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Introduction to Finite and Infinite Dimensional Lie (Super)Algebras

Introduction to Finite and Infinite Dimensional Lie (Super)Algebras
A Book

by Sthanumoorthy Neelacanta

  • Publisher : Academic Press
  • Release : 2016-04-01
  • Pages : 492
  • ISBN : 9780128046753
  • Language : En, Es, Fr & De
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Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Introduction to Finite and Infinite Dimensional Lie Algebras and Superalgebras introduces the theory of Lie superalgebras, their algebras, and their representations. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semi-simple Lie algebras. While discussing all classes of finite and infinite dimensional Lie algebras and Lie superalgebras in terms of their different classes of root systems, the book focuses on Kac-Moody algebras. With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras. Discusses the fundamental structure and all root relationships of Lie algebras and Lie superalgebras and their finite and infinite dimensional representation theory Closely describes BKM Lie superalgebras, their different classes of imaginary root systems, their complete classifications, root-supermultiplicities, and related combinatorial identities Includes numerous tables of the properties of individual Lie algebras and Lie superalgebras Focuses on Kac-Moody algebras

Introduction to Finite and Infinite Dimensional Lie (Super)algebras

Introduction to Finite and Infinite Dimensional Lie (Super)algebras
A Book

by Neelacanta Sthanumoorthy

  • Publisher : Academic Press
  • Release : 2016-04-26
  • Pages : 512
  • ISBN : 012804683X
  • Language : En, Es, Fr & De
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Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Introduction to Finite and Infinite Dimensional Lie Algebras and Superalgebras introduces the theory of Lie superalgebras, their algebras, and their representations. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semi-simple Lie algebras. While discussing all classes of finite and infinite dimensional Lie algebras and Lie superalgebras in terms of their different classes of root systems, the book focuses on Kac-Moody algebras. With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras. Discusses the fundamental structure and all root relationships of Lie algebras and Lie superalgebras and their finite and infinite dimensional representation theory Closely describes BKM Lie superalgebras, their different classes of imaginary root systems, their complete classifications, root-supermultiplicities, and related combinatorial identities Includes numerous tables of the properties of individual Lie algebras and Lie superalgebras Focuses on Kac-Moody algebras

Infinite-dimensional Lie Algebras

Infinite-dimensional Lie Algebras
A Book

by Minoru Wakimoto

  • Publisher : American Mathematical Soc.
  • Release : 2001
  • Pages : 304
  • ISBN : 9780821826546
  • Language : En, Es, Fr & De
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This volume begins with an introduction to the structure of finite-dimensional simple Lie algebras, including the representation of ${\widehat {\mathfrak {sl}}}(2, {\mathbb C})$, root systems, the Cartan matrix, and a Dynkin diagram of a finite-dimensional simple Lie algebra. Continuing on, the main subjects of the book are the structure (real and imaginary root systems) of and the character formula for Kac-Moody superalgebras, which is explained in a very general setting. Only elementary linear algebra and group theory are assumed. Also covered is modular property and asymptotic behavior of integrable characters of affine Lie algebras. The exposition is self-contained and includes examples. The book can be used in a graduate-level course on the topic.

Infinite Dimensional Lie Superalgebras

Infinite Dimensional Lie Superalgebras
A Book

by Yuri Bahturin,Alexander V. Mikhalev,Viktor M. Petrogradsky,Mikhail V. Zaicev

  • Publisher : Walter de Gruyter
  • Release : 1992-01-01
  • Pages : 260
  • ISBN : 3110851202
  • Language : En, Es, Fr & De
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The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany

Bimonoids for Hyperplane Arrangements

Bimonoids for Hyperplane Arrangements
A Book

by Marcelo Aguiar,Swapneel Mahajan

  • Publisher : Cambridge University Press
  • Release : 2020-03-31
  • Pages : 824
  • ISBN : 110849580X
  • Language : En, Es, Fr & De
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Develops a new theory, parallel to the classical theory of connected Hopf algebras, including a real hyperplane arrangement.

Supersymmetries and Infinite-Dimensional Algebras

Supersymmetries and Infinite-Dimensional Algebras
A Book

by N. H. March

  • Publisher : Academic Press
  • Release : 2013-10-22
  • Pages : 628
  • ISBN : 1483288374
  • Language : En, Es, Fr & De
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Recent devopments, particularly in high-energy physics, have projected group theory and symmetry consideration into a central position in theoretical physics. These developments have taken physicists increasingly deeper into the fascinating world of pure mathematics. This work presents important mathematical developments of the last fifteen years in a form that is easy to comprehend and appreciate.

Infinite Dimensional Lie Algebras

Infinite Dimensional Lie Algebras
An Introduction

by Victor G. Kac

  • Publisher : Springer Science & Business Media
  • Release : 2013-11-11
  • Pages : 252
  • ISBN : 1475713827
  • Language : En, Es, Fr & De
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Dualities and Representations of Lie Superalgebras

Dualities and Representations of Lie Superalgebras
A Book

by Shun-Jen Cheng,Weiqiang Wang

  • Publisher : American Mathematical Soc.
  • Release : 2012
  • Pages : 302
  • ISBN : 0821891189
  • Language : En, Es, Fr & De
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This book gives a systematic account of the structure and representation theory of finite-dimensional complex Lie superalgebras of classical type and serves as a good introduction to representation theory of Lie superalgebras. Several folklore results are rigorously proved (and occasionally corrected in detail), sometimes with new proofs. Three important dualities are presented in the book, with the unifying theme of determining irreducible characters of Lie superalgebras. In order of increasing sophistication, they are Schur duality, Howe duality, and super duality. The combinatorics of symmetric functions is developed as needed in connections to Harish-Chandra homomorphism as well as irreducible characters for Lie superalgebras. Schur-Sergeev duality for the queer Lie superalgebra is presented from scratch with complete detail. Howe duality for Lie superalgebras is presented in book form for the first time. Super duality is a new approach developed in the past few years toward understanding the Bernstein-Gelfand-Gelfand category of modules for classical Lie superalgebras. Super duality relates the representation theory of classical Lie superalgebras directly to the representation theory of classical Lie algebras and thus gives a solution to the irreducible character problem of Lie superalgebras via the Kazhdan-Lusztig polynomials of classical Lie algebras.

Lie Groups and Invariant Theory

Lie Groups and Invariant Theory
A Book

by Ėrnest Borisovich Vinberg

  • Publisher : American Mathematical Soc.
  • Release : 2005
  • Pages : 270
  • ISBN : 9780821837337
  • Language : En, Es, Fr & De
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This volume, devoted to the 70th birthday of A. L. Onishchik, contains a collection of articles by participants in the Moscow Seminar on Lie Groups and Invariant Theory headed by E. B. Vinberg and A. L. Onishchik. The book is suitable for graduate students and researchers interested in Lie groups and related topics.

Ring Theory - Proceedings Of The Biennial Ohio State-denison Conference 1992

Ring Theory - Proceedings Of The Biennial Ohio State-denison Conference 1992
A Book

by Jain Surender K,Rizvi Syed Tariq

  • Publisher : World Scientific
  • Release : 1993-09-30
  • Pages : 392
  • ISBN : 9814553123
  • Language : En, Es, Fr & De
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This invaluable book deals with the many-electron theory of the solid state. Mastery of the material in it will equip the reader for research in areas such as high-temperature superconductors and the fractional quantum Hall effect. The whole book has been designed to provide the diligent reader with a wide variety of approaches to many-electron theory.The level of the book is suitable for research workers and higher-degree students in a number of disciplines, embracing theoretical physics, materials science and solid-state chemistry. It should be useful not only to theorists in these areas but also to experimental scientists who desire to orient their programmes to address outstanding questions raised by many-body theory.

Advances in Lie Superalgebras

Advances in Lie Superalgebras
A Book

by Maria Gorelik,Paolo Papi

  • Publisher : Springer Science & Business
  • Release : 2014-04-28
  • Pages : 280
  • ISBN : 3319029525
  • Language : En, Es, Fr & De
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The volume is the outcome of the conference "Lie superalgebras," which was held at the Istituto Nazionale di Alta Matematica, in 2012. The conference gathered many specialists in the subject, and the talks held provided comprehensive insights into the newest trends in research on Lie superalgebras (and related topics like vertex algebras, representation theory and supergeometry). The book contains contributions of many leading esperts in the field and provides a complete account of the newest trends in research on Lie Superalgebras.

Introduction to Vertex Operator Superalgebras and Their Modules

Introduction to Vertex Operator Superalgebras and Their Modules
A Book

by Xiaoping Xu

  • Publisher : Springer Science & Business Media
  • Release : 2013-03-09
  • Pages : 360
  • ISBN : 9401590974
  • Language : En, Es, Fr & De
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This book presents a systematic study on the structures of vertex operator superalgebras and their modules. Related theories of self-dual codes and lattices are included, as well as recent achievements on classifications of certain simple vertex operator superalgebras and their irreducible twisted modules, constructions of simple vertex operator superalgebras from graded associative algebras and their anti-involutions, self-dual codes and lattices. Audience: This book is of interest to researchers and graduate students in mathematics and mathematical physics.

Groups, Rings, Lie and Hopf Algebras

Groups, Rings, Lie and Hopf Algebras
A Book

by Y. Bahturin

  • Publisher : Springer Science & Business Media
  • Release : 2013-12-01
  • Pages : 241
  • ISBN : 1461302358
  • Language : En, Es, Fr & De
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The volume is almost entirely composed of the research and expository papers by the participants of the International Workshop "Groups, Rings, Lie and Hopf Algebras", which was held at the Memorial University of Newfoundland, St. John's, NF, Canada. All four areas from the title of the workshop are covered. In addition, some chapters touch upon the topics, which belong to two or more areas at the same time. Audience: The readership targeted includes researchers, graduate and senior undergraduate students in mathematics and its applications.

Developments and Trends in Infinite-Dimensional Lie Theory

Developments and Trends in Infinite-Dimensional Lie Theory
A Book

by Karl-Hermann Neeb,Arturo Pianzola

  • Publisher : Springer Science & Business Media
  • Release : 2010-10-17
  • Pages : 492
  • ISBN : 0817647414
  • Language : En, Es, Fr & De
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This collection of invited expository articles focuses on recent developments and trends in infinite-dimensional Lie theory, which has become one of the core areas of modern mathematics. The book is divided into three parts: infinite-dimensional Lie (super-)algebras, geometry of infinite-dimensional Lie (transformation) groups, and representation theory of infinite-dimensional Lie groups. Contributors: B. Allison, D. Beltiţă, W. Bertram, J. Faulkner, Ph. Gille, H. Glöckner, K.-H. Neeb, E. Neher, I. Penkov, A. Pianzola, D. Pickrell, T.S. Ratiu, N.R. Scheithauer, C. Schweigert, V. Serganova, K. Styrkas, K. Waldorf, and J.A. Wolf.

Infinite Dimensional Lie Algebras and Groups

Infinite Dimensional Lie Algebras and Groups
Proceedings of the Conference

by V G Kac

  • Publisher : World Scientific
  • Release : 1989-07-01
  • Pages : 640
  • ISBN : 9814663174
  • Language : En, Es, Fr & De
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Contents:Integrable Representation of Kac-Moody Algebras: Results and Open Problems (V Chari & A Pressley)Existence of Certain Components in the Tensor Product of Two Integrable Highest Weight Modules for Kac-Moody Algebras (SKumar)Frobenius Action on the B-Cohomology (O Mathieu)Certain Rank Two Subsystems of Kac-Moody Root Systems (J Morita)Lie Groups Associated to Kac-Moody Lie Algebras: An Analytic Approach (E Rodriguez-Carrington)Almost Split-K-Forms of Kac-Moody Algebras (G Rousseau)Global Representations of the Diffeomorphism Groups of the Circle (F Bien)Path Space Realization of the Basic Representation of An(1) (E Date et al)Boson-Fermion Correspondence Over (C De Concini et al)Classification of Modular Invariant Representations of Affine Algebras (V G Kac & M Wakimoto)Standard Monomial Theory for SL2 (V Lakshmibai & C S Seshadri)Some Results on Modular Invariant Representations (S Lu)Current Algebras in 3+1 Space-Time Dimensions (J Mickelson)Standard Representations of An(1) (M Primc)Representations of the Algebra Uq(sI(2)), q-Orthogonal Polynomials and Invariants of Links (A N Kirillov & N Yu Reshetikhin)Infinite Super Grassmannians and Super Plücker Equations (M J Bergvelt)Drinfeld-Sokolov Hierarchies and t-Functions (H J Imbens)Super Boson-Fermion Correspondence of Type B (V G Kac & J W van de Leur)Prym Varieties and Soliton Equations (T Shiota)Polynomial Solutions of the BKP Hierarchy and Projective Representations of Symmetric Groups (Y You)Toward Generalized Macdonald's Identities (D Bernard)Conformal Theories with Non-Linearly Extended Virasoro Symmetries and Lie Algebra Classification (A Bilal & J-LGervais)Extended Conformal Algebras from Kac-Moody Algebras (P Bouwknegt)Meromorphic Conformal Field Theory (P Goddard)Local Extensions of the U(1) Current Algebra and Their Positive Energy Representations (R R Paunov & I T Todorov)Conformal Field Theory on Moduli Family of Stable Curves with Gauge Symmetries (A Tsuchiya & Y Yamada) Readership: Mathematicians and mathematical physicists

AdS/CFT, (Super-)Virasoro, Affine (Super-)Algebras

AdS/CFT, (Super-)Virasoro, Affine (Super-)Algebras
A Book

by Vladimir K. Dobrev

  • Publisher : Walter de Gruyter GmbH & Co KG
  • Release : 2019-04-01
  • Pages : 246
  • ISBN : 3110609711
  • Language : En, Es, Fr & De
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The De Gruyter Studies in Mathematical Physics are devoted to the publication of monographs and high-level texts in mathematical physics. They cover topics and methods in fields of current interest, with an emphasis on didactical presentation. The series will enable readers to understand, apply and develop further, with sufficient rigor, mathematical methods to given problems in physics. For this reason, works with a few authors are preferred over edited volumes. The works in this series are aimed at advanced students and researchers in mathematical and theoretical physics. They can also serve as secondary reading for lectures and seminars at advanced levels.

Automorphic Forms and Lie Superalgebras

Automorphic Forms and Lie Superalgebras
A Book

by Urmie Ray

  • Publisher : Springer Science & Business Media
  • Release : 2007-03-06
  • Pages : 278
  • ISBN : 1402050100
  • Language : En, Es, Fr & De
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This book provides the reader with the tools to understand the ongoing classification and construction project of Lie superalgebras. It presents the material in as simple terms as possible. Coverage specifically details Borcherds-Kac-Moody superalgebras. The book examines the link between the above class of Lie superalgebras and automorphic form and explains their construction from lattice vertex algebras. It also includes all necessary background information.

Quantum Lie Theory

Quantum Lie Theory
A Multilinear Approach

by Vladislav Kharchenko

  • Publisher : Springer
  • Release : 2015-12-24
  • Pages : 302
  • ISBN : 3319227041
  • Language : En, Es, Fr & De
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This is an introduction to the mathematics behind the phrase “quantum Lie algebra”. The numerous attempts over the last 15-20 years to define a quantum Lie algebra as an elegant algebraic object with a binary “quantum” Lie bracket have not been widely accepted. In this book, an alternative approach is developed that includes multivariable operations. Among the problems discussed are the following: a PBW-type theorem; quantum deformations of Kac--Moody algebras; generic and symmetric quantum Lie operations; the Nichols algebras; the Gurevich--Manin Lie algebras; and Shestakov--Umirbaev operations for the Lie theory of nonassociative products. Opening with an introduction for beginners and continuing as a textbook for graduate students in physics and mathematics, the book can also be used as a reference by more advanced readers. With the exception of the introductory chapter, the content of this monograph has not previously appeared in book form.

Introduction to Vertex Operator Algebras and Their Representations

Introduction to Vertex Operator Algebras and Their Representations
A Book

by James Lepowsky,Haisheng Li

  • Publisher : Springer Science & Business Media
  • Release : 2012-12-06
  • Pages : 318
  • ISBN : 0817681868
  • Language : En, Es, Fr & De
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* Introduces the fundamental theory of vertex operator algebras and its basic techniques and examples. * Begins with a detailed presentation of the theoretical foundations and proceeds to a range of applications. * Includes a number of new, original results and brings fresh perspective to important works of many other researchers in algebra, lie theory, representation theory, string theory, quantum field theory, and other areas of math and physics.

Lie Superalgebras and Enveloping Algebras

Lie Superalgebras and Enveloping Algebras
A Book

by Ian Malcolm Musson

  • Publisher : American Mathematical Soc.
  • Release : 2012-04-04
  • Pages : 488
  • ISBN : 0821868675
  • Language : En, Es, Fr & De
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Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. This book develops the theory of Lie superalgebras, their enveloping algebras, and their representations. The book begins with five chapters on the basic properties of Lie superalgebras, including explicit constructions for all the classical simple Lie superalgebras. Borel subalgebras, which are more subtle in this setting, are studied and described. Contragredient Lie superalgebras are introduced, allowing a unified approach to several results, in particular to the existence of an invariant bilinear form on $\mathfrak{g}$. The enveloping algebra of a finite dimensional Lie superalgebra is studied as an extension of the enveloping algebra of the even part of the superalgebra. By developing general methods for studying such extensions, important information on the algebraic structure is obtained, particularly with regard to primitive ideals. Fundamental results, such as the Poincare-Birkhoff-Witt Theorem, are established. Representations of Lie superalgebras provide valuable tools for understanding the algebras themselves, as well as being of primary interest in applications to other fields. Two important classes of representations are the Verma modules and the finite dimensional representations. The fundamental results here include the Jantzen filtration, the Harish-Chandra homomorphism, the Sapovalov determinant, supersymmetric polynomials, and Schur-Weyl duality. Using these tools, the center can be explicitly described in the general linear and orthosymplectic cases. In an effort to make the presentation as self-contained as possible, some background material is included on Lie theory, ring theory, Hopf algebras, and combinatorics.