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Representation Theory of the Virasoro Algebra

Author: Kenji Iohara

Publisher: Springer Science & Business Media

Published: 2010-11-12

Total Pages: 482

ISBN-13: 0857291602

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The Virasoro algebra is an infinite dimensional Lie algebra that plays an increasingly important role in mathematics and theoretical physics. This book describes some fundamental facts about the representation theory of the Virasoro algebra in a self-contained manner. Topics include the structure of Verma modules and Fock modules, the classification of (unitarizable) Harish-Chandra modules, tilting equivalence, and the rational vertex operator algebras associated to the so-called minimal series representations. Covering a wide range of material, this book has three appendices which provide background information required for some of the chapters. The authors organize fundamental results in a unified way and refine existing proofs. For instance in chapter three, a generalization of Jantzen filtration is reformulated in an algebraic manner, and geometric interpretation is provided. Statements, widely believed to be true, are collated, and results which are known but not verified are proven, such as the corrected structure theorem of Fock modules in chapter eight. This book will be of interest to a wide range of mathematicians and physicists from the level of graduate students to researchers.

Book Synopsis Representation Theory of the Virasoro Algebra by : Kenji Iohara

Download or read book Representation Theory of the Virasoro Algebra written by Kenji Iohara and published by Springer Science & Business Media. This book was released on 2010-11-12 with total page 482 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Virasoro algebra is an infinite dimensional Lie algebra that plays an increasingly important role in mathematics and theoretical physics. This book describes some fundamental facts about the representation theory of the Virasoro algebra in a self-contained manner. Topics include the structure of Verma modules and Fock modules, the classification of (unitarizable) Harish-Chandra modules, tilting equivalence, and the rational vertex operator algebras associated to the so-called minimal series representations. Covering a wide range of material, this book has three appendices which provide background information required for some of the chapters. The authors organize fundamental results in a unified way and refine existing proofs. For instance in chapter three, a generalization of Jantzen filtration is reformulated in an algebraic manner, and geometric interpretation is provided. Statements, widely believed to be true, are collated, and results which are known but not verified are proven, such as the corrected structure theorem of Fock modules in chapter eight. This book will be of interest to a wide range of mathematicians and physicists from the level of graduate students to researchers.

Vertex Operators in Mathematics and Physics

Author: J. Lepowsky

Publisher: Springer Science & Business Media

Published: 2013-03-08

Total Pages: 484

ISBN-13: 146139550X

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James Lepowsky t The search for symmetry in nature has for a long time provided representation theory with perhaps its chief motivation. According to the standard approach of Lie theory, one looks for infinitesimal symmetry -- Lie algebras of operators or concrete realizations of abstract Lie algebras. A central theme in this volume is the construction of affine Lie algebras using formal differential operators called vertex operators, which originally appeared in the dual-string theory. Since the precise description of vertex operators, in both mathematical and physical settings, requires a fair amount of notation, we do not attempt it in this introduction. Instead we refer the reader to the papers of Mandelstam, Goddard-Olive, Lepowsky-Wilson and Frenkel-Lepowsky-Meurman. We have tried to maintain consistency of terminology and to some extent notation in the articles herein. To help the reader we shall review some of the terminology. We also thought it might be useful to supplement an earlier fairly detailed exposition of ours [37] with a brief historical account of vertex operators in mathematics and their connection with affine algebras. Since we were involved in the development of the subject, the reader should be advised that what follows reflects our own understanding. For another view, see [29].1 t Partially supported by the National Science Foundation through the Mathematical Sciences Research Institute and NSF Grant MCS 83-01664. 1 We would like to thank Igor Frenkel for his valuable comments on the first draft of this introduction.

Book Synopsis Vertex Operators in Mathematics and Physics by : J. Lepowsky

Download or read book Vertex Operators in Mathematics and Physics written by J. Lepowsky and published by Springer Science & Business Media. This book was released on 2013-03-08 with total page 484 pages. Available in PDF, EPUB and Kindle. Book excerpt: James Lepowsky t The search for symmetry in nature has for a long time provided representation theory with perhaps its chief motivation. According to the standard approach of Lie theory, one looks for infinitesimal symmetry -- Lie algebras of operators or concrete realizations of abstract Lie algebras. A central theme in this volume is the construction of affine Lie algebras using formal differential operators called vertex operators, which originally appeared in the dual-string theory. Since the precise description of vertex operators, in both mathematical and physical settings, requires a fair amount of notation, we do not attempt it in this introduction. Instead we refer the reader to the papers of Mandelstam, Goddard-Olive, Lepowsky-Wilson and Frenkel-Lepowsky-Meurman. We have tried to maintain consistency of terminology and to some extent notation in the articles herein. To help the reader we shall review some of the terminology. We also thought it might be useful to supplement an earlier fairly detailed exposition of ours [37] with a brief historical account of vertex operators in mathematics and their connection with affine algebras. Since we were involved in the development of the subject, the reader should be advised that what follows reflects our own understanding. For another view, see [29].1 t Partially supported by the National Science Foundation through the Mathematical Sciences Research Institute and NSF Grant MCS 83-01664. 1 We would like to thank Igor Frenkel for his valuable comments on the first draft of this introduction.

Lectures on Representation Theory

Author: Jing-Song Huang

Publisher: World Scientific

Published: 1999

Total Pages: 206

ISBN-13: 9789810237257

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This book is an expanded version of the lectures given at the Nankai Mathematical Summer School in 1997. It provides an introduction to Lie groups, Lie algebras and their representations as well as introduces some directions of current research for graduate students who have little specialized knowledge in representation theory. It only assumes that the reader has a good knowledge of linear algebra and some basic knowledge of abstract algebra.Parts I-III of the book cover the relatively elementary material of representation theory of finite groups, simple Lie algebras and compact Lie groups. These theories are natural continuation of linear algebra. The last chapter of Part III includes some recent results on extension of Weyl's construction to exceptional groups. Part IV covers some advanced material on infinite-dimensional representations of non-compact groups such as the orbit method, minimal representations and dual pair correspondences, which introduces some directions of the current research in representation theory.

Book Synopsis Lectures on Representation Theory by : Jing-Song Huang

Download or read book Lectures on Representation Theory written by Jing-Song Huang and published by World Scientific. This book was released on 1999 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an expanded version of the lectures given at the Nankai Mathematical Summer School in 1997. It provides an introduction to Lie groups, Lie algebras and their representations as well as introduces some directions of current research for graduate students who have little specialized knowledge in representation theory. It only assumes that the reader has a good knowledge of linear algebra and some basic knowledge of abstract algebra.Parts I-III of the book cover the relatively elementary material of representation theory of finite groups, simple Lie algebras and compact Lie groups. These theories are natural continuation of linear algebra. The last chapter of Part III includes some recent results on extension of Weyl's construction to exceptional groups. Part IV covers some advanced material on infinite-dimensional representations of non-compact groups such as the orbit method, minimal representations and dual pair correspondences, which introduces some directions of the current research in representation theory.

The Schrödinger-Virasoro Algebra

Author: Jérémie Unterberger

Publisher: Springer Science & Business Media

Published: 2011-10-25

Total Pages: 334

ISBN-13: 3642227171

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This monograph provides the first up-to-date and self-contained presentation of a recently discovered mathematical structure—the Schrödinger-Virasoro algebra. Just as Poincaré invariance or conformal (Virasoro) invariance play a key rôle in understanding, respectively, elementary particles and two-dimensional equilibrium statistical physics, this algebra of non-relativistic conformal symmetries may be expected to apply itself naturally to the study of some models of non-equilibrium statistical physics, or more specifically in the context of recent developments related to the non-relativistic AdS/CFT correspondence. The study of the structure of this infinite-dimensional Lie algebra touches upon topics as various as statistical physics, vertex algebras, Poisson geometry, integrable systems and supergeometry as well as representation theory, the cohomology of infinite-dimensional Lie algebras, and the spectral theory of Schrödinger operators.

Book Synopsis The Schrödinger-Virasoro Algebra by : Jérémie Unterberger

Download or read book The Schrödinger-Virasoro Algebra written by Jérémie Unterberger and published by Springer Science & Business Media. This book was released on 2011-10-25 with total page 334 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides the first up-to-date and self-contained presentation of a recently discovered mathematical structure—the Schrödinger-Virasoro algebra. Just as Poincaré invariance or conformal (Virasoro) invariance play a key rôle in understanding, respectively, elementary particles and two-dimensional equilibrium statistical physics, this algebra of non-relativistic conformal symmetries may be expected to apply itself naturally to the study of some models of non-equilibrium statistical physics, or more specifically in the context of recent developments related to the non-relativistic AdS/CFT correspondence. The study of the structure of this infinite-dimensional Lie algebra touches upon topics as various as statistical physics, vertex algebras, Poisson geometry, integrable systems and supergeometry as well as representation theory, the cohomology of infinite-dimensional Lie algebras, and the spectral theory of Schrödinger operators.

Topics in Representation Theory

Author:

Publisher: American Mathematical Soc.

Published:

Total Pages: 266

ISBN-13: 9780821873076

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Almost every major mathematical theory, from 19th century classical analysis and geometry to the newest abstract constructions of category theory, have recently acquired a "physical flavour". In the case of representation theory, two new areas of mathematical physics - the theory of completely integrable systems and string theory - have had a great influence. In addition, the idea of supersymmetry has become a general mathematical principle that has had important ramifications in representation theory. Together with this wave of new connections and new trends in representation theory, more traditional activity, dealing mostly with the study of classical objects, has also flourished. The papers in this volume were written by members of the seminar on representation theory at Moscow University, which has been running continuously since 1961. The papers reflect some of the new influences seen in representation theory today. Among the topics included are representation theory of "large" groups, indecomposable representations of the affine unimodular group of the plane, dual objects for certain real reductive Lie groups, and geometrical interpretations of a certain infinite-dimensional Lie algebra.

Book Synopsis Topics in Representation Theory by :

Download or read book Topics in Representation Theory written by and published by American Mathematical Soc.. This book was released on with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: Almost every major mathematical theory, from 19th century classical analysis and geometry to the newest abstract constructions of category theory, have recently acquired a "physical flavour". In the case of representation theory, two new areas of mathematical physics - the theory of completely integrable systems and string theory - have had a great influence. In addition, the idea of supersymmetry has become a general mathematical principle that has had important ramifications in representation theory. Together with this wave of new connections and new trends in representation theory, more traditional activity, dealing mostly with the study of classical objects, has also flourished. The papers in this volume were written by members of the seminar on representation theory at Moscow University, which has been running continuously since 1961. The papers reflect some of the new influences seen in representation theory today. Among the topics included are representation theory of "large" groups, indecomposable representations of the affine unimodular group of the plane, dual objects for certain real reductive Lie groups, and geometrical interpretations of a certain infinite-dimensional Lie algebra.

Representation Theory and Noncommutative Harmonic Analysis I

Author: A.A. Kirillov

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 241

ISBN-13: 3662030020

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This two-part survey provides a short review of the classical part of representation theory, carefully exposing the structure of the theory without overwhelming readers with details, and deals with representations of Virasoro and Kac-Moody algebra. It presents a wealth of recent results on representations of infinite-dimensional groups.

Book Synopsis Representation Theory and Noncommutative Harmonic Analysis I by : A.A. Kirillov

Download or read book Representation Theory and Noncommutative Harmonic Analysis I written by A.A. Kirillov and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 241 pages. Available in PDF, EPUB and Kindle. Book excerpt: This two-part survey provides a short review of the classical part of representation theory, carefully exposing the structure of the theory without overwhelming readers with details, and deals with representations of Virasoro and Kac-Moody algebra. It presents a wealth of recent results on representations of infinite-dimensional groups.

Lectures On Infinite-dimensional Lie Algebra

Author: Minoru Wakimoto

Publisher: World Scientific

Published: 2001-10-26

Total Pages: 456

ISBN-13: 9814494003

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The representation theory of affine Lie algebras has been developed in close connection with various areas of mathematics and mathematical physics in the last two decades. There are three excellent books on it, written by Victor G Kac. This book begins with a survey and review of the material treated in Kac's books. In particular, modular invariance and conformal invariance are explained in more detail. The book then goes further, dealing with some of the recent topics involving the representation theory of affine Lie algebras. Since these topics are important not only in themselves but also in their application to some areas of mathematics and mathematical physics, the book expounds them with examples and detailed calculations.

Book Synopsis Lectures On Infinite-dimensional Lie Algebra by : Minoru Wakimoto

Download or read book Lectures On Infinite-dimensional Lie Algebra written by Minoru Wakimoto and published by World Scientific. This book was released on 2001-10-26 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: The representation theory of affine Lie algebras has been developed in close connection with various areas of mathematics and mathematical physics in the last two decades. There are three excellent books on it, written by Victor G Kac. This book begins with a survey and review of the material treated in Kac's books. In particular, modular invariance and conformal invariance are explained in more detail. The book then goes further, dealing with some of the recent topics involving the representation theory of affine Lie algebras. Since these topics are important not only in themselves but also in their application to some areas of mathematics and mathematical physics, the book expounds them with examples and detailed calculations.

Introduction to Representation Theory

Author: Pavel I. Etingof

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 240

ISBN-13: 0821853511

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Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum field theory. The goal of this book is to give a ``holistic'' introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints. The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra.

Book Synopsis Introduction to Representation Theory by : Pavel I. Etingof

Download or read book Introduction to Representation Theory written by Pavel I. Etingof and published by American Mathematical Soc.. This book was released on 2011 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum field theory. The goal of this book is to give a ``holistic'' introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints. The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra.

Operators and Representation Theory

Author: P.E.T. Jorgensen

Publisher: Elsevier

Published: 1987-12-01

Total Pages: 336

ISBN-13: 9780080872582

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Historically, operator theory and representation theory both originated with the advent of quantum mechanics. The interplay between the subjects has been and still is active in a variety of areas. This volume focuses on representations of the universal enveloping algebra, covariant representations in general, and infinite-dimensional Lie algebras in particular. It also provides new applications of recent results on integrability of finite-dimensional Lie algebras. As a central theme, it is shown that a number of recent developments in operator algebras may be handled in a particularly elegant manner by the use of Lie algebras, extensions, and projective representations. In several cases, this Lie algebraic approach to questions in mathematical physics and C*-algebra theory is new; for example, the Lie algebraic treatment of the spectral theory of curved magnetic field Hamiltonians, the treatment of irrational rotation type algebras, and the Virasoro algebra. Also examined are C*-algebraic methods used (in non-traditional ways) in the study of representations of infinite-dimensional Lie algebras and their extensions, and the methods developed by A. Connes and M.A. Rieffel for the study of the Yang-Mills problem. Cutting across traditional separations between fields of specialization, the book addresses a broad audience of graduate students and researchers.

Book Synopsis Operators and Representation Theory by : P.E.T. Jorgensen

Download or read book Operators and Representation Theory written by P.E.T. Jorgensen and published by Elsevier. This book was released on 1987-12-01 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: Historically, operator theory and representation theory both originated with the advent of quantum mechanics. The interplay between the subjects has been and still is active in a variety of areas. This volume focuses on representations of the universal enveloping algebra, covariant representations in general, and infinite-dimensional Lie algebras in particular. It also provides new applications of recent results on integrability of finite-dimensional Lie algebras. As a central theme, it is shown that a number of recent developments in operator algebras may be handled in a particularly elegant manner by the use of Lie algebras, extensions, and projective representations. In several cases, this Lie algebraic approach to questions in mathematical physics and C*-algebra theory is new; for example, the Lie algebraic treatment of the spectral theory of curved magnetic field Hamiltonians, the treatment of irrational rotation type algebras, and the Virasoro algebra. Also examined are C*-algebraic methods used (in non-traditional ways) in the study of representations of infinite-dimensional Lie algebras and their extensions, and the methods developed by A. Connes and M.A. Rieffel for the study of the Yang-Mills problem. Cutting across traditional separations between fields of specialization, the book addresses a broad audience of graduate students and researchers.

Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras

Author:

Publisher:

Published:

Total Pages:

ISBN-13: 9814507725

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Book Synopsis Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras by :

Download or read book Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras written by and published by . This book was released on with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: