diffeomorphism is determined. The twists of such super Kac-Moody algebras as well as the generalization to non-compact surfaces are partially studied. Finally, the general construction is applied to the study of conformal and superconformal algebras, as well as area-preserving diffeomorphisms algebra and its supersymmetric extension. (author) 65 refs
Compact quantum group C*-algebras as Hopf algebras with approximate unit
International Nuclear Information System (INIS)
Do Ngoc Diep; Phung Ho Hai; Kuku, A.O.
1999-04-01
In this paper, we construct and study the representation theory of a Hopf C*-algebra with approximate unit, which constitutes quantum analogue of a compact group C*-algebra. The construction is done by first introducing a convolution-product on an arbitrary Hopf algebra H with integral, and then constructing the L 2 and C*-envelopes of H (with the new convolution-product) when H is a compact Hopf *-algebra. (author)
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.
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...
Ford, Timothy J
2017-01-01
This book presents a comprehensive introduction to the theory of separable algebras over commutative rings. After a thorough introduction to the general theory, the fundamental roles played by separable algebras are explored. For example, Azumaya algebras, the henselization of local rings, and Galois theory are rigorously introduced and treated. Interwoven throughout these applications is the important notion of étale algebras. Essential connections are drawn between the theory of separable algebras and Morita theory, the theory of faithfully flat descent, cohomology, derivations, differentials, reflexive lattices, maximal orders, and class groups. The text is accessible to graduate students who have finished a first course in algebra, and it includes necessary foundational material, useful exercises, and many nontrivial examples.
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.)
Introduction to algebraic quantum field theory
International Nuclear Information System (INIS)
Horuzhy, S.S.
1990-01-01
This volume presents a systematic introduction to the algebraic approach to quantum field theory. The structure of the contents corresponds to the way the subject has advanced. It is shown how the algebraic approach has developed from the purely axiomatic theory of observables via superselection rules into the dynamical formalism of fields and observables. Chapter one discusses axioms and their consequences -many of which are now classical theorems- and deals, in general, with the axiomatic theory of local observable algebras. The absence of field concepts makes this theory incomplete and, in chapter two, superselection rules are shown to be the key to the reconstruction of fields from observables. Chapter three deals with the algebras of Wightman fields, first unbounded operator algebras, then Von Neumann field algebras (with a special section on wedge region algebras) and finally local algebras of free and generalised free fields. (author). 447 refs.; 4 figs
The algebra of Weyl symmetrised polynomials and its quantum extension
International Nuclear Information System (INIS)
Gelfand, I.M.; Fairlie, D.B.
1991-01-01
The Algebra of Weyl symmetrised polynomials in powers of Hamiltonian operators P and Q which satisfy canonical commutation relations is constructed. This algebra is shown to encompass all recent infinite dimensional algebras acting on two-dimensional phase space. In particular the Moyal bracket algebra and the Poisson bracket algebra, of which the Moyal is the unique one parameter deformation are shown to be different aspects of this infinite algebra. We propose the introduction of a second deformation, by the replacement of the Heisenberg algebra for P, Q with a q-deformed commutator, and construct algebras of q-symmetrised Polynomials. (orig.)
Covariant representations of nuclear *-algebras
International Nuclear Information System (INIS)
Moore, S.M.
1978-01-01
Extensions of the Csup(*)-algebra theory for covariant representations to nuclear *-algebra are considered. Irreducible covariant representations are essentially unique, an invariant state produces a covariant representation with stable vacuum, and the usual relation between ergodic states and covariant representations holds. There exist construction and decomposition theorems and a possible relation between derivations and covariant representations
Infinite dimension algebra and conformal symmetry
International Nuclear Information System (INIS)
Ragoucy-Aubezon, E.
1991-04-01
A generalisation of Kac-Moody algebras (current algebras defined on a circle) to algebras defined on a compact supermanifold of any dimension and with any number of supersymmetries is presented. For such a purpose, we compute all the central extensions of loop algebras defined on this supermanifold, i.e. all the cohomology classes of these loop algebras. Then, we try to extend the relation (i.e. semi-direct sum) that exists between the two dimensional conformal algebras (called Virasoro algebra) and the usual Kac-Moody algebras, by considering the derivation algebra of our extended Kac-Moody algebras. The case of superconformal algebras (used in superstrings theories) is treated, as well as the cases of area-preserving diffeomorphisms (used in membranes theories), and Krichever-Novikov algebras (used for interacting strings). Finally, we present some generalizations of the Sugawara construction to the cases of extended Kac-Moody algebras, and Kac-Moody of superalgebras. These constructions allow us to get new realizations of the Virasoro, and Ramond, Neveu-Schwarz algebras
Chiral algebras for trinion theories
International Nuclear Information System (INIS)
Lemos, Madalena; Peelaers, Wolfger
2015-01-01
It was recently understood that one can identify a chiral algebra in any four-dimensional N=2 superconformal theory. In this note, we conjecture the full set of generators of the chiral algebras associated with the T n theories. The conjecture is motivated by making manifest the critical affine module structure in the graded partition function of the chiral algebras, which is computed by the Schur limit of the superconformal index for T n theories. We also explicitly construct the chiral algebra arising from the T 4 theory. Its null relations give rise to new T 4 Higgs branch chiral ring relations.
On the algebraic structure of self-dual gauge fields and sigma models
International Nuclear Information System (INIS)
Bais, F.A.; Sasaki, R.
1983-01-01
An extensive and detailed analysis of self-dual gauge fields, in particular with axial symmetry, is presented, culminating in a purely algebraic procedure to generate solutions. The method which is particularly suited for the construction of multimonopole solutions for a theory with arbitrary G, is also applicable to a wide class of non-linear sigma models. The relevant symmetries as well as the associated linear problems which underly the exact solubility of the problem, are constructed and discussed in detail. (orig.)
Pure Gravities via Color-Kinematics Duality for Fundamental Matter
Johansson, Henrik
2015-01-01
We give a prescription for the computation of loop-level scattering amplitudes in pure Einstein gravity, and four-dimensional pure supergravities, using the color-kinematics duality. Amplitudes are constructed using double copies of pure (super-)Yang-Mills parts and additional contributions from double copies of fundamental matter, which are treated as ghosts. The opposite-statistics states cancel the unwanted dilaton and axion in the bosonic theory, as well as the extra matter supermultiplets in supergravities. As a spinoff, we obtain a prescription for obtaining amplitudes in supergravities with arbitrary non-self-interacting matter. As a prerequisite, we extend the color-kinematics duality from the adjoint to the fundamental representation of the gauge group. We explain the numerator relations that the fundamental kinematic Lie algebra should satisfy. We give nontrivial evidence supporting our construction using explicit tree and loop amplitudes, as well as more general arguments.
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)
New family of Maxwell like algebras
International Nuclear Information System (INIS)
Concha, P.K.; Durka, R.; Merino, N.; Rodríguez, E.K.
2016-01-01
We introduce an alternative way of closing Maxwell like algebras. We show, through a suitable change of basis, that resulting algebras are given by the direct sums of the AdS and the Maxwell algebras already known in the literature. Casting the result into the S-expansion method framework ensures the straightaway construction of the gravity theories based on a found enlargement.
New family of Maxwell like algebras
Energy Technology Data Exchange (ETDEWEB)
Concha, P.K., E-mail: patillusion@gmail.com [Departamento de Ciencias, Facultad de Artes y Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña del Mar (Chile); Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Casilla 567, Valdivia (Chile); Durka, R., E-mail: remigiuszdurka@gmail.com [Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso (Chile); Merino, N., E-mail: nemerino@gmail.com [Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso (Chile); Rodríguez, E.K., E-mail: everodriguezd@gmail.com [Departamento de Ciencias, Facultad de Artes y Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña del Mar (Chile); Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Casilla 567, Valdivia (Chile)
2016-08-10
We introduce an alternative way of closing Maxwell like algebras. We show, through a suitable change of basis, that resulting algebras are given by the direct sums of the AdS and the Maxwell algebras already known in the literature. Casting the result into the S-expansion method framework ensures the straightaway construction of the gravity theories based on a found enlargement.
Evolution algebras generated by Gibbs measures
International Nuclear Information System (INIS)
Rozikov, Utkir A.; Tian, Jianjun Paul
2009-03-01
In this article we study algebraic structures of function spaces defined by graphs and state spaces equipped with Gibbs measures by associating evolution algebras. We give a constructive description of associating evolution algebras to the function spaces (cell spaces) defined by graphs and state spaces and Gibbs measure μ. For finite graphs we find some evolution subalgebras and other useful properties of the algebras. We obtain a structure theorem for evolution algebras when graphs are finite and connected. We prove that for a fixed finite graph, the function spaces have a unique algebraic structure since all evolution algebras are isomorphic to each other for whichever Gibbs measures are assigned. When graphs are infinite graphs then our construction allows a natural introduction of thermodynamics in studying of several systems of biology, physics and mathematics by theory of evolution algebras. (author)
Energy Technology Data Exchange (ETDEWEB)
Lefrancois, Daniel; Dreuw, Andreas, E-mail: dreuw@uni-heidelberg.de [Interdisciplinary Center for Scientific Computing, Ruprecht-Karls University, Im Neuenheimer Feld 205, 69120 Heidelberg (Germany); Rehn, Dirk R. [Departments of Physics, Chemistry and Biology, Linköping University, SE-581 83 Linköping (Sweden)
2016-08-28
For the calculation of adiabatic singlet-triplet gaps (STG) in diradicaloid systems the spin-flip (SF) variant of the algebraic diagrammatic construction (ADC) scheme for the polarization propagator in third order perturbation theory (SF-ADC(3)) has been applied. Due to the methodology of the SF approach the singlet and triplet states are treated on an equal footing since they are part of the same determinant subspace. This leads to a systematically more accurate description of, e.g., diradicaloid systems than with the corresponding non-SF single-reference methods. Furthermore, using analytical excited state gradients at ADC(3) level, geometry optimizations of the singlet and triplet states were performed leading to a fully consistent description of the systems, leading to only small errors in the calculated STGs ranging between 0.6 and 2.4 kcal/mol with respect to experimental references.
Finite W-algebras and intermediate statistics
International Nuclear Information System (INIS)
Barbarin, F.; Ragoucy, E.; Sorba, P.
1995-01-01
New realizations of finite W-algebras are constructed by relaxing the usual constraint conditions. Then finite W-algebras are recognized in the Heisenberg quantization recently proposed by Leinaas and Myrheim, for a system of two identical particles in d dimensions. As the anyonic parameter is directly associated to the W-algebra involved in the d=1 case, it is natural to consider that the W-algebra framework is well adapted for a possible generalization of the anyon statistics. ((orig.))
Lectures on W algebras and W gravity
International Nuclear Information System (INIS)
Pope, C.N.
1992-01-01
We give a review of the extended conformal algebras, known as W algebras, which contain currents of spins higher than 2 in addition to the energy-momentum tensor. These include the non-linear W N algebras; the linear W ∞ and W 1+∞ algebras; and their super-extensions. We discuss their applications to the construction of W-gravity and W-string theories. (author). 46 refs
Spin-zero mesons and current algebras
International Nuclear Information System (INIS)
Wellner, M.
1977-01-01
Large chiral algebras, using the f and d coefficients of SU(3) can be constructed with spin-1/2 baryons. Such algebras have been found useful in some previous investigations. This article examines under what conditions similar or identical current algebras may be realized with spin-0 mesons. A curious lack of analogy emerges between meson and baryon currents. Second-class currents, made of mesons, are required in some algebras. If meson and baryon currents are to satisfy the same extended SU(3) algebra, four meson nonets are needed, in terms of which we give an explicit construction for the currents
Garrett, Paul B
2007-01-01
Designed for an advanced undergraduate- or graduate-level course, Abstract Algebra provides an example-oriented, less heavily symbolic approach to abstract algebra. The text emphasizes specifics such as basic number theory, polynomials, finite fields, as well as linear and multilinear algebra. This classroom-tested, how-to manual takes a more narrative approach than the stiff formalism of many other textbooks, presenting coherent storylines to convey crucial ideas in a student-friendly, accessible manner. An unusual feature of the text is the systematic characterization of objects by universal
Kolman, Bernard
1985-01-01
College Algebra, Second Edition is a comprehensive presentation of the fundamental concepts and techniques of algebra. The book incorporates some improvements from the previous edition to provide a better learning experience. It provides sufficient materials for use in the study of college algebra. It contains chapters that are devoted to various mathematical concepts, such as the real number system, the theory of polynomial equations, exponential and logarithmic functions, and the geometric definition of each conic section. Progress checks, warnings, and features are inserted. Every chapter c
A New Formalism for Relational Algebra
DEFF Research Database (Denmark)
Schwartzbach, Michael Ignatieff; Larsen, Kim Skak; Schmidt, Erik Meineche
1992-01-01
We present a new formalism for relational algebra, the FC language, which is based on a novel factorization of relations. The acronym stands for factorize and combine. A pure version of this language is equivalent to relational algebra in the sense that semantics preserving translations exist...
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...
Non-relativistic Bondi-Metzner-Sachs algebra
Batlle, Carles; Delmastro, Diego; Gomis, Joaquim
2017-09-01
We construct two possible candidates for non-relativistic bms4 algebra in four space-time dimensions by contracting the original relativistic bms4 algebra. bms4 algebra is infinite-dimensional and it contains the generators of the Poincaré algebra, together with the so-called super-translations. Similarly, the proposed nrbms4 algebras can be regarded as two infinite-dimensional extensions of the Bargmann algebra. We also study a canonical realization of one of these algebras in terms of the Fourier modes of a free Schrödinger field, mimicking the canonical realization of relativistic bms4 algebra using a free Klein-Gordon field.
Quantum algebra of N superspace
International Nuclear Information System (INIS)
Hatcher, Nicolas; Restuccia, A.; Stephany, J.
2007-01-01
We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the κ-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra
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)
W algebra in the SU(3) parafermion model
International Nuclear Information System (INIS)
Ding, X.; Fan, H.; Shi, K.; Wang, P.; Zhu, C.
1993-01-01
A construction of W 3 algebra for the SU(3) parafermion model is proposed, in which a Z algebra technique is used instead of the popular free-field realization. The central charge of the underlying algebra is different from known W algebras
More on the linearization of W-algebras
International Nuclear Information System (INIS)
Krivonos, S.; Sorin, A.
1995-01-01
We show that a wide class of W-(super)algebras, including W N (N-1) , U(N)-superconformal as well as W N nonlinear algebras, can be linearized by embedding them as subalgebras into some linear (super)conformal algebras with finite sets of currents. The general construction is illustrated by the example of W 4 algebra. 16 refs
Quivers of Bound Path Algebras and Bound Path Coalgebras
Directory of Open Access Journals (Sweden)
Dr. Intan Muchtadi
2010-09-01
Full Text Available bras and coalgebras can be represented as quiver (directed graph, and from quiver we can construct algebras and coalgebras called path algebras and path coalgebras. In this paper we show that the quiver of a bound path coalgebra (resp. algebra is the dual quiver of its bound path algebra (resp. coalgebra.
Generalized NLS hierarchies from rational W algebras
International Nuclear Information System (INIS)
Toppan, F.
1993-11-01
Finite rational W algebras are very natural structures appearing in coset constructions when a Kac-Moody subalgebra is factored out. The problem of relating these algebras to integrable hierarchies of equations is studied by showing how to associate to a rational W algebra its corresponding hierarchy. Two examples are worked out, the sl(2)/U(1) coset, leading to the Non-Linear Schroedinger hierarchy, and the U(1) coset of the Polyakov-Bershadsky W algebra, leading to a 3-field representation of the KP hierarchy already encountered in the literature. In such examples a rational algebra appears as algebra of constraints when reducing a KP hierarchy to a finite field representation. This fact arises the natural question whether rational algebras are always associated to such reductions and whether a classification of rational algebras can lead to a classification of the integrable hierarchies. (author). 19 refs
Vertex algebras and mirror symmetry
International Nuclear Information System (INIS)
Borisov, L.A.
2001-01-01
Mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly vertex algebras that correspond to holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in toric varieties. We establish the relation between these vertex algebras for mirror Calabi-Yau manifolds. This should eventually allow us to rewrite the whole story of toric mirror symmetry in the language of sheaves of vertex algebras. Our approach is purely algebraic and involves simple techniques from toric geometry and homological algebra, as well as some basic results of the theory of vertex algebras. Ideas of this paper may also be useful in other problems related to maps from curves to algebraic varieties.This paper could also be of interest to physicists, because it contains explicit description of holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in terms of free bosons and fermions. (orig.)
Algebraic entropy for algebraic maps
International Nuclear Information System (INIS)
Hone, A N W; Ragnisco, Orlando; Zullo, Federico
2016-01-01
We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Bäcklund transformations. (letter)
N=2 current algebra and coset models
International Nuclear Information System (INIS)
Hull, C.M.; Spence, B.
1990-01-01
The N=2 supersymmetric extension of the Kac-Moody algebra and the corresponding Sugawara construction of the N=2 superconformal algebra are discussed both in components and in N=1 superspace. A formulation of the Kac-Moody algebra and Sugawara construction is given in N=2 superspace in terms of supercurrents satisfying a non-linear chiral constraint. The operator product of two supercurrents includes terms that are non-linear in the supercurrents. The N=2 generalization of the GKO coset construction is then given and the conditions found by Kazama and Suzuki are seen to arise from the non-linearity of the algebra. (orig.)
Directory of Open Access Journals (Sweden)
Kushnir V.
2017-12-01
Full Text Available The problem of constructing quadratic equations and systems of equations with parameters using Maple-technology is studied. Today, the "learning tasks of reverse thinking" (V.A. Krutetsky or simply "inverse problems" (P.M.Erdniev are increasingly being introduced into the educational process. The tasks of constructing mathematical tasks in advance of a certain type and certain properties are inverse problems that unfold another aspect of the learning situation and thereby create a "surplus of its vision" (M.M. Bakhtin. The solution of inverse problems develops students’ thinking, imagination and other higher mental functions. However, their introduction into the educational process is still insufficient. One of the reasons for this situation is the insufficient number of benefits with a sufficient number of variants of the same type of tasks. Especially it concerns the construction of problems with parameters. Designing in "manual mode" requires significant temporary cognitive, physical and other efforts, carries the risks of allowing technical and computational errors. In the days of the information society and the digital economy, there are all the possibilities to perform the chain of design actions in a certain ICT environment (we have a Maple-environment. It solves the resulted difficulties of construction, creates a new educational and information environment, allows to produce automatically a sufficient number of different versions of the same type of tasks. Tasks with parameters require creativity from the students, non-standard approaches to the solution. Each task with parameters requires the creation of its own method and algorithm for solving and productive learning. The article is devoted to solving of the above problems.
Characterizations of locally C*-algebras
International Nuclear Information System (INIS)
Mohammad, N.; Somasundaram, S.
1991-08-01
We seek the generalization of the Gelfand-Naimark theorems for locally C*-algebras. Precisely, if A is a unital commutative locally C*-algebra, then it is shown that A is *-isomorphic (topologically and algebraically) to C(Δ). Further, if A is any locally C*-algebra, then it is realized as a closed *-subalgebra of some L(H) up to a topological algebraic *-isomorphism. Also, a brief exposition of the Gelfand-Naimark-Segal construction is given and some of its consequences are discussed. (author). 16 refs
Banana Algebra: Compositional syntactic language extension
DEFF Research Database (Denmark)
Andersen, Jacob; Brabrand, Claus; Christiansen, David Raymond
2013-01-01
We propose an algebra of languages and transformations as a means of compositional syntactic language extension. The algebra provides a layer of high-level abstractions built on top of languages (captured by context-free grammars) and transformations (captured by constructive catamorphisms...... algebra as presented in the paper is implemented as the Banana Algebra Tool which may be used to syntactically extend languages in an incremental and modular fashion via algebraic composition of previously defined languages and transformations. We demonstrate and evaluate the tool via several kinds...
Explicit field realizations of W algebras
International Nuclear Information System (INIS)
Wei Shaowen; Liu Yuxiao; Ren Jirong; Zhang Lijie
2009-01-01
The fact that certain nonlinear W 2,s algebras can be linearized by the inclusion of a spin-1 current can provide a simple way to realize W 2,s algebras from linear W 1,2,s algebras. In this paper, we first construct the explicit field realizations of linear W 1,2,s algebras with double scalar and double spinor, respectively. Then, after a change of basis, the realizations of W 2,s algebras are presented. The results show that all these realizations are Romans-type realizations.
Explicit field realizations of W algebras
Wei, Shao-Wen; Liu, Yu-Xiao; Zhang, Li-Jie; Ren, Ji-Rong
2009-01-01
The fact that certain non-linear $W_{2,s}$ algebras can be linearized by the inclusion of a spin-1 current can provide a simple way to realize $W_{2,s}$ algebras from linear $W_{1,2,s}$ algebras. In this paper, we first construct the explicit field realizations of linear $W_{1,2,s}$ algebras with double-scalar and double-spinor, respectively. Then, after a change of basis, the realizations of $W_{2,s}$ algebras are presented. The results show that all these realizations are Romans-type realiz...
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
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...
International Nuclear Information System (INIS)
MacCallum, M.A.H.
1990-01-01
The implementation of a new computer algebra system is time consuming: designers of general purpose algebra systems usually say it takes about 50 man-years to create a mature and fully functional system. Hence the range of available systems and their capabilities changes little between one general relativity meeting and the next, despite which there have been significant changes in the period since the last report. The introductory remarks aim to give a brief survey of capabilities of the principal available systems and highlight one or two trends. The reference to the most recent full survey of computer algebra in relativity and brief descriptions of the Maple, REDUCE and SHEEP and other applications are given. (author)
Liesen, Jörg
2015-01-01
This self-contained textbook takes a matrix-oriented approach to linear algebra and presents a complete theory, including all details and proofs, culminating in the Jordan canonical form and its proof. Throughout the development, the applicability of the results is highlighted. Additionally, the book presents special topics from applied linear algebra including matrix functions, the singular value decomposition, the Kronecker product and linear matrix equations. The matrix-oriented approach to linear algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore simplifies their use in real world applications. Some of these applications are presented in detailed examples. In several ‘MATLAB-Minutes’ students can comprehend the concepts and results using computational experiments. Necessary basics for the use of MATLAB are presented in a short introduction. Students can also actively work with the material and practice their mathematical skills in more than 300 exerc...
Canonical form of three-fermion pure-states with six single particle states
International Nuclear Information System (INIS)
Chen, Lin; Ž Ðoković, Dragomir; Grassl, Markus; Zeng, Bei
2014-01-01
We construct a canonical form for pure states in ∧ 3 (C 6 ), the three-fermion system with six single particle states, under local unitary (LU) transformations, i.e., the unitary group U(6). We also construct a minimal set of generators of the algebra of polynomial U(6)-invariants on ∧ 3 (C 6 ). It turns out that this algebra is isomorphic to the algebra of polynomial LU-invariants of three-qubits which are additionally invariant under qubit permutations. As a consequence of this surprising fact, we deduce that there is a one-to-one correspondence between the U(6)-orbits of pure three-fermion states in ∧ 3 (C 6 ) and the LU orbits of pure three-qubit states when qubit permutations are allowed. As an important byproduct, we obtain a new canonical form for pure three-qubit states under LU transformations U(2) × U(2) × U(2) (no qubit permutations allowed)
Stoll, R R
1968-01-01
Linear Algebra is intended to be used as a text for a one-semester course in linear algebra at the undergraduate level. The treatment of the subject will be both useful to students of mathematics and those interested primarily in applications of the theory. The major prerequisite for mastering the material is the readiness of the student to reason abstractly. Specifically, this calls for an understanding of the fact that axioms are assumptions and that theorems are logical consequences of one or more axioms. Familiarity with calculus and linear differential equations is required for understand
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
Jacobson, Nathan
2009-01-01
A classic text and standard reference for a generation, this volume and its companion are the work of an expert algebraist who taught at Yale for two decades. Nathan Jacobson's books possess a conceptual and theoretical orientation, and in addition to their value as classroom texts, they serve as valuable references.Volume I explores all of the topics typically covered in undergraduate courses, including the rudiments of set theory, group theory, rings, modules, Galois theory, polynomials, linear algebra, and associative algebra. Its comprehensive treatment extends to such rigorous topics as L
Fractional supersymmetry through generalized anyonic algebra
International Nuclear Information System (INIS)
Douari, Jamila; Abdus Salam International Centre for Theoretical Physics, Trieste; Hassouni, Yassine
2001-01-01
The construction of anyonic operators and algebra is generalized by using quons operators. Therefore, the particular version of fractional supersymmetry is constructed on the two-dimensional lattice by associating two generalized anyons of different kinds. The fractional supersymmetry Hamiltonian operator is obtained on the two-dimensional lattice and the quantum algebra U q (sl 2 ) is realized. (author)
Coherent states for polynomial su(2) algebra
International Nuclear Information System (INIS)
Sadiq, Muhammad; Inomata, Akira
2007-01-01
A class of generalized coherent states is constructed for a polynomial su(2) algebra in a group-free manner. As a special case, the coherent states for the cubic su(2) algebra are discussed. The states so constructed reduce to the usual SU(2) coherent states in the linear limit
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
DEFF Research Database (Denmark)
Schmidt, Thomas Lundsgaard
such a map, generalising the transformation groupoid of a local homeomorphism first introduced by Renault in \\cite{re}. We conduct a detailed study of the relationship between the dynamics of $\\phi$, the properties of these groupoids, the structure of their corresponding reduced groupoid $C^*$-algebras, and......, for certain classes of maps, the K-theory of these algebras. When the map $\\phi$ is transitive, we show that the algebras $C^*_r(\\Gamma_\\phi)$ and $C^*_r(\\Gamma_\\phi^+)$ are purely infinite and satisfy the Universal Coefficient Theorem. Furthermore, we find necessary and sufficient conditions for simplicity...... of these algebras in terms of dynamical properties of $\\phi$. We proceed to consider the situation when the algebras are non-simple, and describe the primitive ideal spectrum in this case. We prove that any irreducible representation factors through the $C^*$-algebra of the reduction of the groupoid to the orbit...
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)
Coxeter groups and Hopf algebras
Aguiar, Marcelo
2011-01-01
An important idea in the work of G.-C. Rota is that certain combinatorial objects give rise to Hopf algebras that reflect the manner in which these objects compose and decompose. Recent work has seen the emergence of several interesting Hopf algebras of this kind, which connect diverse subjects such as combinatorics, algebra, geometry, and theoretical physics. This monograph presents a novel geometric approach using Coxeter complexes and the projection maps of Tits for constructing and studying many of these objects as well as new ones. The first three chapters introduce the necessary backgrou
Linear operators in Clifford algebras
International Nuclear Information System (INIS)
Laoues, M.
1991-01-01
We consider the real vector space structure of the algebra of linear endomorphisms of a finite-dimensional real Clifford algebra (2, 4, 5, 6, 7, 8). A basis of that space is constructed in terms of the operators M eI,eJ defined by x→e I .x.e J , where the e I are the generators of the Clifford algebra and I is a multi-index (3, 7). In particular, it is shown that the family (M eI,eJ ) is exactly a basis in the even case. (orig.)
Directory of Open Access Journals (Sweden)
V. P. Shapeev
2014-01-01
Full Text Available The method of collocations and least residuals (CLR, which was proposed previously for the numerical solution of two-dimensional Navier–Stokes equations governing the stationary flows of a viscous incompressible fluid, is extended here for the three-dimensional case. The solution is sought in the implemented version of the method in the form of an expansion in the basis solenoidal functions. At all stages of the CLR method construction, a computer algebra system (CAS is applied for the derivation and verification of the formulas of the method and for their translation into arithmetic operators of the Fortran language. For accelerating the convergence of iterations a sufficiently universal algorithm is proposed, which is simple in its implementation and is based on the use of the Krylov’s subspaces. The obtained computational formulas of the CLR method were verified on the exact analytic solution of a test problem. Comparisons with the published numerical results of solving the benchmark problem of the 3D driven cubic cavity flow show that the accuracy of the results obtained by the CLR method corresponds to the known high-accuracy solutions.
Knippenberg, Stefan
2016-10-07
Third-order nonlinear optical (NLO) properties of polymethine dyes have been widely studied for applications such as all-optical switching. However, the limited accuracy of the current computational methodologies has prevented a comprehensive understanding of the nature of the lowest excited states and their influence on the molecular optical and NLO properties. Here, attention is paid to the lowest excited-state energies and their energetic ratio, as these characteristics impact the figure-of-merit for all-optical switching. For a series of model polymethines, we compare several algebraic diagrammatic construction (ADC) schemes for the polarization propagator with approximate second-order coupled cluster (CC2) theory, the widely used INDO/MRDCI approach and the symmetry-adapted cluster configuration interaction (SAC-CI) algorithm incorporating singles and doubles linked excitation operators (SAC-CI SD-R). We focus in particular on the ground-to-excited state transition dipole moments and the corresponding state dipole moments, since these quantities are found to be of utmost importance for an effective description of the third-order polarizability γ and two-photon absorption spectra. A sum-overstates expression has been used, which is found to quickly converge. While ADC(3/2) has been found to be the most appropriate method to calculate these properties, CC2 performs poorly.
Magnetic monopoles without strings by Kaehler-Clifford algebra
International Nuclear Information System (INIS)
Maia, A. Jr.; Recami, E.; Rodrigues, W.A. Jr.; Rosa, M.A.F.
1990-01-01
In place of Dirac monopoles with string, this paper presents monopoles without string on the basis of a generalized potential, the sum of a vector A and a pseudovector γB potential. By having recourse to the (graded) Clifford algebra which allows adding together tensors of different ranks (e.g., scalars + pseudoscalars + vectors + pseudovectors + . . .), in a previous paper we succeeded in constructing a Lagrangian and Hamiltonian formalism for interacting monopoles that can be regarded as satisfactory from various points of view. In the present note, after having completed that formalism, the authors put forth a purely geometrical interpretation of it within the Kahler algebra on differential forms, essential ingredients being the natural introduction of a generalized curvature and the Hodge decomposition. The authors thus pave the way for the extension of monopoles without string to non-abelian gauge groups. The analogies of this approach with supersymmetric theories are apparent
Very true operators on MTL-algebras
Directory of Open Access Journals (Sweden)
Wang Jun Tao
2016-01-01
Full Text Available The main goal of this paper is to investigate very true MTL-algebras and prove the completeness of the very true MTL-logic. In this paper, the concept of very true operators on MTL-algebras is introduced and some related properties are investigated. Also, conditions for an MTL-algebra to be an MV-algebra and a Gödel algebra are given via this operator. Moreover, very true filters on very true MTL-algebras are studied. In particular, subdirectly irreducible very true MTL-algebras are characterized and an analogous of representation theorem for very true MTL-algebras is proved. Then, the left and right stabilizers of very true MTL-algebras are introduced and some related properties are given. As applications of stabilizer of very true MTL-algebras, we produce a basis for a topology on very true MTL-algebras and show that the generated topology by this basis is Baire, connected, locally connected and separable. Finally, the corresponding logic very true MTL-logic is constructed and the soundness and completeness of this logic are proved based on very true MTL-algebras.
Indian Academy of Sciences (India)
tion - 6. How Architectural Features Affect. Building During Earthquakes? C VRMurty. 48 Turbulence and Dispersion. K 5 Gandhi. BOOK REVIEWS. 86 Algebraic Topology. Siddhartha Gadgil. Front Cover. - .. ..-.......... -. Back Cover. Two-dimensional vertical section through a turbulent plume. (Courtesy: G S Shat, CAOS, IISc.).
Indian Academy of Sciences (India)
Deligne, Mumford and Artin [DM, Ar2]) and consider algebraic stacks, then we can cons- truct the 'moduli ... the moduli scheme and the moduli stack of vector bundles. First I will give ... 1–31. © Printed in India. 1 ...... Cultura, Spain. References.
(Super)conformal algebra on the (super)torus
International Nuclear Information System (INIS)
Mezincescu, L.; Nepomechie, R.I.; Zachos, C.K.
1989-01-01
A generalization of the Virasoro algebra has recently been introduced by Krichever and Novikov (KN). The KN algebra describes the algebra of general conformal transformations in a basis appropriate to a genus-g Riemann surface. We examine in detail the genus-one KN algebra, and find explicit expressions for the central extension. We, further, construct explicitly the superconformal algebra of the supertorus, which yields supersymmetric generalizations of the genus-one KN algebra. A novel feature of the odd-spin-structure case is that the algebra includes a central element which is anticommuting. We comment on possible applications to string theory. (orig.)
Triangle identity and free differential algebra of massless higher spins
Energy Technology Data Exchange (ETDEWEB)
Vasiliev, M A [AN SSSR, Moscow. Fizicheskij Inst.
1989-09-25
In terms of Berezins's theory of symbols of operators, the integral formulation is suggested for the free differential algebra which gives rise to consistent equations of motion of interacting massless fields of all spins 0{le}s<{infinity} in the frameworks of gravity. In the first nontrivial order of the expansion in powers of curvatures, Frobenius consistency conditions for higher-spin equations of motion are shown to reduce to the simple geometrical fast that there are two ways for splitting any quadrangle in two triangles. To clarify our construction, we illustrate how it works in the simplest case of pure gravity. (orig.).
Ternary q-Virasoro-Witt Hom-Nambu-Lie algebras
International Nuclear Information System (INIS)
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.
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.)
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.
Algebraic characterizations of measure algebras
Czech Academy of Sciences Publication Activity Database
Jech, Thomas
2008-01-01
Roč. 136, č. 4 (2008), s. 1285-1294 ISSN 0002-9939 R&D Projects: GA AV ČR IAA100190509 Institutional research plan: CEZ:AV0Z10190503 Keywords : Von - Neumann * sequential topology * Boolean-algebras * Souslins problem * Submeasures Subject RIV: BA - General Mathematics Impact factor: 0.584, year: 2008
International Nuclear Information System (INIS)
Mohammad, N.; Siddiqui, A.H.
1987-11-01
The notion of a 2-Banach algebra is introduced and its structure is studied. After a short discussion of some fundamental properties of bivectors and tensor product, several classical results of Banach algebras are extended to the 2-Banach algebra case. A condition under which a 2-Banach algebra becomes a Banach algebra is obtained and the relation between algebra of bivectors and 2-normed algebra is discussed. 11 refs
Macdonald index and chiral algebra
Song, Jaewon
2017-08-01
For any 4d N = 2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. We conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type ( A 1 , A 2 n ) and ( A 1 , D 2 n+1) where the chiral algebras are given by Virasoro and \\widehat{su}(2) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.
Paragrassmann analysis and covariant quantum algebras
International Nuclear Information System (INIS)
Filippov, A.T.; Isaev, A.P.; Kurdikov, A.B.; Pyatov, P.N.
1993-01-01
This report is devoted to the consideration from the algebraic point of view the paragrassmann algebras with one and many paragrassmann generators Θ i , Θ p+1 i = 0. We construct the paragrassmann versions of the Heisenberg algebra. For the special case, this algebra is nothing but the algebra for coordinates and derivatives considered in the context of covariant differential calculus on quantum hyperplane. The parameter of deformation q in our case is (p+1)-root of unity. Our construction is nondegenerate only for even p. Taking bilinear combinations of paragrassmann derivatives and coordinates we realize generators for the covariant quantum algebras as tensor products of (p+1) x (p+1) matrices. (orig./HSI)
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)
Algebraic limit cycles in polynomial systems of differential equations
International Nuclear Information System (INIS)
Llibre, Jaume; Zhao Yulin
2007-01-01
Using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4, 5 and 6. We also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3. Moreover, we show that there are polynomial systems of differential equations of arbitrary degree that have algebraic limit cycles of degree 3, as well as give an example of a cubic polynomial system of differential equations with two algebraic limit cycles of degree 4
Mulligan, Jeffrey B.
2017-01-01
A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. The difficulty addressed here is the fact that, because of metamerism, we cannot know with certainty the spectrum that produced a particular color solely on the basis of sensory data. Knowledge of the spectrum is not required to compute additive mixture of colors, but is critical for subtractive (multiplicative) mixture. Therefore, we cannot predict with certainty the multiplicative interactions between colors based solely on sensory data. There are two potential applications of a color algebra: first, to aid modeling phenomena of human visual perception, such as color constancy and transparency; and, second, to provide better models of the interactions of lights and surfaces for computer graphics rendering.
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
International Nuclear Information System (INIS)
Jacob, M.
1967-01-01
The first three chapters of these lecture notes are devoted to generalities concerning current algebra. The weak currents are defined, and their main properties given (V-A hypothesis, conserved vector current, selection rules, partially conserved axial current,...). The SU (3) x SU (3) algebra of Gell-Mann is introduced, and the general properties of the non-leptonic weak Hamiltonian are discussed. Chapters 4 to 9 are devoted to some important applications of the algebra. First one proves the Adler- Weisberger formula, in two different ways, by either the infinite momentum frame, or the near-by singularities method. In the others chapters, the latter method is the only one used. The following topics are successively dealt with: semi leptonic decays of K mesons and hyperons, Kroll- Ruderman theorem, non leptonic decays of K mesons and hyperons ( ΔI = 1/2 rule), low energy theorems concerning processes with emission (or absorption) of a pion or a photon, super-convergence sum rules, and finally, neutrino reactions. (author) [fr
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.)
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.)
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
Algebraic structure of chiral anomalies
International Nuclear Information System (INIS)
Stora, R.
1985-09-01
I will describe first the algebraic aspects of chiral anomalies, exercising however due care about the topological delicacies. I will illustrate the structure and methods in the context of gauge anomalies and will eventually make contact with results obtained from index theory. I will go into two sorts of generalizations: on the one hand, generalizing the algebraic set up yields e.g. gravitational and mixed gauge anomalies, supersymmetric gauge anomalies, anomalies in supergravity theories; on the other hand most constructions applied to the cohomologies which characterize anomalies easily extend to higher cohomologies. Section II is devoted to a description of the general set up as it applies to gauge anomalies. Section III deals with a number of algebraic set ups which characterize more general types of anomalies: gravitational and mixed gauge anomalies, supersymmetric gauge anomalies, anomalies in supergravity theories. It also includes brief remarks on σ models and a reminder on the full BRST algebra of quantized gauge theories
Kleyn, Aleks
2007-01-01
The concept of F-algebra and its representation can be extended to an arbitrary bundle. We define operations of fibered F-algebra in fiber. The paper presents the representation theory of of fibered F-algebra as well as a comparison of representation of F-algebra and of representation of fibered F-algebra.
International Nuclear Information System (INIS)
Aisaka, Yuri; Kazama, Yoichi
2006-01-01
In a previous work, we have constructed a reparametrization invariant worldsheet action from which one can derive the super-Poincare covariant pure spinor formalism for the superstring at the fully quantum level. The main idea was the doubling of the spinor degrees of freedom in the Green-Schwarz formulation together with the introduction of a new compensating local fermionic symmetry. In this paper, we extend this 'double spinor' formalism to the case of the supermembrane in 11 dimensions at the classical level. The basic scheme works in parallel with the string case and we are able to construct the closed algebra of first class constraints which governs the entire dynamics of the system. A notable difference from the string case is that this algebra is first order reducible and the associated BRST operator must be constructed accordingly. The remaining problems which need to be solved for the quantization will also be discussed
An application of the division algebras, Jordan algebras and split composition algebras
International Nuclear Information System (INIS)
Foot, R.; Joshi, G.C.
1992-01-01
It has been established that the covering group of the Lorentz group in D = 3, 4, 6, 10 can be expressed in a unified way, based on the four composition division algebras R, C, Q and O. In this paper, the authors discuss, in this framework, the role of the complex numbers of quantum mechanics. A unified treatment of quantum-mechanical spinors is given. The authors provide an explicit demonstration that the vector and spinor transformations recently constructed from a subgroup of the reduced structure group of the Jordan algebras M n 3 are indeed the Lorentz transformations. The authors also show that if the division algebras in the construction of the covering groups of the Lorentz groups in D = 3, 4, 6, 10 are replaced by the split composition algebras, then the sequence of groups SO(2, 2), SO(3, 3) and SO(5, 5) result. The analysis is presumed to be self-contained as the relevant aspects of the division algebras and Jordan algebras are reviewed. Some applications to physical theory are indicated
Optimizing relational algebra operations using generic equivalence discriminators and lazy products
DEFF Research Database (Denmark)
Henglein, Fritz
2010-01-01
We show how to efficiently evaluate generic map-filter-product queries, generalizations of select-project-join (SPJ) queries in re- lational algebra, based on a combination of two novel techniques: generic discrimination-based joins and lazy (formal) products. Discrimination-based joins are based...... that discriminators can be constructed generically (by structural recursion on equivalence expressions), purely func- tionally, and efficiently (worst-case linear time). The array-based basic multiset discrimination algorithm of Cai and Paige (1995) provides a base discriminator that is both asymptotically and prac...... on relational algebra equalities, without need for a query preprocessing phase. They require no indexes and behave purely functionally. They can be considered a form of symbolic execution of set expressions that automate and encapsulate dynamic program transformation of such expressions and lead to asymptotic...
A bicategorical approach to Morita equivalence for von Neumann algebras
International Nuclear Information System (INIS)
Brouwer, R. M.
2003-01-01
We relate Morita equivalence for von Neumann algebras to the ''Connes fusion'' tensor product between correspondences. In the purely algebraic setting, it is well known that rings are Morita equivalent if they are equivalent objects in a bicategory whose 1-cells are bimodules. We present a similar result for von Neumann algebras. We show that von Neumann algebras form a bicategory, having Connes's correspondences as 1-morphisms, and (bounded) intertwiners as 2-morphisms. Further, we prove that two von Neumann algebras are Morita equivalent iff they are equivalent objects in the bicategory. The proofs make extensive use of the Tomita-Takesaki modular theory
On extensions of superconformal algebras
International Nuclear Information System (INIS)
Nagi, Jasbir
2005-01-01
Starting from vector fields that preserve a differential form on a Riemann sphere with Grassmann variables, one can construct a superconformal algebra by considering central extensions of the algebra of vector fields. In this paper, the N=4 case is analyzed closely, where the presence of weight zero operators in the field theory forces the introduction of noncentral extensions. How this modifies the existing field theory, representation theory, and Gelfand-Fuchs constructions is discussed. It is also discussed how graded Riemann sphere geometry can be used to give a geometrical description of the central charge in the N=1 theory
International Nuclear Information System (INIS)
Dragon, N.
1979-01-01
The possible use of trilinear algebras as symmetry algebras for para-Fermi fields is investigated. The shortcomings of the examples are argued to be a general feature of such generalized algebras. (author)
Bliss, Gilbert Ames
1933-01-01
This book, immediately striking for its conciseness, is one of the most remarkable works ever produced on the subject of algebraic functions and their integrals. The distinguishing feature of the book is its third chapter, on rational functions, which gives an extremely brief and clear account of the theory of divisors.... A very readable account is given of the topology of Riemann surfaces and of the general properties of abelian integrals. Abel's theorem is presented, with some simple applications. The inversion problem is studied for the cases of genus zero and genus unity. The chapter on t
Verifying Process Algebra Proofs in Type Theory
Sellink, M.P.A.
In this paper we study automatic verification of proofs in process algebra. Formulas of process algebra are represented by types in typed λ-calculus. Inhabitants (terms) of these types represent proofs. The specific typed λ-calculus we use is the Calculus of Inductive Constructions as implemented
Generalized W-algebras and integrable hierarchies
International Nuclear Information System (INIS)
Burroughs, N.; De Groot, M.; Hollowood, T.; Miramontes, L.
1992-01-01
We report on generalizations of the KdV-type integrable hierarchies of Drinfel'd and Sokolov. These hierarchies lead to the existence of new classical W-algebras, which arise as the second hamiltonian structure of the hierarchies. In particular, we present a construction of the W n (l) -algebras. (orig.)
Planar Algebra of the Subgroup-Subfactor
Indian Academy of Sciences (India)
The crucial step in this identification is an exhibition of a model for the basic construction tower, and thereafter of the standard invariant of R ⋊ H ⊂ R ⋊ G in terms of operator matrices. We also obtain an identification between the planar algebra of the fixed algebra subfactor R G ⊂ R H and the -invariant planar subalgebra ...
DEFF Research Database (Denmark)
Schmidt, Thomas Lundsgaard; Thomsen, Klaus
2015-01-01
We consider a construction of $C^*$-algebras from continuous piecewise monotone maps on the circle which generalizes the crossed product construction for homeomorphisms and more generally the construction of Renault, Deaconu and Anantharaman-Delaroche for local homeomorphisms. Assuming that the map...... is surjective and not locally injective we give necessary and sufficient conditions for the simplicity of the $C^*$-algebra and show that it is then a Kirchberg algebra. We provide tools for the calculation of the K-theory groups and turn them into an algorithmic method for Markov maps....
Iterated Leavitt Path Algebras
International Nuclear Information System (INIS)
Hazrat, R.
2009-11-01
Leavitt path algebras associate to directed graphs a Z-graded algebra and in their simplest form recover the Leavitt algebras L(1,k). In this note, we introduce iterated Leavitt path algebras associated to directed weighted graphs which have natural ± Z grading and in their simplest form recover the Leavitt algebras L(n,k). We also characterize Leavitt path algebras which are strongly graded. (author)
Finite W-algebras and intermediate statistics
International Nuclear Information System (INIS)
Barbarin, F.; Ragoucy, E.; Sorba, P.
1994-09-01
New realizations of finite W-algebras are constructed by relaxing the usual conditions. Then finite W-algebras are recognized in the Heisenberg quantization recently proposed by Leinaas and Myrheim, for a system of two identical particles in d dimensions. As the anyonic parameter is directly associated to the W-algebra involved in the d=1 case, it is natural to consider that the W-algebra framework is well adapted for a possible generalization of the anyon statistics. (author). 13 refs
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...
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.)
Computations in finite-dimensional Lie algebras
Directory of Open Access Journals (Sweden)
A. M. Cohen
1997-12-01
Full Text Available This paper describes progress made in context with the construction of a general library of Lie algebra algorithms, called ELIAS (Eindhoven Lie Algebra System, within the computer algebra package GAP. A first sketch of the package can be found in Cohen and de Graaf[1]. Since then, in a collaborative effort with G. Ivanyos, the authors have continued to develop algorithms which were implemented in ELIAS by the second author. These activities are part of a bigger project, called ACELA and financed by STW, the Dutch Technology Foundation, which aims at an interactive book on Lie algebras (cf. Cohen and Meertens [2]. This paper gives a global description of the main ways in which to present Lie algebras on a computer. We focus on the transition from a Lie algebra abstractly given by an array of structure constants to a Lie algebra presented as a subalgebra of the Lie algebra of n×n matrices. We describe an algorithm typical of the structure analysis of a finite-dimensional Lie algebra: finding a Levi subalgebra of a Lie algebra.
Grätzer, George
1979-01-01
Universal Algebra, heralded as ". . . the standard reference in a field notorious for the lack of standardization . . .," has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science. Each chapter is followed by an extensive list of exercises and problems. The "state of the art" account also includes new appendices (with contributions from B. Jónsson, R. Quackenbush, W. Taylor, and G. Wenzel) and a well-selected additional bibliography of over 1250 papers and books which makes this a fine work for students, instructors, and researchers in the field. "This book will certainly be, in the years to come, the basic reference to the subject." --- The American Mathematical Monthly (First Edition) "In this reviewer's opinion [the author] has more than succeeded in his aim. The problems at the end of each chapter are well-chosen; there are more than 650 of them. The book is especially sui...
On the Structure of С*-Algebras Generated by Representations of the Elementary Inverse Semigroup
Directory of Open Access Journals (Sweden)
S.A. Grigoryan
2016-06-01
Full Text Available The class of С*-algebras generated by the elementary inverse semigroup and being deformations of the Toeplitz algebra has been introduced and studied. The properties of these algebras have been investigated. All their irreducible representations and automorphism groups have been described. These algebras have been proved to be Z-graded С*-algebras. For a certain class of algebras in the family under consideration the compact quantum semigroup structure has been constructed.
Observable algebras for the rational and trigonometric Euler-Calogero-Moser Models
International Nuclear Information System (INIS)
Avan, J.; Billey, E.
1995-01-01
We construct polynomial Poisson algebras of observables for the classical Euler-Calogero-Moser (ECM) models. Their structure connects them to flavour-indexed non-linear W ∞ algebras, albeit with qualitative differences. The conserved Hamiltonians and symmetry algebras derived in a previous work are subsets of these algebra. We define their linear, N →∞ limits, realizing W ∞ type algebras coupled to current algebras. ((orig.))
Wigner oscillators, twisted Hopf algebras and second quantization
Energy Technology Data Exchange (ETDEWEB)
Castro, P.G.; Toppan, F. [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil)]. E-mails: pgcastro@cbpf.br; toppan@cbpf.br; Chakraborty, B. [S. N. Bose National Center for Basic Sciences, Kolkata (India)]. E-mail: biswajit@bose.res.in
2008-07-01
By correctly identifying the role of central extension in the centrally extended Heisenberg algebra h, we show that it is indeed possible to construct a Hopf algebraic structure on the corresponding enveloping algebra U(h) and eventually deform it through Drinfeld twist. This Hopf algebraic structure and its deformed version U{sup F}(h) is shown to be induced from a more 'fundamental' Hopf algebra obtained from the Schroedinger field/oscillator algebra and its deformed version, provided that the fields/oscillators are regarded as odd-elements of a given superalgebra. We also discuss the possible implications in the context of quantum statistics. (author)
Boundary Lax pairs from non-ultra-local Poisson algebras
International Nuclear Information System (INIS)
Avan, Jean; Doikou, Anastasia
2009-01-01
We consider non-ultra-local linear Poisson algebras on a continuous line. Suitable combinations of representations of these algebras yield representations of novel generalized linear Poisson algebras or 'boundary' extensions. They are parametrized by a boundary scalar matrix and depend, in addition, on the choice of an antiautomorphism. The new algebras are the classical-linear counterparts of the known quadratic quantum boundary algebras. For any choice of parameters, the non-ultra-local contribution of the original Poisson algebra disappears. We also systematically construct the associated classical Lax pair. The classical boundary principal chiral model is examined as a physical example.
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)
Yoneda algebras of almost Koszul algebras
Indian Academy of Sciences (India)
Abstract. Let k be an algebraically closed field, A a finite dimensional connected. (p,q)-Koszul self-injective algebra with p, q ≥ 2. In this paper, we prove that the. Yoneda algebra of A is isomorphic to a twisted polynomial algebra A![t; β] in one inde- terminate t of degree q +1 in which A! is the quadratic dual of A, β is an ...
String field theory-inspired algebraic structures in gauge theories
International Nuclear Information System (INIS)
Zeitlin, Anton M.
2009-01-01
We consider gauge theories in a string field theory-inspired formalism. The constructed algebraic operations lead, in particular, to homotopy algebras of the related Batalin-Vilkovisky theories. We discuss an invariant description of the gauge fixing procedure and special algebraic features of gauge theories coupled to matter fields.
Reflection equation algebras, coideal subalgebras, and their centres
Kolb, S.; Stokman, J.V.
2009-01-01
Reflection equation algebras and related U-q(g)-comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or equivalently via twisting by a cocycle. In this paper we investigate algebraic and representation theoretic properties of such so
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...
Syntactic Language Extension via an Algebra of Languages and Transformations
DEFF Research Database (Denmark)
Andersen, Jacob; Brabrand, Claus
2010-01-01
We propose an algebra of languages and transformations as a means for extending languages syntactically. The algebra provides a layer of high-level abstractions built on top of languages (captured by context-free grammars) and transformations (captured by constructive catamorphisms). The algebra...... is self-contained in that any term of the algebra specifying a transformation can be reduced to a catamorphism, before the transformation is run. Thus, the algebra comes “for free” without sacrificing the strong safety and efficiency properties of constructive catamorphisms. The entire algebra...... as presented in the paper is implemented as the Banana Algebra Tool which may be used to syntactically extend languages in an incremental and modular fashion via algebraic composition of previously defined languages and transformations. We demonstrate and evaluate the tool via several kinds of extensions....
International Nuclear Information System (INIS)
Edelen, Dominic G B
2003-01-01
Local action of the fundamental group SO(a, 4 + k - a) is used to show that any solution of an algebraically closed differential system, that is generated from matrix Lie algebra valued 1-forms on a four-dimensional parameter space, will generate families of immersions of four-dimensional spacetimes R 4 in flat (4 + k)-dimensional spaces M 4+k with compatible signature. The algorithm is shown to work with local action of SO(a, 4 + k - a) replaced by local action of GL(4 + k). Immersions generated by local action of the Poincare group on the target spacetime are also obtained. Evaluations of the line elements, immersion loci and connection and curvature forms of these immersions are algebraic. Families of immersions that depend on one or more arbitrary functions are calculated for 1 ≤ k ≤ 4. Appropriate sections of graphs of the conformal factor for two and three interacting line singularities immersed in M 6 are given in appendix A. The local immersion theorem given in appendix B shows that all local solutions of the immersion problem are obtained by use of this method and an algebraic extension in exceptional cases
Cluster algebras in mathematical physics
International Nuclear Information System (INIS)
Francesco, Philippe Di; Gekhtman, Michael; Kuniba, Atsuo; Yamazaki, Masahito
2014-01-01
This special issue of Journal of Physics A: Mathematical and Theoretical contains reviews and original research articles on cluster algebras and their applications to mathematical physics. Cluster algebras were introduced by S Fomin and A Zelevinsky around 2000 as a tool for studying total positivity and dual canonical bases in Lie theory. Since then the theory has found diverse applications in mathematics and mathematical physics. Cluster algebras are axiomatically defined commutative rings equipped with a distinguished set of generators (cluster variables) subdivided into overlapping subsets (clusters) of the same cardinality subject to certain polynomial relations. A cluster algebra of rank n can be viewed as a subring of the field of rational functions in n variables. Rather than being presented, at the outset, by a complete set of generators and relations, it is constructed from the initial seed via an iterative procedure called mutation producing new seeds successively to generate the whole algebra. A seed consists of an n-tuple of rational functions called cluster variables and an exchange matrix controlling the mutation. Relations of cluster algebra type can be observed in many areas of mathematics (Plücker and Ptolemy relations, Stokes curves and wall-crossing phenomena, Feynman integrals, Somos sequences and Hirota equations to name just a few examples). The cluster variables enjoy a remarkable combinatorial pattern; in particular, they exhibit the Laurent phenomenon: they are expressed as Laurent polynomials rather than more general rational functions in terms of the cluster variables in any seed. These characteristic features are often referred to as the cluster algebra structure. In the last decade, it became apparent that cluster structures are ubiquitous in mathematical physics. Examples include supersymmetric gauge theories, Poisson geometry, integrable systems, statistical mechanics, fusion products in infinite dimensional algebras, dilogarithm
An Invitation to Algebraic Statistics: New Outlook and Opportunities
Çetin, Eyüp
2012-01-01
Algebra, a branch of pure mathematics, now advances statistics and operations research of applied mathematics. This synergy is called algebraic statistics as a new discipline. Algebraic statistics offers statisticians, management scientists, business researchers, econometricians and algebraists new opportunities, horizons and connections to advance their fields and related application areas. In this effort, this young, vibrant, quickly growing, and active discipline is briefly discussed and s...
Said-Houari, Belkacem
2017-01-01
This self-contained, clearly written textbook on linear algebra is easily accessible for students. It begins with the simple linear equation and generalizes several notions from this equation for the system of linear equations and introduces the main ideas using matrices. It then offers a detailed chapter on determinants and introduces the main ideas with detailed proofs. The third chapter introduces the Euclidean spaces using very simple geometric ideas and discusses various major inequalities and identities. These ideas offer a solid basis for understanding general Hilbert spaces in functional analysis. The following two chapters address general vector spaces, including some rigorous proofs to all the main results, and linear transformation: areas that are ignored or are poorly explained in many textbooks. Chapter 6 introduces the idea of matrices using linear transformation, which is easier to understand than the usual theory of matrices approach. The final two chapters are more advanced, introducing t...
'Affine' algebras on the torus
International Nuclear Information System (INIS)
Zakkari, M.
1993-07-01
The analysis of the Kac-Moody ''like'' algebra L-circumflex 2 (G) on the torus is performed. It will be seen that the root systems construction leading to a Cartan matrix is not possible. Different twist of L-circumflex 2 λ (G) are discussed. Connections with known results are done. (author). 10 refs
Clifford algebras, noncommutative tori and universal quantum computers
Vlasov, Alexander Yu.
2001-01-01
Recently author suggested [quant-ph/0010071] an application of Clifford algebras for construction of a "compiler" for universal binary quantum computer together with later development [quant-ph/0012009] of the similar idea for a non-binary base. The non-binary case is related with application of some extension of idea of Clifford algebras. It is noncommutative torus defined by polynomial algebraic relations of order l. For l=2 it coincides with definition of Clifford algebra. Here is presente...
On (co)homology of Frobenius Poisson algebras
Zhu, Can; Van Oystaeyen, Fred; ZHANG, Yinhuo
2014-01-01
In this paper, we study Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between Poisson homology and Poisson cohomology of Frobenius Poisson algebras, similar to that between Hochschild homology and Hochschild cohomology of Frobenius algebras. Then we use the non-degenerate bilinear form on a unimodular Frobenius Poisson algebra to construct a Batalin-Vilkovisky structure on the Poisson cohomology ring making it into a Batalin-Vilkovisk...
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.)
Examples of algebrae with equal dynamic entropy
International Nuclear Information System (INIS)
Narnhofer, H.
1988-01-01
For given dynamical entropy we construct uncountably many examples of corresponding algebras, some of them are quantum K systems, whereas at least one explicit example is not. Consequences for cluster properties are studied. 12 refs. (Author)
An introduction to Kac-Moody algebras and their physical applications
International Nuclear Information System (INIS)
Goddard, P.; Olive, D.
1986-01-01
Kac-Moody algebras, the physical applications and results on their representation theory are surveyed. The Sugawara construction of the Virasoro algebra associated with a Kac-Moody algebra is described and it is used to produce the full discrete series of representations of the Virasoro algebra. The quark model construction of representations of Kac-Moody algebras is also described. Conditions necessary for the equivalence of two-dimensional σ-models to free fermion theories are derived
Dzhumadil'daev, A. S.
2002-01-01
Algebras with identity $(a\\star b)\\star (c\\star d) -(a\\star d)\\star(c\\star b)$ $=(a,b,c)\\star d-(a,d,c)\\star b$ are studied. Novikov algebras under Jordan multiplication and Leibniz dual algebras satisfy this identity. If algebra with such identity has unit, then it is associative and commutative.
A bicategorical approach to Morita equivalence for Von Neumann algebras
R.M. Brouwer (Rachel)
2003-01-01
textabstractWe relate Morita equivalence for von Neumann algebras to the ``Connes fusion'' tensor product between correspondences. In the purely algebraic setting, it is well known that rings are Morita equivalent if and only if they are equivalent objects in a bicategory whose 1-cells are
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
Foulis, David J.; Pulmannov, Sylvia
2018-04-01
Using a representation theorem of Erik Alfsen, Frederic Schultz, and Erling Størmer for special JB-algebras, we prove that a synaptic algebra is norm complete (i.e., Banach) if and only if it is isomorphic to the self-adjoint part of a Rickart C∗-algebra. Also, we give conditions on a Banach synaptic algebra that are equivalent to the condition that it is isomorphic to the self-adjoint part of an AW∗-algebra. Moreover, we study some relationships between synaptic algebras and so-called generalized Hermitian algebras.
A note on the algebraic evaluation of correlators in local chiral conformal field theory
International Nuclear Information System (INIS)
Honecker, A.
1992-09-01
We comment on a program designed for the study of local chiral algebras and their representations in 2D conformal field theory. Based on the algebraic approach described by W. Nahm, this program efficiently calculates arbitrary n-point functions of these algebras. The program is designed such that calculations involving e.g. current algebras, W-algebras and N-Superconformal algebras can be performed. As a non-trivial application we construct an extension of the Virasoro algebra by two fields with spin four and six using the N=1-Super-Virasoro algebra. (orig.)
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)
A type of loop algebra and the associated loop algebras
International Nuclear Information System (INIS)
Tam Honwah; Zhang Yufeng
2008-01-01
A higher-dimensional twisted loop algebra is constructed. As its application, a new Lax pair is presented, whose compatibility gives rise to a Liouville integrable hierarchy of evolution equations by making use of Tu scheme. One of the reduction cases of the hierarchy is an analogous of the well-known AKNS system. Next, the twisted loop algebra, furthermore, is extended to another higher dimensional loop algebra, from which a hierarchy of evolution equations with 11-potential component functions is obtained, whose reduction is just standard AKNS system. Especially, we prove that an arbitrary linear combination of the four Hamiltonian operators directly obtained from the recurrence relations is still a Hamiltonian operator. Therefore, the hierarchy with 11-potential functions possesses 4-Hamiltonian structures. Finally, an integrable coupling of the hierarchy is worked out
Algebraic K-theory of generalized schemes
DEFF Research Database (Denmark)
Anevski, Stella Victoria Desiree
and geometry over the field with one element. It also permits the construction of important Arakelov theoretical objects, such as the completion \\Spec Z of Spec Z. In this thesis, we prove a projective bundle theorem for the eld with one element and compute the Chow rings of the generalized schemes Sp\\ec ZN......Nikolai Durov has developed a generalization of conventional scheme theory in which commutative algebraic monads replace commutative unital rings as the basic algebraic objects. The resulting geometry is expressive enough to encompass conventional scheme theory, tropical algebraic geometry......, appearing in the construction of \\Spec Z....
Surface defects and chiral algebras
Energy Technology Data Exchange (ETDEWEB)
Córdova, Clay [School of Natural Sciences, Institute for Advanced Study,1 Einstein Dr, Princeton, NJ 08540 (United States); Gaiotto, Davide [Perimeter Institute for Theoretical Physics,31 Caroline St N, Waterloo, ON N2L 2Y5 (Canada); Shao, Shu-Heng [School of Natural Sciences, Institute for Advanced Study,1 Einstein Dr, Princeton, NJ 08540 (United States)
2017-05-26
We investigate superconformal surface defects in four-dimensional N=2 superconformal theories. Each such defect gives rise to a module of the associated chiral algebra and the surface defect Schur index is the character of this module. Various natural chiral algebra operations such as Drinfeld-Sokolov reduction and spectral flow can be interpreted as constructions involving four-dimensional surface defects. We compute the index of these defects in the free hypermultiplet theory and Argyres-Douglas theories, using both infrared techniques involving BPS states, as well as renormalization group flows onto Higgs branches. In each case we find perfect agreement with the predicted characters.
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
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).
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
Abrams, Gene; Siles Molina, Mercedes
2017-01-01
This book offers a comprehensive introduction by three of the leading experts in the field, collecting fundamental results and open problems in a single volume. Since Leavitt path algebras were first defined in 2005, interest in these algebras has grown substantially, with ring theorists as well as researchers working in graph C*-algebras, group theory and symbolic dynamics attracted to the topic. Providing a historical perspective on the subject, the authors review existing arguments, establish new results, and outline the major themes and ring-theoretic concepts, such as the ideal structure, Z-grading and the close link between Leavitt path algebras and graph C*-algebras. The book also presents key lines of current research, including the Algebraic Kirchberg Phillips Question, various additional classification questions, and connections to noncommutative algebraic geometry. Leavitt Path Algebras will appeal to graduate students and researchers working in the field and related areas, such as C*-algebras and...
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
Towards a classification of rational Hopf algebras
International Nuclear Information System (INIS)
Fuchs, J.; Ganchev, A.; Vecsernyes, P.
1994-02-01
Rational Hopf algebras, i.e. certain quasitriangular weak quasi-Hopf *-algebras, are expected to describe the quantum symmetry of rational field theories. In this paper methods are developed which allow for a classification of all rational Hopf algebras that are compatible with some prescribed set of fusion rules. The algebras are parametrized by the solutions of the square, pentagon and hexagon identities. As examples, we classify all solutions for fusion rules with not more than three sectors, as well as for the level three affine A 1 (1) fusion rules. We also establish several general properties of rational Hopf algebras and present a graphical description of the coassociator in terms of labelled tetrahedra. The latter construction allows to make contact with conformal field theory fusing matrices and with invariants of three-manifolds and topological lattice field theory. (orig.)
Fixed point algebras for easy quantum groups
DEFF Research Database (Denmark)
Gabriel, Olivier; Weber, Moritz
2016-01-01
Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their K-groups. Building on prior work by the second author,we prove...... that free easy quantum groups satisfy these conditions and we compute the K-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group S+ n,the free orthogonal quantum group O+ n and the quantum reflection groups Hs+ n. Our fixed point......-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups,which are related to Hopf-Galois extensions....
Samuel, Pierre
2008-01-01
Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics - algebraic geometry, in particular.This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Gal
Boicescu, V; Georgescu, G; Rudeanu, S
1991-01-01
The Lukasiewicz-Moisil algebras were created by Moisil as an algebraic counterpart for the many-valued logics of Lukasiewicz. The theory of LM-algebras has developed to a considerable extent both as an algebraic theory of intrinsic interest and in view of its applications to logic and switching theory.This book gives an overview of the theory, comprising both classical results and recent contributions, including those of the authors. N-valued and &THgr;-valued algebras are presented, as well as &THgr;-algebras with negation.Mathematicians interested in lattice theory or symbolic logic, and computer scientists, will find in this monograph stimulating material for further research.
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
Poincare invariant algebra from instant to light-front quantization
International Nuclear Information System (INIS)
Ji, Chueng-Ryong; Mitchell, Chad
2001-01-01
We present the Poincare algebra interpolating between instant and light-front time quantizations. The angular momentum operators satisfying SU(2) algebra are constructed in an arbitrary interpolation angle and shown to be identical to the ordinary angular momentum and Leutwyler-Stern angular momentum in the instant and light-front quantization limits, respectively. The exchange of the dynamical role between the transverse angular mometum and the boost operators is manifest in our newly constructed algebra
Pure states of general quantum-mechanical systems as Kaehler bundles
International Nuclear Information System (INIS)
Abbati, M.C.; Cirelli, R.; Lanzavecchia, P.; Mania, A.
1984-01-01
Pure states of general quantum systems in the Csup(*)-algebraic approach are endowed with a structure both of Kaehler manifold and of projective bundle with uniformity on the total space. The former structure gives a geometric interpretation of transition probabilities and Wigner theorem. The latter is a finer structure which determines Csup(*)-algebras up to sup(*)-isomorphisms. Pure states of Csup(*)-algebras with continuous trace among projective bundles with uniformity are characterized
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
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
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
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
Generalized EMV-Effect Algebras
Borzooei, R. A.; Dvurečenskij, A.; Sharafi, A. H.
2018-04-01
Recently in Dvurečenskij and Zahiri (2017), new algebraic structures, called EMV-algebras which generalize both MV-algebras and generalized Boolean algebras, were introduced. We present equivalent conditions for EMV-algebras. In addition, we define a partial algebraic structure, called a generalized EMV-effect algebra, which is close to generalized MV-effect algebras. Finally, we show that every generalized EMV-effect algebra is either an MV-effect algebra or can be embedded into an MV-effect algebra as a maximal ideal.
Asveld, P.R.J.
1976-01-01
Operaties op formele talen geven aanleiding tot bijbehorende operatoren op families talen. Bepaalde onderwerpen uit de algebra (universele algebra, tralies, partieel geordende monoiden) kunnen behulpzaam zijn in de studie van verzamelingen van dergelijke operatoren.
Rudiments of algebraic geometry
Jenner, WE
2017-01-01
Aimed at advanced undergraduate students of mathematics, this concise text covers the basics of algebraic geometry. Topics include affine spaces, projective spaces, rational curves, algebraic sets with group structure, more. 1963 edition.
Pure energy solutions - pure tomorrows
International Nuclear Information System (INIS)
Allison, J.
2006-01-01
HTC is an energy technology company whose mandate is to deliver 'Carbon Clear Solutions' to address the pending challenges the energy sector is facing in meeting the environmental impact of Greenhouse Gas emissions, and energy security. HTC will speak on its comprehensive suite of technologies including hydrogen production, CO 2 capture and CO 2 sequestration. HTC has patented technologies that produce H 2 from a broad variety of feedstocks such as Natural gas, Diesel, Gasoline, Bio-fuels i.e. ethanol, methanol and Coal Gasification. HTC Hydrogen reformation systems are unique in their method of delivering pure Hydrogen. Dry Reformation Reactor - New catalyst system designed to eliminate contamination problems (i.e. coking) while at the same time operate at a low temperature. Water Gas Shift Reactor - Plus - improved and redesigned catalyst that improves operating temperature and hydrogen production efficiency. Two stage catalyst reactor that provides near balance of the endothermic and exothermic reaction temperatures for efficient energy balance
Cylindric-like algebras and algebraic logic
Ferenczi, Miklós; Németi, István
2013-01-01
Algebraic logic is a subject in the interface between logic, algebra and geometry, it has strong connections with category theory and combinatorics. Tarski’s quest for finding structure in logic leads to cylindric-like algebras as studied in this book, they are among the main players in Tarskian algebraic logic. Cylindric algebra theory can be viewed in many ways: as an algebraic form of definability theory, as a study of higher-dimensional relations, as an enrichment of Boolean Algebra theory, or, as logic in geometric form (“cylindric” in the name refers to geometric aspects). Cylindric-like algebras have a wide range of applications, in, e.g., natural language theory, data-base theory, stochastics, and even in relativity theory. The present volume, consisting of 18 survey papers, intends to give an overview of the main achievements and new research directions in the past 30 years, since the publication of the Henkin-Monk-Tarski monographs. It is dedicated to the memory of Leon Henkin.
Statistical Aspects of Coherent States of the Higgs Algebra
Shreecharan, T.; Kumar, M. Naveen
2018-04-01
We construct and study various aspects of coherent states of a polynomial angular momentum algebra. The coherent states are constructed using a new unitary representation of the nonlinear algebra. The new representation involves a parameter γ that shifts the eigenvalues of the diagonal operator J 0.
Hopf structure and Green ansatz of deformed parastatistics algebras
Energy Technology Data Exchange (ETDEWEB)
Aneva, Boyka [Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, bld. Tsarigradsko chaussee 72, BG-1784 Sofia (Bulgaria); Popov, Todor [Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, bld. Tsarigradsko chaussee 72, BG-1784 Sofia (Bulgaria)
2005-07-22
Deformed parabose and parafermi algebras are revised and endowed with Hopf structure in a natural way. The noncocommutative coproduct allows for construction of parastatistics Fock-like representations, built out of the simplest deformed Bose and Fermi representations. The construction gives rise to quadratic algebras of deformed anomalous commutation relations which define the generalized Green ansatz.
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...
Commutator identities on associative algebras and integrability of nonlinear pde's
Pogrebkov, A. K.
2007-01-01
It is shown that commutator identities on associative algebras generate solutions of linearized integrable equations. Next, a special kind of the dressing procedure is suggested that in a special class of integral operators enables to associate to such commutator identity both nonlinear equation and its Lax pair. Thus problem of construction of new integrable pde's reduces to construction of commutator identities on associative algebras.
Representation theory of lattice current algebras
International Nuclear Information System (INIS)
Alekseev, A.Yu.; Eidgenoessische Technische Hochschule, Zurich; Faddeev, L.D.; Froehlich, L.D.; Schomerus, V.; Kyoto Univ.
1996-04-01
Lattice current algebras were introduced as a regularization of the left-and right moving degrees of freedom in the WZNW model. They provide examples of lattice theories with a local quantum symmetry U q (G). Their representation theory is studied in detail. In particular, we construct all irreducible representations along with a lattice analogue of the fusion product for representations of the lattice current algebra. It is shown that for an arbitrary number of lattice sites, the representation categories of the lattice current algebras agree with their continuum counterparts. (orig.)
Poisson-Hopf limit of quantum algebras
International Nuclear Information System (INIS)
Ballesteros, A; Celeghini, E; Olmo, M A del
2009-01-01
The Poisson-Hopf analogue of an arbitrary quantum algebra U z (g) is constructed by introducing a one-parameter family of quantizations U z,ℎ (g) depending explicitly on ℎ and by taking the appropriate ℎ → 0 limit. The q-Poisson analogues of the su(2) algebra are discussed and the novel su q P (3) case is introduced. The q-Serre relations are also extended to the Poisson limit. This approach opens the perspective for possible applications of higher rank q-deformed Hopf algebras in semiclassical contexts
Dirac calculus for modules over Grassmann algebra
International Nuclear Information System (INIS)
Plyushchay, M.S.; Razumov, A.V.
1983-01-01
The main ideas of the theory of the modules over a Grassmann algebra are given. The presentation is intended for physicists, therefore acquaintance only with main ideas of the linear algebra including the concept of a tensor product is assumed. Proofs of statements are not given as a rule due to their elementariness. The main result of the work the generalization of case of the modules over a Grassmann algebra. As an example of utilization of this formalism the construction of the oherent states for fermions is considered
The planar algebra associated to a Kac algebra
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
Given a Kac algebra H and a II1 factor M, there is a well-known construction of a ... We have attempted to render this paper 'accessible' and self-contained to a person ..... Define Pk(L) to be the vector-space with basis given by the collection of 'k-tangles ..... The assertions (i), (iii) and (iv) are immediate consequences of the ...
Abstract algebra for physicists
International Nuclear Information System (INIS)
Zeman, J.
1975-06-01
Certain recent models of composite hadrons involve concepts and theorems from abstract algebra which are unfamiliar to most theoretical physicists. The algebraic apparatus needed for an understanding of these models is summarized here. Particular emphasis is given to algebraic structures which are not assumed to be associative. (2 figures) (auth)
Combinatorial commutative algebra
Miller, Ezra
2005-01-01
Offers an introduction to combinatorial commutative algebra, focusing on combinatorial techniques for multigraded polynomial rings, semigroup algebras, and determined rings. The chapters in this work cover topics ranging from homological invariants of monomial ideals and their polyhedral resolutions, to tools for studying algebraic varieties.
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.
The algebras of higher order currents of the fermionic Gross-Neveu model
International Nuclear Information System (INIS)
Saltini, Luis Eduardo
1996-01-01
Results are reported from our studies on the following 2-dimensional field theories: the supersymmetric non-linear sigma model and the fermionic Gross-Neveu model. About the supersymmetric non-linear sigma model, an attempt is made to solve the the algebraic problem of finding the non-local conserved charges and the corresponding algebra, extending the methods described in a previous article for the case of the purely bosonic non linear sigma model. For the fermionic Gross-Neveu model, we intend to construct the conserved currents and the respective charges, related to the abelian U(1) symmetry and non-abelian SU(n) symmetry, at the conformal point and calculate the correlation functions between them. From these results at the conformal point, we want to study the effects of perturbation to get a massive but integral theory
Abstract algebra an introductory course
Lee, Gregory T
2018-01-01
This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields. The first two chapters present preliminary topics such as properties of the integers and equivalence relations. The author then explores the first major algebraic structure, the group, progressing as far as the Sylow theorems and the classification of finite abelian groups. An introduction to ring theory follows, leading to a discussion of fields and polynomials that includes sections on splitting fields and the construction of finite fields. The final part contains applications to public key cryptography as well as classical straightedge and compass constructions. Explaining key topics at a gentle pace, this book is aimed at undergraduate students. It assumes no prior knowledge of the subject and contains over 500 exercises, half of which have detailed solutions provided.
Division algebras, extended supersymmetries and applications
International Nuclear Information System (INIS)
Toppan, F.
2001-03-01
I present here some new results which make explicit the role of the division algebras R, C, H, O in the construction and classification of, respectively, N= 1, 2, 4, 8 global supersymmetric quantum mechanical and classical dynamical systems. In particular an N=8 Malcev superaffine algebra is introduced and its relation to the non-associative N = 8 SCA is discussed. A list of present and possible future applications is given. (author)
Division algebras, extended supersymmetries and applications
International Nuclear Information System (INIS)
Toppan, F.
2001-01-01
I present here some new results which make explicit the role of the division algebras R, C, H, O in the construction and classification of, respectively, N = 1, 2, 4, 8 global supersymmetric quantum mechanical and classical dynamical systems. In particular an N = 8 Malcev superaffine algebra is introduced and its relation to the non-associative N = 8 SCA is discussed. A list of present and possible future applications is given
Meijer, Alko R
2016-01-01
This textbook provides an introduction to the mathematics on which modern cryptology is based. It covers not only public key cryptography, the glamorous component of modern cryptology, but also pays considerable attention to secret key cryptography, its workhorse in practice. Modern cryptology has been described as the science of the integrity of information, covering all aspects like confidentiality, authenticity and non-repudiation and also including the protocols required for achieving these aims. In both theory and practice it requires notions and constructions from three major disciplines: computer science, electronic engineering and mathematics. Within mathematics, group theory, the theory of finite fields, and elementary number theory as well as some topics not normally covered in courses in algebra, such as the theory of Boolean functions and Shannon theory, are involved. Although essentially self-contained, a degree of mathematical maturity on the part of the reader is assumed, corresponding to his o...
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
Kimura, Taro; Pestun, Vasily
2018-04-01
We introduce quiver gauge theory associated with the non-simply laced type fractional quiver and define fractional quiver W-algebras by using construction of Kimura and Pestun (Lett Math Phys, 2018. https://doi.org/10.1007/s11005-018-1072-1; Lett Math Phys, 2018. https://doi.org/10.1007/s11005-018-1073-0) with representation of fractional quivers.
Symmetric vectors and algebraic classification
International Nuclear Information System (INIS)
Leibowitz, E.
1980-01-01
The concept of symmetric vector field in Riemannian manifolds, which arises in the study of relativistic cosmological models, is analyzed. Symmetric vectors are tied up with the algebraic properties of the manifold curvature. A procedure for generating a congruence of symmetric fields out of a given pair is outlined. The case of a three-dimensional manifold of constant curvature (''isotropic universe'') is studied in detail, with all its symmetric vector fields being explicitly constructed
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)
Algorithms in Algebraic Geometry
Dickenstein, Alicia; Sommese, Andrew J
2008-01-01
In the last decade, there has been a burgeoning of activity in the design and implementation of algorithms for algebraic geometric computation. Some of these algorithms were originally designed for abstract algebraic geometry, but now are of interest for use in applications and some of these algorithms were originally designed for applications, but now are of interest for use in abstract algebraic geometry. The workshop on Algorithms in Algebraic Geometry that was held in the framework of the IMA Annual Program Year in Applications of Algebraic Geometry by the Institute for Mathematics and Its
Computer algebra and operators
Fateman, Richard; Grossman, Robert
1989-01-01
The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.
Lectures on algebraic statistics
Drton, Mathias; Sullivant, Seth
2009-01-01
How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics". In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of these lectures is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations: many important statistical models correspond to algebraic or semi-algebraic sets of parameters; the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures; computational algebraic geometry can be used to study parameter spaces and other features of statistical models.
International Nuclear Information System (INIS)
Goddard, Peter
1990-01-01
The algebra of the group of conformal transformations in two dimensions consists of two commuting copies of the Virasoro algebra. In many mathematical and physical contexts, the representations of ν which are relevant satisfy two conditions: they are unitary and they have the ''positive energy'' property that L o is bounded below. In an irreducible unitary representation the central element c takes a fixed real value. In physical contexts, the value of c is a characteristic of a theory. If c < 1, it turns out that the conformal algebra is sufficient to ''solve'' the theory, in the sense of relating the calculation of the infinite set of physically interesting quantities to a finite subset which can be handled in principle. For c ≥ 1, this is no longer the case for the algebra alone and one needs some sort of extended conformal algebra, such as the superconformal algebra. It is these algebras that this paper aims at addressing. (author)
The Boolean algebra and central Galois algebras
Directory of Open Access Journals (Sweden)
George Szeto
2001-01-01
Full Text Available Let B be a Galois algebra with Galois group G, Jg={b∈B∣bx=g(xb for all x∈B} for g∈G, and BJg=Beg for a central idempotent eg. Then a relation is given between the set of elements in the Boolean algebra (Ba,≤ generated by {0,eg∣g∈G} and a set of subgroups of G, and a central Galois algebra Be with a Galois subgroup of G is characterized for an e∈Ba.
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
Nonflexible Lie-admissible algebras
International Nuclear Information System (INIS)
Myung, H.C.
1978-01-01
We discuss the structure of Lie-admissible algebras which are defined by nonflexible identities. These algebras largely arise from the antiflexible algebras, 2-varieties and associator dependent algebras. The nonflexible Lie-admissible algebras in our discussion are in essence byproducts of the study of nonassociative algebras defined by identities of degree 3. The main purpose is to discuss the classification of simple Lie-admissible algebras of nonflexible type
Logarithmic sℓ-hat (2) CFT models from Nichols algebras: I
International Nuclear Information System (INIS)
Semikhatov, A M; Tipunin, I Yu
2013-01-01
We construct chiral algebras that centralize rank-2 Nichols algebras with at least one fermionic generator. This gives ‘logarithmic’ W-algebra extensions of a fractional-level sℓ-hat (2) algebra. We discuss crucial aspects of the emerging general relation between Nichols algebras and logarithmic conformal field theory (CFT) models: (i) the extra input, beyond the Nichols algebra proper, needed to uniquely specify a conformal model; (ii) a relation between the CFT counterparts of Nichols algebras connected by Weyl groupoid maps; and (iii) the common double bosonization U(X) of such Nichols algebras. For an extended chiral algebra, candidates for its simple modules that are counterparts of the U(X) simple modules are proposed, as a first step toward a functorial relation between U(X) and W-algebra representation categories. (paper)
Extended Virasoro algebra and algebra of area preserving diffeomorphisms
International Nuclear Information System (INIS)
Arakelyan, T.A.
1990-01-01
The algebra of area preserving diffeomorphism plays an important role in the theory of relativistic membranes. It is pointed out that the relation between this algebra and the extended Virasoro algebra associated with the generalized Kac-Moody algebras G(T 2 ). The highest weight representation of these infinite-dimensional algebras as well as of their subalgebras is studied. 5 refs
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.
Scaling algebras and renormalization group in algebraic quantum field theory
International Nuclear Information System (INIS)
Buchholz, D.; Verch, R.
1995-01-01
For any given algebra of local observables in Minkowski space an associated scaling algebra is constructed on which renormalization group (scaling) transformations act in a canonical manner. The method can be carried over to arbitrary spacetime manifolds and provides a framework for the systematic analysis of the short distance properties of local quantum field theories. It is shown that every theory has a (possibly non-unique) scaling limit which can be classified according to its classical or quantum nature. Dilation invariant theories are stable under the action of the renormalization group. Within this framework the problem of wedge (Bisognano-Wichmann) duality in the scaling limit is discussed and some of its physical implications are outlined. (orig.)
Towers of algebras in rational conformal field theories
International Nuclear Information System (INIS)
Gomez, C.; Sierra, G.
1991-01-01
This paper reports on Jones fundamental construction applied to rational conformal field theories. The Jones algebra which emerges in this application is realized in terms of duality operations. The generators of the algebra are an open version of Verlinde's operators. The polynomial equations appear in this context as sufficient conditions for the existence of Jones algebra. The ADE classification of modular invariant partition functions is put in correspondence with Jones classification of subfactors
The Universal C*-Algebra of the Electromagnetic Field
Buchholz, Detlev; Ciolli, Fabio; Ruzzi, Giuseppe; Vasselli, Ezio
2016-02-01
A universal C*-algebra of the electromagnetic field is constructed. It is represented in any quantum field theory which incorporates electromagnetism and expresses basic features of the field such as Maxwell's equations, Poincaré covariance and Einstein causality. Moreover, topological properties of the field resulting from Maxwell's equations are encoded in the algebra, leading to commutation relations with values in its center. The representation theory of the algebra is discussed with focus on vacuum representations, fixing the dynamics of the field.
The use of ultraproducts in commutative algebra
Schoutens, Hans
2010-01-01
In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra.
Borzooei, R. A.; Dudek, W. A.; Koohestani, N.
2006-01-01
We study hyper BCC-algebras which are a common generalization of BCC-algebras and hyper BCK-algebras. In particular, we investigate different types of hyper BCC-ideals and describe the relationship among them. Next, we calculate all nonisomorphic 22 hyper BCC-algebras of order 3 of which only three are not hyper BCK-algebras.
Directory of Open Access Journals (Sweden)
R. A. Borzooei
2006-01-01
Full Text Available We study hyper BCC-algebras which are a common generalization of BCC-algebras and hyper BCK-algebras. In particular, we investigate different types of hyper BCC-ideals and describe the relationship among them. Next, we calculate all nonisomorphic 22 hyper BCC-algebras of order 3 of which only three are not hyper BCK-algebras.
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...
Twisted classical Poincare algebras
International Nuclear Information System (INIS)
Lukierski, J.; Ruegg, H.; Tolstoy, V.N.; Nowicki, A.
1993-11-01
We consider the twisting of Hopf structure for classical enveloping algebra U(g), where g is the inhomogeneous rotations algebra, with explicite formulae given for D=4 Poincare algebra (g=P 4 ). The comultiplications of twisted U F (P 4 ) are obtained by conjugating primitive classical coproducts by F element of U(c)xU(c), where c denotes any Abelian subalgebra of P 4 , and the universal R-matrices for U F (P 4 ) are triangular. As an example we show that the quantum deformation of Poincare algebra recently proposed by Chaichian and Demiczev is a twisted classical Poincare algebra. The interpretation of twisted Poincare algebra as describing relativistic symmetries with clustered 2-particle states is proposed. (orig.)
Learning abstract algebra with ISETL
Dubinsky, Ed
1994-01-01
Most students in abstract algebra classes have great difficulty making sense of what the instructor is saying. Moreover, this seems to remain true almost independently of the quality of the lecture. This book is based on the constructivist belief that, before students can make sense of any presentation of abstract mathematics, they need to be engaged in mental activities which will establish an experiential base for any future verbal explanation. No less, they need to have the opportunity to reflect on their activities. This approach is based on extensive theoretical and empirical studies as well as on the substantial experience of the authors in teaching astract algebra. The main source of activities in this course is computer constructions, specifically, small programs written in the mathlike programming language ISETL; the main tool for reflections is work in teams of 2-4 students, where the activities are discussed and debated. Because of the similarity of ISETL expressions to standard written mathematics...
Directory of Open Access Journals (Sweden)
Frank Roumen
2017-01-01
Full Text Available We will define two ways to assign cohomology groups to effect algebras, which occur in the algebraic study of quantum logic. The first way is based on Connes' cyclic cohomology. The resulting cohomology groups are related to the state space of the effect algebra, and can be computed using variations on the Kunneth and Mayer-Vietoris sequences. The second way involves a chain complex of ordered abelian groups, and gives rise to a cohomological characterization of state extensions on effect algebras. This has applications to no-go theorems in quantum foundations, such as Bell's theorem.
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
Solomon, Alan D
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Boolean Algebra includes set theory, sentential calculus, fundamental ideas of Boolean algebras, lattices, rings and Boolean algebras, the structure of a Boolean algebra, and Boolean
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
Nonlinear evolution equations and solving algebraic systems: the importance of computer algebra
International Nuclear Information System (INIS)
Gerdt, V.P.; Kostov, N.A.
1989-01-01
In the present paper we study the application of computer algebra to solve the nonlinear polynomial systems which arise in investigation of nonlinear evolution equations. We consider several systems which are obtained in classification of integrable nonlinear evolution equations with uniform rank. Other polynomial systems are related with the finding of algebraic curves for finite-gap elliptic potentials of Lame type and generalizations. All systems under consideration are solved using the method based on construction of the Groebner basis for corresponding polynomial ideals. The computations have been carried out using computer algebra systems. 20 refs
ASYS: a computer algebra package for analysis of nonlinear algebraic equations systems
International Nuclear Information System (INIS)
Gerdt, V.P.; Khutornoj, N.V.
1992-01-01
A program package ASYS for analysis of nonlinear algebraic equations based on the Groebner basis technique is described. The package is written in REDUCE computer algebra language. It has special facilities to treat polynomial ideals of positive dimension, corresponding to algebraic systems with infinitely many solutions. Such systems can be transformed to an equivalent set of subsystems with reduced number of variables in completely automatic way. It often allows to construct the explicit form of a solution set in many problems of practical importance. Some examples and results of comparison with the standard Reduce package GROEBNER and special-purpose systems FELIX and A1PI are given. 21 refs.; 2 tabs
The Virasoro algebra from harmonic superspace
International Nuclear Information System (INIS)
Saidi, E.H.; Zakkari, M.
1990-08-01
Using harmonic superspace techniques, we construct a new field realization of the Virasoro algebra. The main conformal objects are U(1) Cartan tensors instead of the U(1) Lorentz ones. The new conformal model, which admits moreover a d=2 (4,0) global supersymmetry is constructed out of the infinitely relaxed HST and FS hypermultiplets. The conformal current T 4+ together with the harmonic superspace OPE rules are given. The Virasoro algebra and the harmonic superspace Schwarzian derivative S 4+ are also derived. (author). 14 refs
Concurrence classes for general pure multipartite states
International Nuclear Information System (INIS)
Heydari, Hoshang
2005-01-01
We propose concurrence classes for general pure multipartite states based on an orthogonal complement of a positive operator-valued measure on quantum phase. In particular, we construct W m class, GHZ m , and GHZ m-1 class concurrences for general pure m-partite states. We give explicit expressions for W 3 and GHZ 3 class concurrences for general pure three-partite states and for W 4 , GHZ 4 and GHZ 3 class concurrences for general pure four-partite states
Algebraic computability and enumeration models recursion theory and descriptive complexity
Nourani, Cyrus F
2016-01-01
This book, Algebraic Computability and Enumeration Models: Recursion Theory and Descriptive Complexity, presents new techniques with functorial models to address important areas on pure mathematics and computability theory from the algebraic viewpoint. The reader is first introduced to categories and functorial models, with Kleene algebra examples for languages. Functorial models for Peano arithmetic are described toward important computational complexity areas on a Hilbert program, leading to computability with initial models. Infinite language categories are also introduced to explain descriptive complexity with recursive computability with admissible sets and urelements. Algebraic and categorical realizability is staged on several levels, addressing new computability questions with omitting types realizably. Further applications to computing with ultrafilters on sets and Turing degree computability are examined. Functorial models computability is presented with algebraic trees realizing intuitionistic type...
Uniqueness of closure of the constraint algebra for quantum gravity
International Nuclear Information System (INIS)
Christodoulakis, T.; Korfiatis, E.
1989-08-01
We investigate the closure of the quantum algebra for the constraints of pure gravity considering a wide class of regularisation assumptions. We thus establish that the only regularisation assumption, within this class, that closes the algebra is the one introduced by one of the authors (T.C. with Jorge Zanelli) in earlier publications and that the closure is a result of both the regularisation and the introduction of a tensor distribution. (author). 15 refs
Uniqueness of closure of the constraint algebra for quantum gravity
International Nuclear Information System (INIS)
Christodoulakis, T.; Korfiatis, E.
1991-01-01
Considering a wide class of regularization assumptions, the closure of the quantum algebra is investigated for the constraints of pure gravity. It is thus established that the only regularization assumption, within this class, that closes the algebra is the one introduced by one of the authors (TC with J. Zanelli) in earlier publications and that the closure is a result of both the regularization and the introduction of a tensor distribution
The ABCDEFG of instantons and W-algebras
Keller, Christoph A.; Mekareeya, Noppadol; Song, Jaewon; Tachikawa, Yuji
2012-03-01
For arbitrary gauge groups, we check at the one-instanton level that the Nekrasov partition function of pure mathcal{N} = {2} super Yang-Mills is equal to the norm of a certain coherent state of the corresponding W-algebra. For non-simply-laced gauge groups, we confirm in particular that the coherent state is in the twisted sector of a simply-laced W-algebra.
The ABCDEFG of instantons and W-algebras
Keller, Christoph A.; Mekareeya, Noppadol; Song, Jaewon; Tachikawa, Yuji
2012-01-01
For arbitrary gauge groups, we check at the one-instanton level that the Nekrasov partition function of pure N = 2 super Yang-Mills is equal to the norm of a certain coherent state of the corresponding W-algebra. For non-simply-laced gauge groups, we confirm in particular that the coherent state is in the twisted sector of a simply-laced W-algebra.
International Nuclear Information System (INIS)
Tsarev, Andrei V
2009-01-01
Pure subrings of finite rank in the Z-adic completion of the ring of integers and in its homomorphic images are considered. Certain properties of these rings are studied (existence of an identity element, decomposability into a direct sum of essentially indecomposable ideals, condition for embeddability into a csp-ring, etc.). Additive groups of these rings and conditions under which these rings are subrings of algebraic number fields are described. Bibliography: 12 titles.
Contemporary developments in algebraic K-theory
International Nuclear Information System (INIS)
Karoubi, M.; Kuku, A.O.; Pedrini, C.
2003-01-01
The School and Conference on Algebraic K-theory which took place at ICTP July 8-26, 2002 was a follow-up to the earlier one in 1997, and like its predecessor, the 2002 meeting endeavoured to emphasise the multidisciplinary aspects of the subject. However, one special feature of the 2002 School and Conference is that the whole activity was dedicated to H. Bass, one of the founders of Algebraic K-theory, on the occasion of his seventieth birthday. The School during the first two weeks, July 8 to 19 was devoted to expository lectures meant to explore and highlight connections between K-theory and several other areas of mathematics - Algebraic Topology, Number theory, Algebraic Geometry, Representation theory, and Non-commutative Geometry. This volume, constituting the Proceedings of the School, is dedicated to H. Bass. The Proceedings of the Conference during the last week July 22 - 26, which will appear in Special issues of K-theory, is also dedicated to H. Bass. The opening contribution by M. Karoubi to this volume consists of a comprehensive survey of developments in K-theory in the last forty-five years, and covers a very broad spectrum of the subject, including Topological K-theory, Atiyah-Singer index theorem, K-theory of Banach algebras, Higher Algebraic K-theory, Cyclic Homology etc. J. Berrick's contribution on 'Algebraic K-theory and Algebraic Topology' treats the various topological constructions of Algebraic K-theory together with the underlying homotopy theory. Topics covered include the plus construction together with its various ramifications and applications, Topological Hochschild and Cyclic Homology as well as K-theory of the ring of integers. The contributions by M. Kolster titled 'K-theory and Arithmetics' includes such topics as values of zeta functions and relations to K-theory, K-theory of integers in number fields and associated conjectures, Etale cohomology, Iwasawa theory etc. A.O. Kuku's contributions on 'K-theory and Representation theory
Division algebras and extended super KdVs
International Nuclear Information System (INIS)
Toppan, F.
2001-05-01
The division algebras R, C, H, O are used to construct and analyze the N = 1, 2, 4, 8 supersymmetric extensions of the KdV hamiltonian equation. In particular a global N = 8 super-KdV system is introduced and shown to admit a Poisson bracket structure given by the 'Non-Associate N = 8 Superconformal Algebra'. (author)
Bochvar-McCarthy logic and process algebra
Bergstra, J.A.; Ponse, A.
1998-01-01
We propose a combination of Bochvar's strict three-valued logic, McCarthy's sequential three-valued logic, and process algebra via the conditional guard construct. This combination entails the introduction of a new constant meaningless in process algebra. We present an operational semantics
Elementary Algebra + Student-Written Web Illustrations = Math Mastery.
Veteto, Bette R.
This project focuses on the construction and use of a student-made elementary algebra tutorial World Wide Web page at the University of Memphis (Tennessee), how this helps students further explore the topics studied in elementary algebra, and how students can publish their work on the class Web page for use by other students. Practical,…
Limit algebras of differential forms in non-commutative geometry
Indian Academy of Sciences (India)
The holomorphic functional calculus closure of Connes' non- commutative de Rham algebra. ∗. D. (p. 549 of [C]) leads to a couple of operator algebras which are briefly discussed in this section. In §5, which contains the main contributions of the paper, quantized integrals are constructed on ∞A by using Dixmier trace ...
Campbell-Hausdorff Formula and Algebras with Operator
International Nuclear Information System (INIS)
Khudaverdyan, O.M.
1994-01-01
Some new classes of algebras are introduced and in these algebras Campbell-Hausdorff like formula is established. The application of these constructions to the problem of the connectivity of the Feynman graphs corresponding to the Green functions in Quantum Field Theory is described. 9 refs
On alphabetic presentations of Clifford algebras and their possible applications
Toppan, F.; Verbeek, P.W.
2009-01-01
In this paper, we address the problem of constructing a class of representations of Clifford algebras that can be named “alphabetic (re)presentations.” The Clifford algebra generators are expressed as m-letter words written with a three-character or a four-character alphabet. We formulate the
On algebraically special perturbations of black holes
International Nuclear Information System (INIS)
Chandrasekhar, S.
1984-01-01
Algebraically special perturbations of black holes excite gravitational waves that are either purely ingoing or purely outgoing. Solutions, appropriate to such perturbations of the Kerr, the Schwarzschild, and the Reissner-Nordstroem black-holes, are obtained in explicit forms by different methods. The different methods illustrate the remarkable inner relations among different facets of the mathematical theory. In the context of the Kerr black-hole they derive from the different ways in which the explicit value of the Starobinsky constant emerges, and in the context of the Schwarzschild and the Reissner-Nordstroem black-holes they derive from the potential barriers surrounding them belonging to a special class. (author)
(Quasi-)Poisson enveloping algebras
Yang, Yan-Hong; Yao, Yuan; Ye, Yu
2010-01-01
We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this dissertation,…
Learning Activity Package, Algebra.
Evans, Diane
A set of ten teacher-prepared Learning Activity Packages (LAPs) in beginning algebra and nine in intermediate algebra, these units cover sets, properties of operations, number systems, open expressions, solution sets of equations and inequalities in one and two variables, exponents, factoring and polynomials, relations and functions, radicals,…
Herriott, Scott R.; Dunbar, Steven R.
2009-01-01
The common understanding within the mathematics community is that the role of the college algebra course is to prepare students for calculus. Though exceptions are emerging, the curriculum of most college algebra courses and the content of most textbooks on the market both reflect that assumption. This article calls that assumption into question…
Hayden, Dunstan; Cuevas, Gilberto
The pre-algebra lexicon is a set of classroom exercises designed to teach the technical words and phrases of pre-algebra mathematics, and includes the terms most commonly found in related mathematics courses. The lexicon has three parts, each with its own introduction. The first introduces vocabulary items in three groups forming a learning…
International Nuclear Information System (INIS)
Calmet, J.
1982-01-01
A survey of applications based either on fundamental algorithms in computer algebra or on the use of a computer algebra system is presented. Recent work in biology, chemistry, physics, mathematics and computer science is discussed. In particular, applications in high energy physics (quantum electrodynamics), celestial mechanics and general relativity are reviewed. (Auth.)
Algebraic Description of Motion
Davidon, William C.
1974-01-01
An algebraic definition of time differentiation is presented and used to relate independent measurements of position and velocity. With this, students can grasp certain essential physical, geometric, and algebraic properties of motion and differentiation before undertaking the study of limits. (Author)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
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...
Forward error correction based on algebraic-geometric theory
A Alzubi, Jafar; M Chen, Thomas
2014-01-01
This book covers the design, construction, and implementation of algebraic-geometric codes from Hermitian curves. Matlab simulations of algebraic-geometric codes and Reed-Solomon codes compare their bit error rate using different modulation schemes over additive white Gaussian noise channel model. Simulation results of Algebraic-geometric codes bit error rate performance using quadrature amplitude modulation (16QAM and 64QAM) are presented for the first time and shown to outperform Reed-Solomon codes at various code rates and channel models. The book proposes algebraic-geometric block turbo codes. It also presents simulation results that show an improved bit error rate performance at the cost of high system complexity due to using algebraic-geometric codes and Chase-Pyndiah’s algorithm simultaneously. The book proposes algebraic-geometric irregular block turbo codes (AG-IBTC) to reduce system complexity. Simulation results for AG-IBTCs are presented for the first time.
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...
International Nuclear Information System (INIS)
Smirnov, Yu.F.; Tolstoi, V.N.; Kharitonov, Yu.I.
1993-01-01
The tree technique for the quantum algebra su q (2) developed in an earlier study is used to construct the q analog of the algebra of irreducible tensor operators. The adjoint action of the algebra su q (2) on irreducible tensor operators is discussed, and the adjoint R matrix is introduced. A set of expressions is obtained for the matrix elements of various irreducible tensor operators and combinations of them. As an application, the recursion relations for the Clebsch-Gordan and Racah coefficients of the algebra su q (2) are derived. 16 refs
Shekhah, Osama; Swaidan, Raja; Belmabkhout, Youssef; Du Plessis, Marike; Jacobs, Tia; Barbour, Leonard J.; Pinnau, Ingo; Eddaoudi, Mohamed
2014-01-01
The liquid-phase epitaxy (LPE) method was effectively implemented to deliberately grow/construct ultrathin (0.5-1 μm) continuous and defect-free ZIF-8 membranes. Permeation properties of different gas pair systems (O 2-N2, H2-CO2, CO2-CH 4, C3H6-C3H8, CH 4-n-C4H10) were studied using the time lag technique. This journal is © The Royal Society of Chemistry.
Elements of mathematics algebra
Bourbaki, Nicolas
2003-01-01
This is a softcover reprint of the English translation of 1990 of the revised and expanded version of Bourbaki's, Algèbre, Chapters 4 to 7 (1981). This completes Algebra, 1 to 3, by establishing the theories of commutative fields and modules over a principal ideal domain. Chapter 4 deals with polynomials, rational fractions and power series. A section on symmetric tensors and polynomial mappings between modules, and a final one on symmetric functions, have been added. Chapter 5 was entirely rewritten. After the basic theory of extensions (prime fields, algebraic, algebraically closed, radical extension), separable algebraic extensions are investigated, giving way to a section on Galois theory. Galois theory is in turn applied to finite fields and abelian extensions. The chapter then proceeds to the study of general non-algebraic extensions which cannot usually be found in textbooks: p-bases, transcendental extensions, separability criterions, regular extensions. Chapter 6 treats ordered groups and fields and...
Reinventing Fractions and Division as They Are Used in Algebra: The Power of Preformal Productions
Peck, Frederick; Matassa, Michael
2016-01-01
In this paper, we explore algebra students' mathematical realities around fractions and division, and the ways in which students reinvented mathematical productions involving fractions and division. We find that algebra students' initial realities do not include the fraction-as-quotient sub-construct. This can be problematic because in algebra,…
Non-critical pure spinor superstrings
International Nuclear Information System (INIS)
Adam, Ido; Grassi, Pietro Antonio; Mazzucato, Luca; Oz, Yaron; Yankielowicz, Shimon
2007-01-01
We construct non-critical pure spinor superstrings in two, four and six dimensions. We find explicitly the map between the RNS variables and the pure spinor ones in the linear dilaton background. The RNS variables map onto a patch of the pure spinor space and the holomorphic top form on the pure spinor space is an essential ingredient of the mapping. A basic feature of the map is the requirement of doubling the superspace, which we analyze in detail. We study the structure of the non-critical pure spinor space, which is different from the ten-dimensional one, and its quantum anomalies. We compute the pure spinor lowest lying BRST cohomology and find an agreement with the RNS spectra. The analysis is generalized to curved backgrounds and we construct as an example the non-critical pure spinor type IIA superstring on AdS 4 with RR 4-form flux
Cubic forms algebra, geometry, arithmetic
Manin, Yu I
1986-01-01
Since this book was first published in English, there has been important progress in a number of related topics. The class of algebraic varieties close to the rational ones has crystallized as a natural domain for the methods developed and expounded in this volume. For this revised edition, the original text has been left intact (except for a few corrections) and has been brought up to date by the addition of an Appendix and recent references.The Appendix sketches some of the most essential new results, constructions and ideas, including the solutions of the Luroth and Zariski problems, the th
Krichever correspondence for algebraic varieties
International Nuclear Information System (INIS)
Osipov, D V
2001-01-01
We construct new acyclic resolutions of quasicoherent sheaves. These resolutions are connected with multidimensional local fields. The resolutions obtained are applied to construct a generalization of the Krichever map to algebraic varieties of any dimension. This map canonically produces two k-subspaces B contained in k((z 1 ))...((z n )) and W contained in k((z 1 ))...((z n )) o+r from the following data: an arbitrary algebraic n-dimensional Cohen-Macaulay projective integral scheme X over a field k, a flag of closed integral subschemes X=Y 0 contains Y 1 contains ... contains Y n such that Y i is an ample Cartier divisor on Y i-1 and Y n is a smooth point on all Y i , formal local parameters of this flag at the point Y n , a rank r vector bundle F on X, and a trivialization of F in the formal neighbourhood of the point Y n where the n-dimensional local field B contained in k((z 1 ))...((z n )) is associated with the flag Y 0 contains Y 1 contains ... contains Y n . In addition, the map constructed is injective, that is, one can uniquely reconstruct all the original geometric data. Moreover, given the subspace B, we can explicitly write down a complex which calculates the cohomology of the sheaf O X on X and, given the subspace W, we can explicitly write down a complex which calculates the cohomology of F on X
Cluster algebras bases on vertex operator algebras
Czech Academy of Sciences Publication Activity Database
Zuevsky, Alexander
2016-01-01
Roč. 30, 28-29 (2016), č. článku 1640030. ISSN 0217-9792 Institutional support: RVO:67985840 Keywords : cluster alegbras * vertex operator algebras * Riemann surfaces Subject RIV: BA - General Mathematics Impact factor: 0.736, year: 2016 http://www.worldscientific.com/doi/abs/10.1142/S0217979216400300
Riemann surfaces, Clifford algebras and infinite dimensional groups
International Nuclear Information System (INIS)
Carey, A.L.; Eastwood, M.G.; Hannabuss, K.C.
1990-01-01
We introduce of class of Riemann surfaces which possess a fixed point free involution and line bundles over these surfaces with which we can associate an infinite dimensional Clifford algebra. Acting by automorphisms of this algebra is a 'gauge' group of meromorphic functions on the Riemann surface. There is a natural Fock representation of the Clifford algebra and an associated projective representation of this group of meromorphic functions in close analogy with the construction of the basic representation of Kac-Moody algebras via a Fock representation of the Fermion algebra. In the genus one case we find a form of vertex operator construction which allows us to prove a version of the Boson-Fermion correspondence. These results are motivated by the analysis of soliton solutions of the Landau-Lifshitz equation and are rather distinct from recent developments in quantum field theory on Riemann surfaces. (orig.)
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
C*-algebras over topological spaces
DEFF Research Database (Denmark)
Meyer, Ralf; Nest, Ryszard
2012-01-01
We define the filtrated K-theory of a C*-algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated K-theory. For finite spaces with a totally ordered lattice of open subsets, this spectral...
Hamiltonian structure of linearly extended Virasoro algebra
International Nuclear Information System (INIS)
Arakelyan, T.A.; Savvidi, G.K.
1991-01-01
The Hamiltonian structure of linearly extended Virasoro algebra which admits free bosonic field representation is described. An example of a non-trivial extension is found. The hierarchy of integrable non-linear equations corresponding to this Hamiltonian structure is constructed. This hierarchy admits the Lax representation by matrix Lax operator of second order
Algebraic solution of an anisotropic nonquadratic potential
International Nuclear Information System (INIS)
Boschi Filho, H.; Vaidya, A.N.
1990-06-01
We show that an anisotropic nonquadratic potential, for which a path integral treatment had been recently discussed in the literature, possesses the (SO(2,1)xSO(2,1))ΛSO(2,1) dynamical symmetry and constructs its Green function algebraically. A particular case which generates new eigenvalues and eigenfunctions is also discussed. (author). 11 refs
Connecting Functions in Geometry and Algebra
Steketee, Scott; Scher, Daniel
2016-01-01
One goal of a mathematics education is that students make significant connections among different branches of mathematics. Connections--such as those between arithmetic and algebra, between two-dimensional and three-dimensional geometry, between compass-and-straight-edge constructions and transformations, and between calculus and analytic…
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.)
Some new classes of division algebras and potential applications to space-time block coding
Steele, Andrew
2014-01-01
In this thesis we study some new classes of nonassociative division algebras. First we introduce a generalisation of both associative cyclic algebras and of Waterhouse's nonassociative quaternions. An important aspect of these algebras is the simplicity of their construction, which is a modification of the classical definition of associative cyclic algebras. By taking the parameter used in the classical definition from a larger field, we lose the property of associativity but gain many new ex...
Springer, T A
1998-01-01
"[The first] ten chapters...are an efficient, accessible, and self-contained introduction to affine algebraic groups over an algebraically closed field. The author includes exercises and the book is certainly usable by graduate students as a text or for self-study...the author [has a] student-friendly style… [The following] seven chapters... would also be a good introduction to rationality issues for algebraic groups. A number of results from the literature…appear for the first time in a text." –Mathematical Reviews (Review of the Second Edition) "This book is a completely new version of the first edition. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field. Reading that book, many people entered the research field of linear algebraic groups. The present book has a wider scope. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Again, the author keeps the treatment of prerequisites self-contained. The material of t...
q-deformed conformal and Poincare algebras on quantum 4-spinors
International Nuclear Information System (INIS)
Kobayashi, Tatsuo; Uematsu, Tsuneo
1993-01-01
We investigate quantum deformation of conformal algebras by constructing the quantum space for sl q (4). The differential calculus on the quantum space and the action of the quantum generators are studied. We derive deformed su(2, 2) algebra from the deformed sl(4) algebra using the quantum 4-spinor and its conjugate spinor. The quantum 6-vector in so q (4, 2) is constructed as a tensor product of two sets of 4-spinors. We obtain the q-deformed conformal algebra with the suitable assignment of the generators which satisfy the reality condition. The deformed Poincare algebra is derived through a contraction procedure. (orig.)
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.
An algebraic approach to the non-symmetric Macdonald polynomial
International Nuclear Information System (INIS)
Nishino, Akinori; Ujino, Hideaki; Wadati, Miki
1999-01-01
In terms of the raising and lowering operators, we algebraically construct the non-symmetric Macdonald polynomials which are simultaneous eigenfunctions of the commuting Cherednik operators. We also calculate Cherednik's scalar product of them
Commuting quantum traces: the case of reflection algebras
Energy Technology Data Exchange (ETDEWEB)
Avan, Jean [Laboratory of Theoretical Physics and Modelization, University of Cergy, 5 mail Gay-Lussac, Neuville-sur-Oise, F-95031, Cergy-Pontoise Cedex (France); Doikou, Anastasia [Theoretical Physics Laboratory of Annecy-Le-Vieux, LAPTH, BP 110, Annecy-Le-Vieux, F-74941 (France)
2004-02-06
We formulate a systematic construction of commuting quantum traces for reflection algebras. This is achieved by introducing two dual sets of generalized reflection equations with associated consistent fusion procedures. Products of their respective solutions yield commuting quantum traces.
Jordan's algebra of a facially homogeneous autopolar cone
International Nuclear Information System (INIS)
Bellissard, Jean; Iochum, Bruno
1979-01-01
It is shown that a Jordan-Banach algebra with predual may be canonically associated with a facially homogeneous autopolar cone. This construction generalizes the case where a trace vector exists in the cone [fr
Quantitative Algebraic Reasoning
DEFF Research Database (Denmark)
Mardare, Radu Iulian; Panangaden, Prakash; Plotkin, Gordon
2016-01-01
We develop a quantitative analogue of equational reasoning which we call quantitative algebra. We deﬁne an equality relation indexed by rationals: a =ε b which we think of as saying that “a is approximately equal to b up to an error of ε”. We have 4 interesting examples where we have a quantitative...... equational theory whose free algebras correspond to well known structures. In each case we have ﬁnitary and continuous versions. The four cases are: Hausdorﬀ metrics from quantitive semilattices; pWasserstein metrics (hence also the Kantorovich metric) from barycentric algebras and also from pointed...
Chatterjee, D
2007-01-01
About the Book: This book provides exposition of the subject both in its general and algebraic aspects. It deals with the notions of topological spaces, compactness, connectedness, completeness including metrizability and compactification, algebraic aspects of topological spaces through homotopy groups and homology groups. It begins with the basic notions of topological spaces but soon going beyond them reaches the domain of algebra through the notions of homotopy, homology and cohomology. How these approaches work in harmony is the subject matter of this book. The book finally arrives at the
Adaptive algebraic reconstruction technique
International Nuclear Information System (INIS)
Lu Wenkai; Yin Fangfang
2004-01-01
Algebraic reconstruction techniques (ART) are iterative procedures for reconstructing objects from their projections. It is proven that ART can be computationally efficient by carefully arranging the order in which the collected data are accessed during the reconstruction procedure and adaptively adjusting the relaxation parameters. In this paper, an adaptive algebraic reconstruction technique (AART), which adopts the same projection access scheme in multilevel scheme algebraic reconstruction technique (MLS-ART), is proposed. By introducing adaptive adjustment of the relaxation parameters during the reconstruction procedure, one-iteration AART can produce reconstructions with better quality, in comparison with one-iteration MLS-ART. Furthermore, AART outperforms MLS-ART with improved computational efficiency
Cohen, A.M.; Liu, S.
2011-01-01
For each n>0, we define an algebra having many properties that one might expect to hold for a Brauer algebra of type Bn. It is defined by means of a presentation by generators and relations. We show that this algebra is a subalgebra of the Brauer algebra of type Dn+1 and point out a cellular
DEFF Research Database (Denmark)
Starrfelt, Randi; Ólafsdóttir, Rannveig Rós; Arendt, Ida-Marie
2013-01-01
that pure alexia was an easy target for rehabilitation efforts. We review the literature on rehabilitation of pure alexia from 1990 to the present, and find that patients differ widely on several dimensions like alexia severity, and associated deficits. Many patients reported to have pure alexia......-designed and controlled studies of rehabilitation of pure alexia....
Covariant representation theory of the Poincaré algebra and some of its extensions
Boels, Rutger
2010-01-01
There has been substantial calculational progress in the last few years for gauge theory amplitudes which involve massless four dimensional particles. One of the central ingredients in this has been the ability to keep precise track of the Poincaré algebra quantum numbers of the particles involved. Technically, this is most easily done using the well-known four dimensional spinor helicity method. In this article a natural generalization to all dimensions higher than four is obtained based on a covariant version of the representation theory of the Poincaré algebra. Covariant expressions for all possible polarization states, both bosonic and fermionic, are constructed. For the fermionic states the analysis leads directly to pure spinors. The natural extension to the representation theory of the on-shell supersymmetry algebra results in an elementary derivation of the supersymmetry Ward identities for scattering amplitudes with massless or massive legs in any integer dimension from four onwards. As a proof-of-concept application a higher dimensional analog of the vanishing helicity-equal amplitudes in four dimensions is presented in (super) Yang-Mills theory, Einstein (super-)gravity and superstring theory in a flat background.
Workshop on Non-Associative & Non-Commutative Algebra and Operator Theory
Molina, Mercedes
2016-01-01
Presenting the collaborations of over thirty international experts in the latest developments in pure and applied mathematics, this volume serves as an anthology of research with a common basis in algebra, functional analysis and their applications. Special attention is devoted to non-commutative algebras, non-associative algebras, operator theory and ring and module theory. These themes are relevant in research and development in coding theory, cryptography and quantum mechanics. The topics in this volume were presented at the Workshop on Non-Associative & Non-Commutative Algebra and Operator Theory, held May 23—25, 2014 at Cheikh Anta Diop University in Dakar, Senegal in honor of Professor Amin Kaidi. The workshop was hosted by the university's Laboratory of Algebra, Cryptology, Algebraic Geometry and Applications, in cooperation with the University of Almería and the University of Málaga. Dr. Kaidi's work focuses on non-associative rings and algebras, operator theory and functional analysis, and he...
Filtrated K-theory for real rank zero C*-algebras
DEFF Research Database (Denmark)
Arklint, Sara Esther; Restorff, Gunnar; Ruiz, Efren
2012-01-01
The smallest primitive ideal spaces for which there exist counterexamples to the classification of non-simple, purely infinite, nuclear, separable C*-algebras using filtrated K-theory, are four-point spaces. In this article, we therefore restrict to real rank zero C*-algebras with four-point prim...
Primary fields in a unitary representation of Virasoro algebras
International Nuclear Information System (INIS)
Sasaki, R.; Yamanaka, I.
1985-08-01
A unitary representation of Virasoro algebras with the central charge c = 1 - 6/(N + 1)(N + 2) is constructed explicitly in terms of a colored (two color) coset space (the complex projective space CP sup(N-1)) quark model. By utilizing the explicit forms of the Virasoro generators Lsub(m), we derive a general method of constructing the primary fields (fields with well-defined conformal transformation properties) of the above Virasoro algebras. (author)
Differential equations from the algebraic standpoint
Ritt, Joseph Fels
1932-01-01
This book can be viewed as a first attempt to systematically develop an algebraic theory of nonlinear differential equations, both ordinary and partial. The main goal of the author was to construct a theory of elimination, which "will reduce the existence problem for a finite or infinite system of algebraic differential equations to the application of the implicit function theorem taken with Cauchy's theorem in the ordinary case and Riquier's in the partial." In his 1934 review of the book, J. M. Thomas called it "concise, readable, original, precise, and stimulating", and his words still rema
Classical impurities associated to high rank algebras
Energy Technology Data Exchange (ETDEWEB)
Doikou, Anastasia, E-mail: A.Doikou@hw.ac.uk [Department of Mathematics, Heriot–Watt University, EH14 4AS, Edinburgh (United Kingdom); Department of Computer Engineering and Informatics, University of Patras, Patras GR-26500 (Greece)
2014-07-15
Classical integrable impurities associated with high rank (gl{sub N}) algebras are investigated. A particular prototype, i.e. the vector non-linear Schrödinger (NLS) model, is chosen as an example. A systematic construction of local integrals of motion as well as the time components of the corresponding Lax pairs is presented based on the underlying classical algebra. Suitable gluing conditions compatible with integrability are also extracted. The defect contribution is also examined in the case where non-trivial integrable conditions are implemented. It turns out that the integrable boundaries may drastically alter the bulk behavior, and in particular the defect contribution.
Hilbert schemes of points and Heisenberg algebras
International Nuclear Information System (INIS)
Ellingsrud, G.; Goettsche, L.
2000-01-01
Let X [n] be the Hilbert scheme of n points on a smooth projective surface X over the complex numbers. In these lectures we describe the action of the Heisenberg algebra on the direct sum of the cohomologies of all the X [n] , which has been constructed by Nakajima. In the second half of the lectures we study the relation of the Heisenberg algebra action and the ring structures of the cohomologies of the X [n] , following recent work of Lehn. In particular we study the Chern and Segre classes of tautological vector bundles on the Hilbert schemes X [n] . (author)
Fock representations of exchange algebras with involution
International Nuclear Information System (INIS)
Liguori, A.; Mintchev, M.; Rossi, M.
1997-01-01
An associative algebra scr(A) R with exchange properties generalizing the canonical (anti)commutation relations is considered. We introduce a family of involutions in scr(A) R and construct the relative Fock representations, examining the positivity of the metric. As an application of the general results, we rigorously prove unitarity of the scattering operator of integrable models in 1+1 space-time dimensions. In this context the possibility of adopting various involutions in the Zamolodchikov endash Faddeev algebra is also explored. copyright 1997 American Institute of Physics
Classical impurities associated to high rank algebras
International Nuclear Information System (INIS)
Doikou, Anastasia
2014-01-01
Classical integrable impurities associated with high rank (gl N ) algebras are investigated. A particular prototype, i.e. the vector non-linear Schrödinger (NLS) model, is chosen as an example. A systematic construction of local integrals of motion as well as the time components of the corresponding Lax pairs is presented based on the underlying classical algebra. Suitable gluing conditions compatible with integrability are also extracted. The defect contribution is also examined in the case where non-trivial integrable conditions are implemented. It turns out that the integrable boundaries may drastically alter the bulk behavior, and in particular the defect contribution
Pseudo-Riemannian Novikov algebras
Energy Technology Data Exchange (ETDEWEB)
Chen Zhiqi; Zhu Fuhai [School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071 (China)], E-mail: chenzhiqi@nankai.edu.cn, E-mail: zhufuhai@nankai.edu.cn
2008-08-08
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. Pseudo-Riemannian Novikov algebras denote Novikov algebras with non-degenerate invariant symmetric bilinear forms. In this paper, we find that there is a remarkable geometry on pseudo-Riemannian Novikov algebras, and give a special class of pseudo-Riemannian Novikov algebras.
International Nuclear Information System (INIS)
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
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.
Indian Academy of Sciences (India)
algebraic geometry but also in related fields like number theory. ... every vector bundle on the affine space is trivial. (equivalently ... les on a compact Riemann surface to unitary rep- ... tial geometry and topology and was generalised in.
Axler, Sheldon
2015-01-01
This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions. No prerequisites are assumed other than the ...
Algebraic Semantics for Narrative
Kahn, E.
1974-01-01
This paper uses discussion of Edmund Spenser's "The Faerie Queene" to present a theoretical framework for explaining the semantics of narrative discourse. The algebraic theory of finite automata is used. (CK)
Leamer, Micah J.
2004-01-01
Let K be a field and Q a finite directed multi-graph. In this paper I classify all path algebras KQ and admissible orders with the property that all of their finitely generated ideals have finite Groebner bases. MS
Pure quasifree states of the Dirac field from the fermionic projector
International Nuclear Information System (INIS)
Fewster, Christopher J; Lang, Benjamin
2015-01-01
We consider the quantized free massive Dirac field on oriented and globally hyperbolic ultrastatic slab spacetimes with compact spatial section and demonstrate how a gauge invariant, pure and quasifree state on the C*-completion of the self-dual CAR-algebra can be extracted from the fermionic projector construction of Finster and Reintjes. This state is analogous to the ‘SJ-state’ of the free scalar field recently discussed in the literature. We prove that this state generically fails to be Hadamard. However, we also show how a modified version of the construction, inspired by the work of Brum and Fredenhagen, yields states that are Hadamard. We also relate the Hadamard condition to the finiteness of fluctuations of Wick polynomials. (paper)
Institute of Scientific and Technical Information of China (English)
Antonio AIZPURU; Antonio GUTI(E)RREZ-D(A)VILA
2004-01-01
In this paper we will study some families and subalgebras ( ) of ( )(N) that let us characterize the unconditional convergence of series through the weak convergence of subseries ∑i∈A xi, A ∈ ( ).As a consequence, we obtain a new version of the Orlicz-Pettis theorem, for Banach spaces. We also study some relationships between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces.
Intermediate algebra & analytic geometry
Gondin, William R
1967-01-01
Intermediate Algebra & Analytic Geometry Made Simple focuses on the principles, processes, calculations, and methodologies involved in intermediate algebra and analytic geometry. The publication first offers information on linear equations in two unknowns and variables, functions, and graphs. Discussions focus on graphic interpretations, explicit and implicit functions, first quadrant graphs, variables and functions, determinate and indeterminate systems, independent and dependent equations, and defective and redundant systems. The text then examines quadratic equations in one variable, system
Introduction to abstract algebra
Nicholson, W Keith
2012-01-01
Praise for the Third Edition ". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."-Zentralblatt MATH The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately be
Certain algebraic structures and their applications to physics
International Nuclear Information System (INIS)
Salingaros, N.A.
1978-01-01
The aim of this thesis is to understand internal and external symmetries in Physics as arising from the same algebra by different processes, while the algebra itself arises out of the geometry of space-time. The result obtained is the Associative Generalized Algebra of Tensor Types. This algebra is constructed from the differential forms of spacetime, and is an algebra in the mathematical sense, describing all tensor types together. It is associative, and therefore very easy to use. A calculational formalism is developed that simplifies algebraic manipulations. The construction allows a classification of algebras that appear useful in Physics. The geometry excludes self-dual Minkowski bivector fields, but allows self-dual Euclidean bivector fields, a result, with important consequences in the theory of solutions of Yang-Mills gauge fields are demonstrated. There is only one bivector field, and every other bivector field, such as the electromagnetic field, is isomorphic to it. An exhaustive classification of the transformations of all fields in space-time yields the result that the only transformations of the electromagnetic field are the Lorentz transformations and the duality rotation. A fundamental asymmetry between the electric and magnetic fields are demonstrated. The derivative in the algebra is associative, and combines the Cartan exterior derivative with the coderivative of Hodge. The simplest derivative equations satisfied by a field in flat space-time are precisely the Maxwell equations
Division algebras with integral elements
International Nuclear Information System (INIS)
Koca, M.; Ozdes, N.
1988-06-01
Pairing two elements of a given division algebra furnished with a multiplication rule leads to an algebra of higher dimension restricted by 8. This fact is used to obtain the roots of SO(4) and SP(2) from the roots ±1 of SU(2) and the weights ±1/2 of its spinor representation. The root lattice of SO(8) described by 24 integral quaternions are obtained by pairing two sets of roots of SP(2). The root system of F 4 is constructed in terms of 24 integral and 24 ''half-integral'' quaternions. The root lattice of E 8 expressed as 240 integral octonions are obtained by pairing two sets of roots of F 4 . 24 integral quaternions of SO(8) forming a discrete subgroup of SU(2) is shown to be the automorphism group of the root lattices of SO(8), F 4 and E 8 . The roots of maximal subgroups SO(16), E 7 XSU(2), E 6 XSU(3), SU(9) and SU(5)XSU(5) of E 8 are identified with a simple method. Subsets of the discrete subgroup of SU(2) leaving maximal subgroups of E 8 are obtained. Constructions of E 8 root lattice with integral octonions in 7 distinct ways are made. Magic square of integral lattices of Goddard, Nahm, Olive, Ruegg and Schwimmer are derived. Possible physical applications are suggested. (author). 16 refs, 6 figs, 5 tabs
The Boolean algebra of Galois algebras
Directory of Open Access Journals (Sweden)
Lianyong Xue
2003-02-01
Full Text Available Let B be a Galois algebra with Galois group G, Jg={bÃ¢ÂˆÂˆB|bx=g(xbÃ¢Â€Â‰for allÃ¢Â€Â‰xÃ¢ÂˆÂˆB} for each gÃ¢ÂˆÂˆG, and BJg=Beg for a central idempotent eg, Ba the Boolean algebra generated by {0,eg|gÃ¢ÂˆÂˆG}, e a nonzero element in Ba, and He={gÃ¢ÂˆÂˆG|eeg=e}. Then, a monomial e is characterized, and the Galois extension Be, generated by e with Galois group He, is investigated.
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
Current algebra, statistical mechanics and quantum models
Vilela Mendes, R.
2017-11-01
Results obtained in the past for free boson systems at zero and nonzero temperatures are revisited to clarify the physical meaning of current algebra reducible functionals which are associated to systems with density fluctuations, leading to observable effects on phase transitions. To use current algebra as a tool for the formulation of quantum statistical mechanics amounts to the construction of unitary representations of diffeomorphism groups. Two mathematical equivalent procedures exist for this purpose. One searches for quasi-invariant measures on configuration spaces, the other for a cyclic vector in Hilbert space. Here, one argues that the second approach is closer to the physical intuition when modelling complex systems. An example of application of the current algebra methodology to the pairing phenomenon in two-dimensional fermion systems is discussed.
International Nuclear Information System (INIS)
Freidel, L.; Maillet, J.M.
1992-09-01
Using a geometrical approach to the quantum Yang-Baxter equation, the quantum algebra U h (sl 2 ) and its universal quantum R-matrix are explicitly constructed as functionals of the associated classical r-matrix. In this framework, the quantum algebra U h (sl 2 ) is naturally imbedded in the universal enveloping algebra of the sl 2 current algebra. (author) 13 refs
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.)
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...
Quantum complexity of graph and algebraic problems
International Nuclear Information System (INIS)
Doern, Sebastian
2008-01-01
This thesis is organized as follows: In Chapter 2 we give some basic notations, definitions and facts from linear algebra, graph theory, group theory and quantum computation. In Chapter 3 we describe three important methods for the construction of quantum algorithms. We present the quantum search algorithm by Grover, the quantum amplitude amplification and the quantum walk search technique by Magniez et al. These three tools are the basis for the development of our new quantum algorithms for graph and algebra problems. In Chapter 4 we present two tools for proving quantum query lower bounds. We present the quantum adversary method by Ambainis and the polynomial method introduced by Beals et al. The quantum adversary tool is very useful to prove good lower bounds for many graph and algebra problems. The part of the thesis containing the original results is organized in two parts. In the first part we consider the graph problems. In Chapter 5 we give a short summary of known quantum graph algorithms. In Chapter 6 to 8 we study the complexity of our new algorithms for matching problems, graph traversal and independent set problems on quantum computers. In the second part of our thesis we present new quantum algorithms for algebraic problems. In Chapter 9 to 10 we consider group testing problems and prove quantum complexity bounds for important problems from linear algebra. (orig.)
Quantum complexity of graph and algebraic problems
Energy Technology Data Exchange (ETDEWEB)
Doern, Sebastian
2008-02-04
This thesis is organized as follows: In Chapter 2 we give some basic notations, definitions and facts from linear algebra, graph theory, group theory and quantum computation. In Chapter 3 we describe three important methods for the construction of quantum algorithms. We present the quantum search algorithm by Grover, the quantum amplitude amplification and the quantum walk search technique by Magniez et al. These three tools are the basis for the development of our new quantum algorithms for graph and algebra problems. In Chapter 4 we present two tools for proving quantum query lower bounds. We present the quantum adversary method by Ambainis and the polynomial method introduced by Beals et al. The quantum adversary tool is very useful to prove good lower bounds for many graph and algebra problems. The part of the thesis containing the original results is organized in two parts. In the first part we consider the graph problems. In Chapter 5 we give a short summary of known quantum graph algorithms. In Chapter 6 to 8 we study the complexity of our new algorithms for matching problems, graph traversal and independent set problems on quantum computers. In the second part of our thesis we present new quantum algorithms for algebraic problems. In Chapter 9 to 10 we consider group testing problems and prove quantum complexity bounds for important problems from linear algebra. (orig.)
Algebraic Modeling of Topological and Computational Structures and Applications
Theodorou, Doros; Stefaneas, Petros; Kauffman, Louis
2017-01-01
This interdisciplinary book covers a wide range of subjects, from pure mathematics (knots, braids, homotopy theory, number theory) to more applied mathematics (cryptography, algebraic specification of algorithms, dynamical systems) and concrete applications (modeling of polymers and ionic liquids, video, music and medical imaging). The main mathematical focus throughout the book is on algebraic modeling with particular emphasis on braid groups. The research methods include algebraic modeling using topological structures, such as knots, 3-manifolds, classical homotopy groups, and braid groups. The applications address the simulation of polymer chains and ionic liquids, as well as the modeling of natural phenomena via topological surgery. The treatment of computational structures, including finite fields and cryptography, focuses on the development of novel techniques. These techniques can be applied to the design of algebraic specifications for systems modeling and verification. This book is the outcome of a w...
Special set linear algebra and special set fuzzy linear algebra
Kandasamy, W. B. Vasantha; Smarandache, Florentin; Ilanthenral, K.
2009-01-01
The authors in this book introduce the notion of special set linear algebra and special set fuzzy Linear algebra, which is an extension of the notion set linear algebra and set fuzzy linear algebra. These concepts are best suited in the application of multi expert models and cryptology. This book has five chapters. In chapter one the basic concepts about set linear algebra is given in order to make this book a self contained one. The notion of special set linear algebra and their fuzzy analog...
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
Hecke algebras with unequal parameters
Lusztig, G
2003-01-01
Hecke algebras arise in representation theory as endomorphism algebras of induced representations. One of the most important classes of Hecke algebras is related to representations of reductive algebraic groups over p-adic or finite fields. In 1979, in the simplest (equal parameter) case of such Hecke algebras, Kazhdan and Lusztig discovered a particular basis (the KL-basis) in a Hecke algebra, which is very important in studying relations between representation theory and geometry of the corresponding flag varieties. It turned out that the elements of the KL-basis also possess very interesting combinatorial properties. In the present book, the author extends the theory of the KL-basis to a more general class of Hecke algebras, the so-called algebras with unequal parameters. In particular, he formulates conjectures describing the properties of Hecke algebras with unequal parameters and presents examples verifying these conjectures in particular cases. Written in the author's precise style, the book gives rese...
A C*-algebra formulation of the quantization of the electromagnetic field
International Nuclear Information System (INIS)
Carey, A.L.; Gaffney, J.M.; Hurst, C.A.
1977-01-01
A presentation of the Fermi, Gupta--Bleuler, and radiation gauge methods for quantizing the free electromagnetic field is given in the Weyl algebra formalism for quantum field theory first introduced by Segal. The abstract Weyl algebra of the vector potential is defined using the formalism of Manuceau. Then the Fermi and Gupta--Bleuler methods are given as schemes for constructing representations of the algebra. The algebra of the physical photons is shown to be a factor algebra of a certain subalgebra of the original algebra of the vector potential. In this formalism, the application of the supplementary condition in the Fermi method, and the supplementary condition and indefinite metric in the Gupta--Bleuler method, can be interpreted as the means by which a representation of this factor algebra is obtained. The Weyl algebra of the physical photons is the Weyl algebra associated with the radiation gauge method. It is also shown that in the Fock representation of the Weyl algebra given by the Fermi method, automorphisms of the algebra corresponding to Lorentz transformations cannot always be implemented by unitary transformations. This leads us to construct a new representation of the Weyl algebra which provides a covariant representation for the vector potential
Noncommutative Cantor–Bendixson derivatives and scattered C⁎-algebras
Czech Academy of Sciences Publication Activity Database
Ghasemi, Saeed; Koszmider, P.
2018-01-01
Roč. 240, 15 May (2018), s. 183-209 ISSN 0166-8641 Institutional support: RVO:67985840 Keywords : C*-algebras * Cantor-Bendixson derivative * scattered locally compact spaces Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 0.377, year: 2016 https://www.sciencedirect.com/science/ article /pii/S0166864118301688?via%3Dihub
Noncommutative Cantor–Bendixson derivatives and scattered C⁎-algebras
Czech Academy of Sciences Publication Activity Database
Ghasemi, Saeed; Koszmider, P.
2018-01-01
Roč. 240, 15 May (2018), s. 183-209 ISSN 0166-8641 Institutional support: RVO:67985840 Keywords : C*-algebras * Cantor-Bendixson derivative * scattered locally compact spaces Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 0.377, year: 2016 https://www.sciencedirect.com/science/article/pii/S0166864118301688?via%3Dihub
On W1+∞ 3-algebra and integrable system
Directory of Open Access Journals (Sweden)
Min-Ru Chen
2015-02-01
Full Text Available We construct the W1+∞ 3-algebra and investigate its connection with the integrable systems. Since the W1+∞ 3-algebra with a fixed generator W00 in the operator Nambu 3-bracket recovers the W1+∞ algebra, it is intrinsically related to the KP hierarchy. For the general case of the W1+∞ 3-algebra, we directly derive the KP and KdV equations from the Nambu–Poisson evolution equation with the different Hamiltonian pairs of the KP hierarchy. Due to the Nambu–Poisson evolution equation involves two Hamiltonians, the deep relationship between the Hamiltonian pairs of KP hierarchy is revealed. Furthermore we give a realization of the W1+∞ 3-algebra in terms of a complex bosonic field. Based on the Nambu 3-brackets of the complex bosonic field, we derive the (generalized nonlinear Schrödinger equation and give an application in optical soliton.
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.)
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
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
An introduction to central simple algebras and their applications to wireless communication
Berhuy, Gre�gory
2013-01-01
Central simple algebras arise naturally in many areas of mathematics. They are closely connected with ring theory, but are also important in representation theory, algebraic geometry and number theory. Recently, surprising applications of the theory of central simple algebras have arisen in the context of coding for wireless communication. The exposition in the book takes advantage of this serendipity, presenting an introduction to the theory of central simple algebras intertwined with its applications to coding theory. Many results or constructions from the standard theory are presented in classical form, but with a focus on explicit techniques and examples, often from coding theory. Topics covered include quaternion algebras, splitting fields, the Skolem-Noether Theorem, the Brauer group, crossed products, cyclic algebras and algebras with a unitary involution. Code constructions give the opportunity for many examples and explicit computations. This book provides an introduction to the theory of central alg...
Coherent states for a polynomial su(1, 1) algebra and a conditionally solvable system
International Nuclear Information System (INIS)
Sadiq, Muhammad; Inomata, Akira; Junker, Georg
2009-01-01
In a previous paper (2007 J. Phys. A: Math. Theor. 40 11105), we constructed a class of coherent states for a polynomially deformed su(2) algebra. In this paper, we first prepare the discrete representations of the nonlinearly deformed su(1, 1) algebra. Then we extend the previous procedure to construct a discrete class of coherent states for a polynomial su(1, 1) algebra which contains the Barut-Girardello set and the Perelomov set of the SU(1, 1) coherent states as special cases. We also construct coherent states for the cubic algebra related to the conditionally solvable radial oscillator problem.
How to Produce S-Tense Operators on Lattice Effect Algebras
Chajda, Ivan; Janda, Jiří; Paseka, Jan
2014-07-01
Tense operators in effect algebras play a key role for the representation of the dynamics of formally described physical systems. For this, it is important to know how to construct them on a given effect algebra and how to compute all possible pairs of tense operators on . However, we firstly need to derive a time frame which enables these constructions and computations. Hence, we usually apply a suitable set of states of the effect algebra in question. To approximate physical reality in quantum mechanics, we use only the so-called Jauch-Piron states on in our paper. To realize our constructions, we are restricted on lattice effect algebras only.
Davidson, Kenneth R
1996-01-01
The subject of C*-algebras received a dramatic revitalization in the 1970s by the introduction of topological methods through the work of Brown, Douglas, and Fillmore on extensions of C*-algebras and Elliott's use of K-theory to provide a useful classification of AF algebras. These results were the beginning of a marvelous new set of tools for analyzing concrete C*-algebras. This book is an introductory graduate level text which presents the basics of the subject through a detailed analysis of several important classes of C*-algebras. The development of operator algebras in the last twenty yea
Algebra II workbook for dummies
Sterling, Mary Jane
2014-01-01
To succeed in Algebra II, start practicing now Algebra II builds on your Algebra I skills to prepare you for trigonometry, calculus, and a of myriad STEM topics. Working through practice problems helps students better ingest and retain lesson content, creating a solid foundation to build on for future success. Algebra II Workbook For Dummies, 2nd Edition helps you learn Algebra II by doing Algebra II. Author and math professor Mary Jane Sterling walks you through the entire course, showing you how to approach and solve the problems you encounter in class. You'll begin by refreshing your Algebr
Srinivas, V
1996-01-01
Algebraic K-Theory has become an increasingly active area of research. With its connections to algebra, algebraic geometry, topology, and number theory, it has implications for a wide variety of researchers and graduate students in mathematics. The book is based on lectures given at the author's home institution, the Tata Institute in Bombay, and elsewhere. A detailed appendix on topology was provided in the first edition to make the treatment accessible to readers with a limited background in topology. The second edition also includes an appendix on algebraic geometry that contains the required definitions and results needed to understand the core of the book; this makes the book accessible to a wider audience. A central part of the book is a detailed exposition of the ideas of Quillen as contained in his classic papers "Higher Algebraic K-Theory, I, II." A more elementary proof of the theorem of Merkujev--Suslin is given in this edition; this makes the treatment of this topic self-contained. An application ...
Regularity of C*-algebras and central sequence algebras
DEFF Research Database (Denmark)
Christensen, Martin S.
The main topic of this thesis is regularity properties of C*-algebras and how these regularity properties are re ected in their associated central sequence algebras. The thesis consists of an introduction followed by four papers [A], [B], [C], [D]. In [A], we show that for the class of simple...... Villadsen algebra of either the rst type with seed space a nite dimensional CW complex, or the second type, tensorial absorption of the Jiang-Su algebra is characterized by the absence of characters on the central sequence algebra. Additionally, in a joint appendix with Joan Bosa, we show that the Villadsen...... algebra of the second type with innite stable rank fails the corona factorization property. In [B], we consider the class of separable C*-algebras which do not admit characters on their central sequence algebra, and show that it has nice permanence properties. We also introduce a new divisibility property...
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...
Kac-Moody algebras and string theory
International Nuclear Information System (INIS)
Cleaver, G.B.
1993-01-01
The focus of this thesis is on (1) the role of Kac-Moody algebras in string theory and the development of techniques for systematically building string theory models based on a higher level (K ≥ 2) KM algebras and (2) fractional superstrings, a new class of solutions based on SU(2) K /U(1) conformal field theories. The content of this thesis is as follows. In chapter two they review KM algebras and their role in string theory. In the next chapter they present two results concerning the construction of modular invariant partition functions for conformal field theories build by tensoring together other conformal field theories. First they show how the possible modular invariants for the tensor product theory are constrained if the allowed modular invariants of the individuals conformal field theory factors have been classified. They illustrate the use of these constraints for theories of the type SU(2) KA direct-product SU(2) KB , finding all consistent theories for K A and K B odd. Second they show how known diagonal modular invariants can be used to construct inherently asymmetric invariants where the holomorphic and anti-holomorphic theories do not share the same chiral algebra. Explicit examples are given. Next, in chapter four they investigate some issues relating to recently proposed fractional superstring theories with D critical K/4 K/4 , as source of spacetime fermions, is demonstrated
Expander graphs in pure and applied mathematics
Lubotzky, Alexander
2012-01-01
Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics.
Schematic limits of rank 4 Azumaya bundles are the locally-Witt algebras
International Nuclear Information System (INIS)
Venkata Balaji, T.E.
2002-07-01
It is shown that the schematic image of the scheme of Azumaya algebra structures on a vector bundle of rank 4 over any base scheme is separated, of finite type, smooth of relative dimension 13 and geometrically irreducible over that base and that this construction base-changes well. This generalises the main theorem of Part I of an earlier work and clarifies it by showing that the algebraic operation of forming the even Clifford algebra (=Witt algebra) of a rank 3 quadratic module essentially translates to performing the geometric operation of taking the schematic image of the scheme of Azumaya algebra structures. (author)
Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras
Directory of Open Access Journals (Sweden)
Lothar Schlafer
2008-05-01
Full Text Available C*-algebraic Weyl quantization is extended by allowing also degenerate pre-symplectic forms for the Weyl relations with infinitely many degrees of freedom, and by starting out from enlarged classical Poisson algebras. A powerful tool is found in the construction of Poisson algebras and non-commutative twisted Banach-*-algebras on the stage of measures on the not locally compact test function space. Already within this frame strict deformation quantization is obtained, but in terms of Banach-*-algebras instead of C*-algebras. Fourier transformation and representation theory of the measure Banach-*-algebras are combined with the theory of continuous projective group representations to arrive at the genuine C*-algebraic strict deformation quantization in the sense of Rieffel and Landsman. Weyl quantization is recognized to depend in the first step functorially on the (in general infinite dimensional, pre-symplectic test function space; but in the second step one has to select a family of representations, indexed by the deformation parameter h. The latter ambiguity is in the present investigation connected with the choice of a folium of states, a structure, which does not necessarily require a Hilbert space representation.
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.
Algebraic quantum field theory
International Nuclear Information System (INIS)
Foroutan, A.
1996-12-01
The basic assumption that the complete information relevant for a relativistic, local quantum theory is contained in the net structure of the local observables of this theory results first of all in a concise formulation of the algebraic structure of the superselection theory and an intrinsic formulation of charge composition, charge conjugation and the statistics of an algebraic quantum field theory. In a next step, the locality of massive particles together with their spectral properties are wed for the formulation of a selection criterion which opens the access to the massive, non-abelian quantum gauge theories. The role of the electric charge as a superselection rule results in the introduction of charge classes which in term lead to a set of quantum states with optimum localization properties. Finally, the asymptotic observables of quantum electrodynamics are investigated within the framework of algebraic quantum field theory. (author)
Kollár, János
1997-01-01
This volume contains the lectures presented at the third Regional Geometry Institute at Park City in 1993. The lectures provide an introduction to the subject, complex algebraic geometry, making the book suitable as a text for second- and third-year graduate students. The book deals with topics in algebraic geometry where one can reach the level of current research while starting with the basics. Topics covered include the theory of surfaces from the viewpoint of recent higher-dimensional developments, providing an excellent introduction to more advanced topics such as the minimal model program. Also included is an introduction to Hodge theory and intersection homology based on the simple topological ideas of Lefschetz and an overview of the recent interactions between algebraic geometry and theoretical physics, which involve mirror symmetry and string theory.
Launey, Warwick De
2011-01-01
Combinatorial design theory is a source of simply stated, concrete, yet difficult discrete problems, with the Hadamard conjecture being a prime example. It has become clear that many of these problems are essentially algebraic in nature. This book provides a unified vision of the algebraic themes which have developed so far in design theory. These include the applications in design theory of matrix algebra, the automorphism group and its regular subgroups, the composition of smaller designs to make larger designs, and the connection between designs with regular group actions and solutions to group ring equations. Everything is explained at an elementary level in terms of orthogonality sets and pairwise combinatorial designs--new and simple combinatorial notions which cover many of the commonly studied designs. Particular attention is paid to how the main themes apply in the important new context of cocyclic development. Indeed, this book contains a comprehensive account of cocyclic Hadamard matrices. The book...
Peternell, Thomas; Schneider, Michael; Schreyer, Frank-Olaf
1992-01-01
The Bayreuth meeting on "Complex Algebraic Varieties" focussed on the classification of algebraic varieties and topics such as vector bundles, Hodge theory and hermitian differential geometry. Most of the articles in this volume are closely related to talks given at the conference: all are original, fully refereed research articles. CONTENTS: A. Beauville: Annulation du H(1) pour les fibres en droites plats.- M. Beltrametti, A.J. Sommese, J.A. Wisniewski: Results on varieties with many lines and their applications to adjunction theory.- G. Bohnhorst, H. Spindler: The stability of certain vector bundles on P(n) .- F. Catanese, F. Tovena: Vector bundles, linear systems and extensions of (1).- O. Debarre: Vers uns stratification de l'espace des modules des varietes abeliennes principalement polarisees.- J.P. Demailly: Singular hermitian metrics on positive line bundles.- T. Fujita: On adjoint bundles of ample vector bundles.- Y. Kawamata: Moderate degenerations of algebraic surfaces.- U. Persson: Genus two fibra...
Wadsworth, A R
2017-01-01
This is a book of problems in abstract algebra for strong undergraduates or beginning graduate students. It can be used as a supplement to a course or for self-study. The book provides more variety and more challenging problems than are found in most algebra textbooks. It is intended for students wanting to enrich their learning of mathematics by tackling problems that take some thought and effort to solve. The book contains problems on groups (including the Sylow Theorems, solvable groups, presentation of groups by generators and relations, and structure and duality for finite abelian groups); rings (including basic ideal theory and factorization in integral domains and Gauss's Theorem); linear algebra (emphasizing linear transformations, including canonical forms); and fields (including Galois theory). Hints to many problems are also included.
Bloch, Spencer J
2000-01-01
This book is the long-awaited publication of the famous Irvine lectures. Delivered in 1978 at the University of California at Irvine, these lectures turned out to be an entry point to several intimately-connected new branches of arithmetic algebraic geometry, such as regulators and special values of L-functions of algebraic varieties, explicit formulas for them in terms of polylogarithms, the theory of algebraic cycles, and eventually the general theory of mixed motives which unifies and underlies all of the above (and much more). In the 20 years since, the importance of Bloch's lectures has not diminished. A lucky group of people working in the above areas had the good fortune to possess a copy of old typewritten notes of these lectures. Now everyone can have their own copy of this classic work.
Olver, Peter J
2018-01-01
This textbook develops the essential tools of linear algebra, with the goal of imparting technique alongside contextual understanding. Applications go hand-in-hand with theory, each reinforcing and explaining the other. This approach encourages students to develop not only the technical proficiency needed to go on to further study, but an appreciation for when, why, and how the tools of linear algebra can be used across modern applied mathematics. Providing an extensive treatment of essential topics such as Gaussian elimination, inner products and norms, and eigenvalues and singular values, this text can be used for an in-depth first course, or an application-driven second course in linear algebra. In this second edition, applications have been updated and expanded to include numerical methods, dynamical systems, data analysis, and signal processing, while the pedagogical flow of the core material has been improved. Throughout, the text emphasizes the conceptual connections between each application and the un...
Blyth, T S
2002-01-01
Basic Linear Algebra is a text for first year students leading from concrete examples to abstract theorems, via tutorial-type exercises. More exercises (of the kind a student may expect in examination papers) are grouped at the end of each section. The book covers the most important basics of any first course on linear algebra, explaining the algebra of matrices with applications to analytic geometry, systems of linear equations, difference equations and complex numbers. Linear equations are treated via Hermite normal forms which provides a successful and concrete explanation of the notion of linear independence. Another important highlight is the connection between linear mappings and matrices leading to the change of basis theorem which opens the door to the notion of similarity. This new and revised edition features additional exercises and coverage of Cramer's rule (omitted from the first edition). However, it is the new, extra chapter on computer assistance that will be of particular interest to readers:...
Deo, Satya
2018-01-01
This book presents the first concepts of the topics in algebraic topology such as the general simplicial complexes, simplicial homology theory, fundamental groups, covering spaces and singular homology theory in greater detail. Originally published in 2003, this book has become one of the seminal books. Now, in the completely revised and enlarged edition, the book discusses the rapidly developing field of algebraic topology. Targeted to undergraduate and graduate students of mathematics, the prerequisite for this book is minimal knowledge of linear algebra, group theory and topological spaces. The book discusses about the relevant concepts and ideas in a very lucid manner, providing suitable motivations and illustrations. All relevant topics are covered, including the classical theorems like the Brouwer’s fixed point theorem, Lefschetz fixed point theorem, Borsuk-Ulam theorem, Brouwer’s separation theorem and the theorem on invariance of the domain. Most of the exercises are elementary, but sometimes chal...
Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction
Wasserman, Nicholas H.
2016-01-01
This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…
Converting nested algebra expressions into flat algebra expressions
Paredaens, J.; Van Gucht, D.
1992-01-01
Nested relations generalize ordinary flat relations by allowing tuple values to be either atomic or set valued. The nested algebra is a generalization of the flat relational algebra to manipulate nested relations. In this paper we study the expressive power of the nested algebra relative to its
Symplectic Maps from Cluster Algebras
Directory of Open Access Journals (Sweden)
Allan P. Fordy
2011-09-01
Full Text Available We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are applied to a set of affine coordinates (the cluster variables. Fordy and Marsh recently provided a complete classification of all such quivers that have a certain periodicity property under sequences of mutations. This periodicity implies that a suitable sequence of cluster mutations is precisely equivalent to iteration of a nonlinear recurrence relation. Here we explain briefly how to introduce a symplectic structure in this setting, which is preserved by a corresponding birational map (possibly on a space of lower dimension. We give examples of both integrable and non-integrable maps that arise from this construction. We use algebraic entropy as an approach to classifying integrable cases. The degrees of the iterates satisfy a tropical version of the map.
Higher-spin extended conformal algebras and W-gravities
International Nuclear Information System (INIS)
Hull, C.M.
1991-01-01
The construction of classical W 3 gravity is reviewed. It is suggested that the hidden symmetry for quantum W 3 gravity in the chiral gauge is not SL(3, R) but a group contraction of this, ISL(2, R). This is extended to W N gravity, and the case of W 4 gravity is presented in detail. The gauge transformations are realized on D free bosons, with the spin-n conserved current (2 ≤ n ≤ N) taking the form d sub(i i ...i n ) δ + Φ sup(i 1 ) δ + Φ sup(i n ) for some constant tensor d sub(i i ...i n ). The d-tensors must satisfy N-2 non-linear algebraic constraints and these constraints are shown to be satisfied if the d-tensors are taken to be the structure-tensors of an Nth degree Jordan algebra. The relation with Jordan algebras is used to give solutions of the d-tensor constraints for any value of D, N. The free-boson construction of the W N algebras is generalized to give a Sugaware-type construction of a large class of classical extended conformal algebras. The chiral gauging of any classical extended conformal algebra is shown to require only a linear Noether coupling to world-sheet gauge-fields, while gauging a non-chiral algebra in general leads to a non-polynomial action. A number of examples are examined, including W ∞ W-supergravity, Knizhnik-Berschadsky supergravity and 'W N/M ' algebras. Theories of higher-spin W-gravity of the type described are only possible in one and two space-time dimensions, and the one-dimensional cases is briefly discussed. The covariant formulation of W-gravity is briefly discussed and the relation between classical and quantum extended conformal algebras is analyzed. (orig.)
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 ...
Computer Program For Linear Algebra
Krogh, F. T.; Hanson, R. J.
1987-01-01
Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.
Algebra for Gifted Third Graders.
Borenson, Henry
1987-01-01
Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)
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.
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.
Dynamical systems and linear algebra
Colonius, Fritz (Prof.)
2007-01-01
Dynamical systems and linear algebra / F. Colonius, W. Kliemann. - In: Handbook of linear algebra / ed. by Leslie Hogben. - Boca Raton : Chapman & Hall/CRC, 2007. - S. 56,1-56,22. - (Discrete mathematics and its applications)
Contractions of quantum algebraic structures
International Nuclear Information System (INIS)
Doikou, A.; Sfetsos, K.
2010-01-01
A general framework for obtaining certain types of contracted and centrally extended algebras is reviewed. The whole process relies on the existence of quadratic algebras, which appear in the context of boundary integrable models. (Abstract Copyright [2010], Wiley Periodicals, Inc.)
Polynomial Heisenberg algebras
International Nuclear Information System (INIS)
Carballo, Juan M; C, David J Fernandez; Negro, Javier; Nieto, Luis M
2004-01-01
Polynomial deformations of the Heisenberg algebra are studied in detail. Some of their natural realizations are given by the higher order susy partners (and not only by those of first order, as is already known) of the harmonic oscillator for even-order polynomials. Here, it is shown that the susy partners of the radial oscillator play a similar role when the order of the polynomial is odd. Moreover, it will be proved that the general systems ruled by such kinds of algebras, in the quadratic and cubic cases, involve Painleve transcendents of types IV and V, respectively
Classical algebraic chromodynamics
International Nuclear Information System (INIS)
Adler, S.L.
1978-01-01
I develop an extension of the usual equations of SU(n) chromodynamics which permits the consistent introduction of classical, noncommuting quark source charges. The extension involves adding a singlet gluon, giving a U(n) -based theory with outer product P/sup a/(u,v) = (1/2)(d/sup a/bc + if/sup a/bc)(u/sup b/v/sup c/ - v/sup b/u/sup c/) which obeys the Jacobi identity, inner product S (u,v) = (1/2)(u/sup a/v/sup a/ + v/sup a/u/sup a/), and with the n 2 gluon fields elevated to algebraic fields over the quark color charge C* algebra. I show that provided the color charge algebra satisfies the condition S (P (u,v),w) = S (u,P (v,w)) for all elements u,v,w of the algebra, all the standard derivations of Lagrangian chromodynamics continue to hold in the algebraic chromodynamics case. I analyze in detail the color charge algebra in the two-particle (qq, qq-bar, q-barq-bar) case and show that the above consistency condition is satisfied for the following unique (and, interestingly, asymmetric) choice of quark and antiquark charges: Q/sup a//sub q/ = xi/sup a/, Q/sup a//sub q/ = xi-bar/sup a/ + delta/sup a/0(n/2)/sup 3/2/1, with xi/sup a/xi/sup b/ = (1/2)(d/sup a/bc + if/sup a/bc) xi/sup c/, xi-bar/sup a/xi-bar/sup b/ = -(1/2)(d/sup a/bc - if/sup a/bc) xi-bar/sup c/. The algebraic structure of the two-particle U(n) force problem, when expressed on an appropriately diagonalized basis, leads for all n to a classical dynamics problem involving an ordinary SU(2) Yang-Mills field with uniquely specified classical source charges which are nonparallel in the color-singlet state. An explicit calculation shows that local algebraic U(n) gauge transformations lead only to a rigid global rotation of axes in the overlying classical SU(2) problem, which implies that the relative orientations of the classical source charges have physical significance
Weiss, Edwin
1998-01-01
Careful organization and clear, detailed proofs characterize this methodical, self-contained exposition of basic results of classical algebraic number theory from a relatively modem point of view. This volume presents most of the number-theoretic prerequisites for a study of either class field theory (as formulated by Artin and Tate) or the contemporary treatment of analytical questions (as found, for example, in Tate's thesis).Although concerned exclusively with algebraic number fields, this treatment features axiomatic formulations with a considerable range of applications. Modem abstract te
Hohn, Franz E
2012-01-01
This complete and coherent exposition, complemented by numerous illustrative examples, offers readers a text that can teach by itself. Fully rigorous in its treatment, it offers a mathematically sound sequencing of topics. The work starts with the most basic laws of matrix algebra and progresses to the sweep-out process for obtaining the complete solution of any given system of linear equations - homogeneous or nonhomogeneous - and the role of matrix algebra in the presentation of useful geometric ideas, techniques, and terminology.Other subjects include the complete treatment of the structur
Principles of algebraic geometry
Griffiths, Phillip A
1994-01-01
A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special top
Energy Technology Data Exchange (ETDEWEB)
Christian, J M; McDonald, G S [Joule Physics Laboratory, School of Computing, Science and Engineering, Materials and Physics Research Centre, University of Salford, Salford M5 4WT (United Kingdom); Chamorro-Posada, P, E-mail: j.christian@salford.ac.u [Departamento de Teoria de la Senal y Comunicaciones e Ingenieria Telematica, Universidad de Valladolid, ETSI Telecomunicacion, Campus Miguel Delibes s/n, 47011 Valladolid (Spain)
2010-02-26
We report, to the best of our knowledge, the first exact analytical algebraic solitons of a generalized cubic-quintic Helmholtz equation. This class of governing equation plays a key role in photonics modelling, allowing a full description of the propagation and interaction of broad scalar beams. New conservation laws are presented, and the recovery of paraxial results is discussed in detail. The stability properties of the new solitons are investigated by combining semi-analytical methods and computer simulations. In particular, new general stability regimes are reported for algebraic bright solitons.
Endomorphisms of graph algebras
DEFF Research Database (Denmark)
Conti, Roberto; Hong, Jeong Hee; Szymanski, Wojciech
2012-01-01
We initiate a systematic investigation of endomorphisms of graph C*-algebras C*(E), extending several known results on endomorphisms of the Cuntz algebras O_n. Most but not all of this study is focused on endomorphisms which permute the vertex projections and globally preserve the diagonal MASA D...... that the restriction to the diagonal MASA of an automorphism which globally preserves both D_E and the core AF-subalgebra eventually commutes with the corresponding one-sided shift. Secondly, we exhibit several properties of proper endomorphisms, investigate invertibility of localized endomorphisms both on C...
Algebraic curves and cryptography
Murty, V Kumar
2010-01-01
It is by now a well-known paradigm that public-key cryptosystems can be built using finite Abelian groups and that algebraic geometry provides a supply of such groups through Abelian varieties over finite fields. Of special interest are the Abelian varieties that are Jacobians of algebraic curves. All of the articles in this volume are centered on the theme of point counting and explicit arithmetic on the Jacobians of curves over finite fields. The topics covered include Schoof's \\ell-adic point counting algorithm, the p-adic algorithms of Kedlaya and Denef-Vercauteren, explicit arithmetic on
Kendig, Keith
2015-01-01
Designed to make learning introductory algebraic geometry as easy as possible, this text is intended for advanced undergraduates and graduate students who have taken a one-year course in algebra and are familiar with complex analysis. This newly updated second edition enhances the original treatment's extensive use of concrete examples and exercises with numerous figures that have been specially redrawn in Adobe Illustrator. An introductory chapter that focuses on examples of curves is followed by a more rigorous and careful look at plane curves. Subsequent chapters explore commutative ring th
International Nuclear Information System (INIS)
Christian, J M; McDonald, G S; Chamorro-Posada, P
2010-01-01
We report, to the best of our knowledge, the first exact analytical algebraic solitons of a generalized cubic-quintic Helmholtz equation. This class of governing equation plays a key role in photonics modelling, allowing a full description of the propagation and interaction of broad scalar beams. New conservation laws are presented, and the recovery of paraxial results is discussed in detail. The stability properties of the new solitons are investigated by combining semi-analytical methods and computer simulations. In particular, new general stability regimes are reported for algebraic bright solitons.
Hogben, Leslie
2013-01-01
With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and updates throughout, the second edition of this bestseller includes 20 new chapters.New to the Second EditionSeparate chapters on Schur complements, additional types of canonical forms, tensors, matrix polynomials, matrix equations, special types of
Algebra & trigonometry I essentials
REA, Editors of
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Algebra & Trigonometry I includes sets and set operations, number systems and fundamental algebraic laws and operations, exponents and radicals, polynomials and rational expressions, eq
Algebra & trigonometry super review
2012-01-01
Get all you need to know with Super Reviews! Each Super Review is packed with in-depth, student-friendly topic reviews that fully explain everything about the subject. The Algebra and Trigonometry Super Review includes sets and set operations, number systems and fundamental algebraic laws and operations, exponents and radicals, polynomials and rational expressions, equations, linear equations and systems of linear equations, inequalities, relations and functions, quadratic equations, equations of higher order, ratios, proportions, and variations. Take the Super Review quizzes to see how much y
Linear Algebra Thoroughly Explained
Vujičić, Milan
2008-01-01
Linear Algebra Thoroughly Explained provides a comprehensive introduction to the subject suitable for adoption as a self-contained text for courses at undergraduate and postgraduate level. The clear and comprehensive presentation of the basic theory is illustrated throughout with an abundance of worked examples. The book is written for teachers and students of linear algebra at all levels and across mathematics and the applied sciences, particularly physics and engineering. It will also be an invaluable addition to research libraries as a comprehensive resource book for the subject.
States and Measures on Hyper BCK-Algebras
Directory of Open Access Journals (Sweden)
Xiao-Long Xin
2014-01-01
Full Text Available We define the notions of Bosbach states and inf-Bosbach states on a bounded hyper BCK-algebra (H,∘,0,e and derive some basic properties of them. We construct a quotient hyper BCK-algebra via a regular congruence relation. We also define a ∘-compatibled regular congruence relation θ and a θ-compatibled inf-Bosbach state s on (H,∘,0,e. By inducing an inf-Bosbach state s^ on the quotient structure H/[0]θ, we show that H/[0]θ is a bounded commutative BCK-algebra which is categorically equivalent to an MV-algebra. In addition, we introduce the notions of hyper measures (states/measure morphisms/state morphisms on hyper BCK-algebras, and present a relation between hyper state-morphisms and Bosbach states. Then we construct a quotient hyper BCK-algebra H/Ker(m by a reflexive hyper BCK-ideal Ker(m. Further, we prove that H/Ker(m is a bounded commutative BCK-algebra.
Introduction to Banach spaces and algebras
Allan, Graham
2010-01-01
Banach spaces and algebras are a key topic of pure mathematics. Graham Allan's careful and detailed introductory account will prove essential reading for anyone wishing to specialise in functional analysis and is aimed at final year undergraduates or masters level students. Based on the author's lectures to fourth year students at Cambridge University, the book assumes knowledge typical of first degrees in mathematics, including metric spaces, analytic topology, and complexanalysis. However, readers are not expected to be familiar with the Lebesgue theory of measure and integration.The text be
The theory of algebraic numbers
Pollard, Harry
1998-01-01
An excellent introduction to the basics of algebraic number theory, this concise, well-written volume examines Gaussian primes; polynomials over a field; algebraic number fields; and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture. 1975 edition. References. List of Symbols. Index.
Classification of non-simple C*-algebras of real rank zero
DEFF Research Database (Denmark)
Arklint, Sara Esther
for real rank zero graph algebras over primitive ideal spaces that admit classification. As a consequence of completeness of filtered K-theory combined with this range result, one can conclude that real rank zero extensions of stabilized Cuntz-Krieger algebras are stabilized Cuntz-Krieger algebras......This thesis deals with classification of nonsimple C-algebras of real rank zero, and whether filtered K-theory is a suitable invariant for this purpose. As a consequence of the result of E. Kirchberg for purely infinite, nuclear C-algebras with a finite primitive ideal space, it suffices to lift...... the thesis is the following: is it possible to achieve the desired classification result for arbitrary finite primitive ideal spaces by restricting to C-algebras of real rank zero that possibly satisfy further restrictions on K-theory? The thesis consists of an account of the relevant theory and the relevant...
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.
Kac and new determinants for fractional superconformal algebras
International Nuclear Information System (INIS)
Kakushadze, Z.; Tye, S.H.
1994-01-01
We derive the Kac and new determinant formulas for an arbitrary (integer) level K fractional superconformal algebra using the BRST cohomology techniques developed in conformal field theory. In particular, we reproduce the Kac determinants for the Virasoro (K=1) and superconformal (K=2) algebras. For K≥3 there always exist modules where the Kac determinant factorizes into a product of more fundamental new determinants. Using our results for general K, we sketch the nonunitarity proof for the SU(2) minimal series; as expected, the only unitary models are those already known from the coset construction. We apply the Kac determinant formulas for the spin 4/3 parafermion current algebra (i.e., the K=4 fractional superconformal algebra) to the recently constructed three-dimensional flat Minkowski space-time representation of the spin-4/3 fractional superstring
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.)
Noncommutative Chern-Connes characters of some noncompact quantum algebras
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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)
Algebraic Verification Method for SEREs Properties via Groebner Bases Approaches
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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.
Towards classical spectrum generating algebras for f-deformations
Kullock, Ricardo; Latini, Danilo
2016-01-01
In this paper we revise the classical analog of f-oscillators, a generalization of q-oscillators given in Man'ko et al. (1997) [8], in the framework of classical spectrum generating algebras (SGA) introduced in Kuru and Negro (2008) [9]. We write down the deformed Poisson algebra characterizing the entire family of non-linear oscillators and construct its general solution algebraically. The latter, covering the full range of f-deformations, shows an energy dependence both in the amplitude and the frequency of the motion.
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
An algebra of reversible computation.
Wang, Yong
2016-01-01
We design an axiomatization for reversible computation called reversible ACP (RACP). It has four extendible modules: basic reversible processes algebra, algebra of reversible communicating processes, recursion and abstraction. Just like process algebra ACP in classical computing, RACP can be treated as an axiomatization foundation for reversible computation.
Thomys, Janus; Zhang, Xiaohong
2013-01-01
We describe weak-BCC-algebras (also called BZ-algebras) in which the condition (x∗y)∗z = (x∗z)∗y is satisfied only in the case when elements x, y belong to the same branch. We also characterize ideals, nilradicals, and nilpotent elements of such algebras. PMID:24311983
Assessing Elementary Algebra with STACK
Sangwin, Christopher J.
2007-01-01
This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS…
Process Algebra and Markov Chains
Brinksma, Hendrik; Hermanns, H.; Brinksma, Hendrik; Hermanns, H.; Katoen, Joost P.
This paper surveys and relates the basic concepts of process algebra and the modelling of continuous time Markov chains. It provides basic introductions to both fields, where we also study the Markov chains from an algebraic perspective, viz. that of Markov chain algebra. We then proceed to study
Process algebra and Markov chains
Brinksma, E.; Hermanns, H.; Brinksma, E.; Hermanns, H.; Katoen, J.P.
2001-01-01
This paper surveys and relates the basic concepts of process algebra and the modelling of continuous time Markov chains. It provides basic introductions to both fields, where we also study the Markov chains from an algebraic perspective, viz. that of Markov chain algebra. We then proceed to study
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
On left Hopf algebras within the framework of inhomogeneous quantum groups for particle algebras
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Rodriguez-Romo, Suemi [Facultad de Estudios Superiores Cuautitlan, Universidad Nacional Autonoma de Mexico (Mexico)
2012-10-15
We deal with some matters needed to construct concrete left Hopf algebras for inhomogeneous quantum groups produced as noncommutative symmetries of fermionic and bosonic creation/annihilation operators. We find a map for the bidimensional fermionic case, produced as in Manin's [Quantum Groups and Non-commutative Hopf Geometry (CRM Univ. de Montreal, 1988)] seminal work, named preantipode that fulfills all the necessary requirements to be left but not right on the generators of the algebra. Due to the complexity and importance of the full task, we consider our result as an important step that will be extended in the near future.
First-Order Polynomial Heisenberg Algebras and Coherent States
International Nuclear Information System (INIS)
Castillo-Celeita, M; Fernández C, D J
2016-01-01
The polynomial Heisenberg algebras (PHA) are deformations of the Heisenberg- Weyl algebra characterizing the underlying symmetry of the supersymmetric partners of the Harmonic oscillator. When looking for the simplest system ruled by PHA, however, we end up with the harmonic oscillator. In this paper we are going to realize the first-order PHA through the harmonic oscillator. The associated coherent states will be also constructed, which turn out to be the well known even and odd coherent states. (paper)
On massless representations of the Q-deformed Poincare algebra
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Ogievetsky, O.; Pillin, M.; Schmidke, W.B.; Wess, J.
1993-01-01
This talk is devoted to the construction of massless representations of the q-deformed Poincare algebra. In section 2 we give Hilbert space representations of the SL q (2, C)-covariant quantum space. We then show in the next section how the generators of the q-Poincare algebra can be expressed in terms of operators which live in the light cone. The q-deformed massless one-particle states are considered in section 4. (orig.)
Bergstra, J.A.; Middelburg, C.A.
2015-01-01
We add probabilistic features to basic thread algebra and its extensions with thread-service interaction and strategic interleaving. Here, threads represent the behaviours produced by instruction sequences under execution and services represent the behaviours exhibited by the components of execution
Indian Academy of Sciences (India)
BOOK REVIEW ... To the Indian reader, the word discourse, evokes a respected ... I dug a bit deeper with Google trans- late, and ... published in a journal of mathematics educa- tion. ... The article on Shafarevich's work elsewhere ... goal then, is to develop the basics of algebra in ... ometric Greeks, and works like a magician.
Thinking Visually about Algebra
Baroudi, Ziad
2015-01-01
Many introductions to algebra in high school begin with teaching students to generalise linear numerical patterns. This article argues that this approach needs to be changed so that students encounter variables in the context of modelling visual patterns so that the variables have a meaning. The article presents sample classroom activities,…
The algebraic collective model
International Nuclear Information System (INIS)
Rowe, D.J.; Turner, P.S.
2005-01-01
A recently proposed computationally tractable version of the Bohr collective model is developed to the extent that we are now justified in describing it as an algebraic collective model. The model has an SU(1,1)xSO(5) algebraic structure and a continuous set of exactly solvable limits. Moreover, it provides bases for mixed symmetry collective model calculations. However, unlike the standard realization of SU(1,1), used for computing beta wave functions and their matrix elements in a spherical basis, the algebraic collective model makes use of an SU(1,1) algebra that generates wave functions appropriate for deformed nuclei with intrinsic quadrupole moments ranging from zero to any large value. A previous paper focused on the SO(5) wave functions, as SO(5) (hyper-)spherical harmonics, and computation of their matrix elements. This paper gives analytical expressions for the beta matrix elements needed in applications of the model and illustrative results to show the remarkable gain in efficiency that is achieved by using such a basis in collective model calculations for deformed nuclei
Benjamin, Carl; And Others
Presented are student performance objectives, a student progress chart, and assignment sheets with objective and diagnostic measures for the stated performance objectives in College Algebra I. Topics covered include: sets; vocabulary; linear equations; inequalities; real numbers; operations; factoring; fractions; formulas; ratio, proportion, and…
Swan, R G
1968-01-01
From the Introduction: "These notes are taken from a course on algebraic K-theory [given] at the University of Chicago in 1967. They also include some material from an earlier course on abelian categories, elaborating certain parts of Gabriel's thesis. The results on K-theory are mostly of a very general nature."
Bergstra, J.A.; Baeten, J.C.M.
1993-01-01
The real time process algebra of Baeten and Bergstra [Formal Aspects of Computing, 3, 142-188 (1991)] is extended to real space by requiring the presence of spatial coordinates for each atomic action, in addition to the required temporal attribute. It is found that asynchronous communication
Contemporary developments in algebraic K-theory
Energy Technology Data Exchange (ETDEWEB)
Karoubi, M [Univ. Paris (France); Kuku, A O [Abdus Salam International Centre for Theoretical Physics, Trieste (Italy); Pedrini, C [Univ. Genova (Italy)
2003-09-15
The School and Conference on Algebraic K-theory which took place at ICTP July 8-26, 2002 was a follow-up to the earlier one in 1997, and like its predecessor, the 2002 meeting endeavoured to emphasise the multidisciplinary aspects of the subject. However, one special feature of the 2002 School and Conference is that the whole activity was dedicated to H. Bass, one of the founders of Algebraic K-theory, on the occasion of his seventieth birthday. The School during the first two weeks, July 8 to 19 was devoted to expository lectures meant to explore and highlight connections between K-theory and several other areas of mathematics - Algebraic Topology, Number theory, Algebraic Geometry, Representation theory, and Non-commutative Geometry. This volume, constituting the Proceedings of the School, is dedicated to H. Bass. The Proceedings of the Conference during the last week July 22 - 26, which will appear in Special issues of K-theory, is also dedicated to H. Bass. The opening contribution by M. Karoubi to this volume consists of a comprehensive survey of developments in K-theory in the last forty-five years, and covers a very broad spectrum of the subject, including Topological K-theory, Atiyah-Singer index theorem, K-theory of Banach algebras, Higher Algebraic K-theory, Cyclic Homology etc. J. Berrick's contribution on 'Algebraic K-theory and Algebraic Topology' treats the various topological constructions of Algebraic K-theory together with the underlying homotopy theory. Topics covered include the plus construction together with its various ramifications and applications, Topological Hochschild and Cyclic Homology as well as K-theory of the ring of integers. The contributions by M. Kolster titled 'K-theory and Arithmetics' includes such topics as values of zeta functions and relations to K-theory, K-theory of integers in number fields and associated conjectures, Etale cohomology, Iwasawa theory etc. A.O. Kuku's contributions on 'K-theory and Representation theory
Projective Connections and the Algebra of Densities
International Nuclear Information System (INIS)
George, Jacob
2008-01-01
Projective connections first appeared in Cartan's papers in the 1920's. Since then they have resurfaced periodically in, for example, integrable systems and perhaps most recently in the context of so called projectively equivariant quantisation. We recall the notion of projective connection and describe its relation with the algebra of densities on a manifold. In particular, we construct a Laplace-type operator on functions using a Thomas projective connection and a symmetric contravariant tensor of rank 2 ('upper metric').
Operator algebras and topology
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Schick, T.
2002-01-01
These notes, based on three lectures on operator algebras and topology at the 'School on High Dimensional Manifold Theory' at the ICTP in Trieste, introduce a new set of tools to high dimensional manifold theory, namely techniques coming from the theory of operator algebras, in particular C*-algebras. These are extensively studied in their own right. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. A central pillar of work in the theory of C*-algebras is the Baum-Connes conjecture. This is an isomorphism conjecture, as discussed in the talks of Luck, but with a certain special flavor. Nevertheless, it has important direct applications to the topology of manifolds, it implies e.g. the Novikov conjecture. In the first chapter, the Baum-Connes conjecture will be explained and put into our context. Another application of the Baum-Connes conjecture is to the positive scalar curvature question. This will be discussed by Stephan Stolz. It implies the so-called 'stable Gromov-Lawson-Rosenberg conjecture'. The unstable version of this conjecture said that, given a closed spin manifold M, a certain obstruction, living in a certain (topological) K-theory group, vanishes if and only M admits a Riemannian metric with positive scalar curvature. It turns out that this is wrong, and counterexamples will be presented in the second chapter. The third chapter introduces another set of invariants, also using operator algebra techniques, namely L 2 -cohomology, L 2 -Betti numbers and other L 2 -invariants. These invariants, their basic properties, and the central questions about them, are introduced in the third chapter. (author)
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
Advanced modern algebra part 2
Rotman, Joseph J
2017-01-01
This book is the second part of the new edition of Advanced Modern Algebra (the first part published as Graduate Studies in Mathematics, Volume 165). Compared to the previous edition, the material has been significantly reorganized and many sections have been rewritten. The book presents many topics mentioned in the first part in greater depth and in more detail. The five chapters of the book are devoted to group theory, representation theory, homological algebra, categories, and commutative algebra, respectively. The book can be used as a text for a second abstract algebra graduate course, as a source of additional material to a first abstract algebra graduate course, or for self-study.
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
Free-field realisations of the BMS_3 algebra and its extensions
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Banerjee, Nabamita; Jatkar, Dileep P.; Mukhi, Sunil; Neogi, Turmoli
2016-01-01
We construct an explicit realisation of the BMS_3 algebra with nonzero central charges using holomorphic free fields. This can be extended by the addition of chiral matter to a realisation having arbitrary values for the two independent central charges. Via the introduction of additional free fields, we extend our construction to the minimally supersymmetric BMS_3 algebra and to the nonlinear higher-spin BMS_3-W_3 algebra. We also describe an extended system that realises both the SU(2) current algebra as well as BMS_3 via the Wakimoto representation, though in this case introducing a central extension also brings in new non-central operators.
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
Canonical formulation of the self-dual Yang-Mills system: Algebras and hierarchies
International Nuclear Information System (INIS)
Chau, L.; Yamanaka, I.
1992-01-01
We construct a canonical formulation of the self-dual Yang-Mills system formulated in the gauge-invariant group-valued J fields and derive their Hamiltonian and the quadratic algebras of the fundamental Dirac brackets. We also show that the quadratic algebras satisfy Jacobi identities and their structure matrices satisfy modified Yang-Baxter equations. From these quadratic algebras, we construct Kac-Moody-like and Virasoro-like algebras. We also discuss their related symmetries, involutive conserved quantities, and hierarchies of nonlinear and linear equations
From affine Hecke algebras to boundary symmetries
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Doikou, Anastasia
2005-01-01
Motivated by earlier works we employ appropriate realizations of the affine Hecke algebra and we recover previously known non-diagonal solutions of the reflection equation for the U q (gl n -bar ) case. The corresponding N site spin chain with open boundary conditions is then constructed and boundary non-local charges associated to the non-diagonal solutions of the reflection equation are derived, as coproduct realizations of the reflection algebra. With the help of linear intertwining relations involving the aforementioned solutions of the reflection equation, the symmetry of the open spin chain with the corresponding boundary conditions is exhibited, being essentially a remnant of the U q (gl n -bar ) algebra. More specifically, we show that representations of certain boundary non-local charges commute with the generators of the affine Hecke algebra and with the local Hamiltonian of the open spin chain for a particular choice of boundary conditions. Furthermore, we are able to show that the transfer matrix of the open spin chain commutes with a certain number of boundary non-local charges, depending on the choice of boundary conditions
On identities of free finitely generated alternative algebras over a field of characteristic 3
International Nuclear Information System (INIS)
Pchelintsev, S V
2001-01-01
In 1981 Filippov solved in the affirmative Shestakov's problem on the strictness of the inclusions in the chains of varieties generated by free alternative and Mal'cev algebras of finite rank over a field of characteristic distinct from 2 and 3. In the present paper an analogous result is proved for alternative algebras over a field of characteristic 3. The proof is based on the construction of three families of identities that hold on the algebras of the corresponding rank. A disproof of the identities on algebras of larger rank is carried out with the help of a prime commutative alternative algebra. It is also proved that in varieties of alternative algebras of finite basis rank over a field of characteristic 3 every soluble algebra is nilpotent
Noncommutative o*(N) and usp*(2N) algebras and the corresponding gauge field theories
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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)
On Dunkl angular momenta algebra
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Feigin, Misha [School of Mathematics and Statistics, University of Glasgow,15 University Gardens, Glasgow G12 8QW (United Kingdom); Hakobyan, Tigran [Yerevan State University,1 Alex Manoogian, 0025 Yerevan (Armenia); Tomsk Polytechnic University,Lenin Ave. 30, 634050 Tomsk (Russian Federation)
2015-11-17
We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincaré-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl(N) version of the subalgebra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.
S-duality and the prepotential in N=2⋆ theories (I): the ADE algebras
International Nuclear Information System (INIS)
Billó, M.; Frau, M.; Fucito, F.; Lerda, A.; Morales, J.F.
2015-01-01
The prepotential of N=2 ⋆ supersymmetric theories with unitary gauge groups in an Ω background satisfies a modular anomaly equation that can be recursively solved order by order in an expansion for small mass. By requiring that S-duality acts on the prepotential as a Fourier transform we generalise this result to N=2 ⋆ theories with gauge algebras of the D and E type and show that their prepotentials can be written in terms of quasi-modular forms of SL(2,ℤ). The results are checked against microscopic multi-instanton calculus based on localization for the A and D series and reproduce the known 1-instanton prepotential of the pure N=2 theories for any gauge group of ADE type. Our results can also be used to obtain the multi-instanton terms in the exceptional theories for which the microscopic instanton calculus and the ADHM construction are not available.