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Yang Baxter Equation And Quantum Enveloping Algebras

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Yang-Baxter Equation and Quantum Enveloping Algebras

Yang-Baxter Equation and Quantum Enveloping Algebras [Pdf/ePub] eBook Author: Zhong-Qi Ma
Editor: World Scientific
ISBN-10: 9814504262
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Yang-Baxter Equation and Quantum Enveloping Algebras by Zhong-Qi Ma Book Summary:

The exact solution of C N Yang's one-dimensional many-body problem with repulsive delta-function interactions and R J Baxter's eight-vertex statistical model are brilliant achievements in many-body statistical physics. A nonlinear equation, now known as the Yang-Baxter equation, is the key to the solution of both problems. The Yang-Baxter equation has also come to play an important role in such diverse topics as completely integrable statistical models, conformal and topological field theories, knots and links, braid groups and quantum enveloping algebras. This pioneering textbook attempts to make accessible results in this rapidly-growing area of research. The author presents the mathematical fundamentals at the outset, then develops an intuitive understanding of Hopf algebras, quantisation of Lie bialgebras and quantum enveloping algebras. The historical derivation of the Yang-Baxter equation from statistical models is recounted, and the interpretation and solution of the equation are systematically discussed. Throughout, emphasis is placed on acquiring calculation skills through physical understanding rather than achieving mathematical rigour. Originating from the author's own research experience and lectures, this book will prove both an excellent graduate text and a useful work of reference. Contents:Mathematical PreliminariesHistorical Origin of the Yang-Baxter EquationClassical Yang-Baxter EquationQuantum Enveloping AlgebrasQuantum Clebsch-Gordan CoefficientsSimple Yang-Baxter EquationTrigonometric and Rational SolutionsNon-Generic q Values Readership: Physicists and mathematicians. keywords:Jones Polynomials;Link Polynomials;Hopf Algebras;Quantization of Lie Bialgebras;Quantum Enveloping Algebras;Yang-Baxter Equation;Representations;Braid Groups;R-Matrices;Quantum Clebsch-Gordan Coefficients;Quantum Racah Coefficients;Trigonometric Solutions;Fusion

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Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach

Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach [Pdf/ePub] eBook Author: L.A. Lambe,D.E. Radford
Editor: Springer Science & Business Media
ISBN-10: 1461541093
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Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach by L.A. Lambe,D.E. Radford Book Summary:

Chapter 1 The algebraic prerequisites for the book are covered here and in the appendix. This chapter should be used as reference material and should be consulted as needed. A systematic treatment of algebras, coalgebras, bialgebras, Hopf algebras, and represen tations of these objects to the extent needed for the book is given. The material here not specifically cited can be found for the most part in [Sweedler, 1969] in one form or another, with a few exceptions. A great deal of emphasis is placed on the coalgebra which is the dual of n x n matrices over a field. This is the most basic example of a coalgebra for our purposes and is at the heart of most algebraic constructions described in this book. We have found pointed bialgebras useful in connection with solving the quantum Yang-Baxter equation. For this reason we develop their theory in some detail. The class of examples described in Chapter 6 in connection with the quantum double consists of pointed Hopf algebras. We note the quantized enveloping algebras described Hopf algebras. Thus for many reasons pointed bialgebras are elsewhere are pointed of fundamental interest in the study of the quantum Yang-Baxter equation and objects quantum groups.

Download or read Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). Chapter 1 The algebraic prerequisites for the book are covered here and in the appendix. This chapter should be used as reference material and should be consulted as needed. A systematic treatment of algebras, coalgebras, bialgebras, Hopf algebras, and represen tations of these objects to the extent needed for the book is given. The material here not specifically cited can be found for the most part in [Sweedler, 1969] in one form or another, with a few exceptions. A great deal of emphasis is placed on the coalgebra which is the dual of n x n matrices over a field. This is the most basic example of a coalgebra for our purposes and is at the heart of most algebraic constructions described in this book. We have found pointed bialgebras useful in connection with solving the quantum Yang-Baxter equation. For this reason we develop their theory in some detail. The class of examples described in Chapter 6 in connection with the quantum double consists of pointed Hopf algebras. We note the quantized enveloping algebras described Hopf algebras. Thus for many reasons pointed bialgebras are elsewhere are pointed of fundamental interest in the study of the quantum Yang-Baxter equation and objects quantum groups.


Quantum Groups

Quantum Groups [Pdf/ePub] eBook Author: Christian Kassel
Editor: Springer Science & Business Media
ISBN-10: 1461207835
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Quantum Groups by Christian Kassel Book Summary:

Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Coverage also focuses on Hopf algebras that produce solutions of the Yang-Baxter equation and provides an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations.

Download or read Quantum Groups book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Coverage also focuses on Hopf algebras that produce solutions of the Yang-Baxter equation and provides an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations.


Knots, Topology And Quantum Field Theory: Proceedings Of The 13th Johns Hopkins Workshop

Knots, Topology And Quantum Field Theory: Proceedings Of The 13th Johns Hopkins Workshop [Pdf/ePub] eBook Author: Lusanna Luca
Editor: World Scientific
ISBN-10: 9814611956
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Knots, Topology And Quantum Field Theory: Proceedings Of The 13th Johns Hopkins Workshop by Lusanna Luca Book Summary:

This book fills a gap in literature for the important interdisciplinary area of biochemical physics, adopting the chemist's view of this topic in the process. The present status of the theory of electron spin effects in fundamental processes such as spin exchange, dipole-dipole interactions, electron transfer, triplet-triplet energy transfer, and annihilation intersystem crossing is reviewed. These effects form a basis for the understanding of the molecular mechanisms essential to chemical and biological reactions including photosynthesis and magnetic field influence, and for the creation of advanced organic magnets and catalysts, as well as the development of new methods of studying the structural and molecular dynamics of biological and non-biological objects.

Download or read Knots, Topology And Quantum Field Theory: Proceedings Of The 13th Johns Hopkins Workshop book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). This book fills a gap in literature for the important interdisciplinary area of biochemical physics, adopting the chemist's view of this topic in the process. The present status of the theory of electron spin effects in fundamental processes such as spin exchange, dipole-dipole interactions, electron transfer, triplet-triplet energy transfer, and annihilation intersystem crossing is reviewed. These effects form a basis for the understanding of the molecular mechanisms essential to chemical and biological reactions including photosynthesis and magnetic field influence, and for the creation of advanced organic magnets and catalysts, as well as the development of new methods of studying the structural and molecular dynamics of biological and non-biological objects.


Yang-Baxter Equation in Integrable Systems

Yang-Baxter Equation in Integrable Systems [Pdf/ePub] eBook Author: Michio Jimbo
Editor: World Scientific
ISBN-10: 9814507067
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Yang-Baxter Equation in Integrable Systems by Michio Jimbo Book Summary:

' This volume will be the first reference book devoted specially to the Yang-Baxter equation. The subject relates to broad areas including solvable models in statistical mechanics, factorized S matrices, quantum inverse scattering method, quantum groups, knot theory and conformal field theory. The articles assembled here cover major works from the pioneering papers to classical Yang-Baxter equation, its quantization, variety of solutions, constructions and recent generalizations to higher genus solutions. Contents:Some Exact Results for the Many-Body Problem in One Dimension with Repulsive Delta-Function Interaction (C N Yang)S matrix for the One Dimensional N-Body Problem with Repulsive or Attractive δ-Function Interaction (C N Yang)Partition Function of the Eight-Vertex Lattice Model (R J Baxter)Solutions of the Classical Yang-Baxter Equation and Simple Lie Algebras (A A Belavin & V G Drinfel'd)Some Algebraic Structures Connected with the Yang-Baxter Equation (E K Sklyanin)Quantization of Lie Groups and Lie Algebras (L D Faddeev, N Yu Reshetikhin & L A Takhtajan)Families of Commuting Transfer Matrices in q-State Vertex Models in Non-Linear Integrable Systems — Classical Theory and Quantum Theory (J H H Perk & C L Schultz)Self-Dual Solutions of the Star-Triangle Relations in ZN Models (V A Fateev & A B Zamolodchikov)Solvable Lattice Models Related to the Vector Representation of Classical Simple Lie Algebras (M Jimbo, T Miwa & M Okado)Exactly Solvable SOS Models. II: Proof of the Star-Triangle and Combinatorial Identities (E Date et al.)New Solutions of the Star-Triangle Relations for the Chiral Potts Model (R J Baxter, J H H Perk & H Au-Yang)and other papers Readership: Physicists and mathematicians. Keywords:Yang-Baxter Equation;Star-Triangle Relation;Tetrahedron Equation;R Matrix;Classical R Matrix;Solvable Lattice Model;Factorized Scattering;Quantum Inverse Method;Quantum Groups;Lie Algebra“The collection serves a dual purpose: it provides the physicist or mathematician who works in a different field with an overview of the subject; furthermore, it provides those who work in the subject with a compendium of basic references put conveniently together in one volume.”Mathematical Reveiws “Thus the book gives a good survey of results in one of the hottest points of mathematical physics from the first hands.”Mathematics Abstracts “The second volume is such an excellent, representative collection of articles on the very rich field centered around the Yang-Baxter equation that is should have its place on the shelves of every good library. It is also warmly recommended for people wishing to join this active research area as well as for those who just want to learn the main developments.”Acta Sci. Math. (Szeged) '

Download or read Yang-Baxter Equation in Integrable Systems book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). ' This volume will be the first reference book devoted specially to the Yang-Baxter equation. The subject relates to broad areas including solvable models in statistical mechanics, factorized S matrices, quantum inverse scattering method, quantum groups, knot theory and conformal field theory. The articles assembled here cover major works from the pioneering papers to classical Yang-Baxter equation, its quantization, variety of solutions, constructions and recent generalizations to higher genus solutions. Contents:Some Exact Results for the Many-Body Problem in One Dimension with Repulsive Delta-Function Interaction (C N Yang)S matrix for the One Dimensional N-Body Problem with Repulsive or Attractive δ-Function Interaction (C N Yang)Partition Function of the Eight-Vertex Lattice Model (R J Baxter)Solutions of the Classical Yang-Baxter Equation and Simple Lie Algebras (A A Belavin & V G Drinfel'd)Some Algebraic Structures Connected with the Yang-Baxter Equation (E K Sklyanin)Quantization of Lie Groups and Lie Algebras (L D Faddeev, N Yu Reshetikhin & L A Takhtajan)Families of Commuting Transfer Matrices in q-State Vertex Models in Non-Linear Integrable Systems — Classical Theory and Quantum Theory (J H H Perk & C L Schultz)Self-Dual Solutions of the Star-Triangle Relations in ZN Models (V A Fateev & A B Zamolodchikov)Solvable Lattice Models Related to the Vector Representation of Classical Simple Lie Algebras (M Jimbo, T Miwa & M Okado)Exactly Solvable SOS Models. II: Proof of the Star-Triangle and Combinatorial Identities (E Date et al.)New Solutions of the Star-Triangle Relations for the Chiral Potts Model (R J Baxter, J H H Perk & H Au-Yang)and other papers Readership: Physicists and mathematicians. Keywords:Yang-Baxter Equation;Star-Triangle Relation;Tetrahedron Equation;R Matrix;Classical R Matrix;Solvable Lattice Model;Factorized Scattering;Quantum Inverse Method;Quantum Groups;Lie Algebra“The collection serves a dual purpose: it provides the physicist or mathematician who works in a different field with an overview of the subject; furthermore, it provides those who work in the subject with a compendium of basic references put conveniently together in one volume.”Mathematical Reveiws “Thus the book gives a good survey of results in one of the hottest points of mathematical physics from the first hands.”Mathematics Abstracts “The second volume is such an excellent, representative collection of articles on the very rich field centered around the Yang-Baxter equation that is should have its place on the shelves of every good library. It is also warmly recommended for people wishing to join this active research area as well as for those who just want to learn the main developments.”Acta Sci. Math. (Szeged) '


Deformation Theory and Quantum Groups with Applications to Mathematical Physics

Deformation Theory and Quantum Groups with Applications to Mathematical Physics [Pdf/ePub] eBook Author: Murray Gerstenhaber
Editor: American Mathematical Soc.
ISBN-10: 0821851411
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Deformation Theory and Quantum Groups with Applications to Mathematical Physics by Murray Gerstenhaber Book Summary:

Quantum groups are not groups at all, but special kinds of Hopf algebras of which the most important are closely related to Lie groups and play a central role in the statistical and wave mechanics of Baxter and Yang. Those occurring physically can be studied as essentially algebraic and closely related to the deformation theory of algebras (commutative, Lie, Hopf, and so on). One of the oldest forms of algebraic quantization amounts to the study of deformations of a commutative algebra $A$ (of classical observables) to a noncommutative algebra $A_h$ (of operators) with the infinitesimal deformation given by a Poisson bracket on the original algebra $A$. This volume grew out of an AMS-IMS-SIAM Joint Summer Research Conference, held in June 1990 at the University of Massachusetts at Amherst. The conference brought together leading researchers in the several areas mentioned and in areas such as ``$q$ special functions'', which have their origins in the last century but whose relevance to modern physics has only recently been understood. Among the advances taking place during the conference was Majid's reconstruction theorem for Drinfeld's quasi-Hopf algebras. Readers will appreciate this snapshot of some of the latest developments in the mathematics of quantum groups and deformation theory.

Download or read Deformation Theory and Quantum Groups with Applications to Mathematical Physics book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). Quantum groups are not groups at all, but special kinds of Hopf algebras of which the most important are closely related to Lie groups and play a central role in the statistical and wave mechanics of Baxter and Yang. Those occurring physically can be studied as essentially algebraic and closely related to the deformation theory of algebras (commutative, Lie, Hopf, and so on). One of the oldest forms of algebraic quantization amounts to the study of deformations of a commutative algebra $A$ (of classical observables) to a noncommutative algebra $A_h$ (of operators) with the infinitesimal deformation given by a Poisson bracket on the original algebra $A$. This volume grew out of an AMS-IMS-SIAM Joint Summer Research Conference, held in June 1990 at the University of Massachusetts at Amherst. The conference brought together leading researchers in the several areas mentioned and in areas such as ``$q$ special functions'', which have their origins in the last century but whose relevance to modern physics has only recently been understood. Among the advances taking place during the conference was Majid's reconstruction theorem for Drinfeld's quasi-Hopf algebras. Readers will appreciate this snapshot of some of the latest developments in the mathematics of quantum groups and deformation theory.


Noncommutative Rings

Noncommutative Rings [Pdf/ePub] eBook Author: Susan Montgomery,Lance Small
Editor: Springer Science & Business Media
ISBN-10: 1461397367
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Noncommutative Rings by Susan Montgomery,Lance Small Book Summary:

This volume collects some of the survey lectures delivered at the Micro program on Noncommutative Rings held at MSRI, July 10-21, 1989. While the program was concerned with recent advances in ring theory, it also had as an important component lectures on related areas of mathematics where ring theory might be expected to have an impact. Thus, there are lectures of S. P. Smith on quantum groups and Marc Ri effel on algebraic aspects of quantum field theory. Martin Lorenz and Don ald Passman consider in their lectures various aspects of crossed products: homological and K-theoretic to group actions. Kenneth Brown presents the "modern" theory of Noetherian rings and localization. These contributions as well as the others not presented here show that ring theory remains a vigorous and useful area. The planning and organization of the program were done by the under signed and the late Robert Warfield. His illness prevented his attendance at the meeting. It is to him we dedicate this volume. The organizers wish to extend their thanks to Irving Kaplansky, Director of MSRI, and the staff for all of their efforts in making this conference such a success. Susan Montgomery Lance Small vii NONCOMMUTATIVE RINGS TABLE OF CONTENTS PREFACE. . . . . . . . . . . . . . . . . . . . . . .. . . . Vll . . . . . . . . . . K. A. Brown THE REPRESENTATION THEORY OF NOETHERIAN RINGS 1 A. Joseph SOME RING THEORETIC TECHNIQUES AND OPEN PROBLEMS IN ENVELOPING ALGEBRAS. . . . . . . . . . . 27 . . . M. Lorenz CROSSED PRODUCTS: CHARACTERS, CYCLIC HOMOLOGY, AND GROTHENDIECK GROUPS . . . . . . . . . . . . . . . 69 . . . . . .

Download or read Noncommutative Rings book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). This volume collects some of the survey lectures delivered at the Micro program on Noncommutative Rings held at MSRI, July 10-21, 1989. While the program was concerned with recent advances in ring theory, it also had as an important component lectures on related areas of mathematics where ring theory might be expected to have an impact. Thus, there are lectures of S. P. Smith on quantum groups and Marc Ri effel on algebraic aspects of quantum field theory. Martin Lorenz and Don ald Passman consider in their lectures various aspects of crossed products: homological and K-theoretic to group actions. Kenneth Brown presents the "modern" theory of Noetherian rings and localization. These contributions as well as the others not presented here show that ring theory remains a vigorous and useful area. The planning and organization of the program were done by the under signed and the late Robert Warfield. His illness prevented his attendance at the meeting. It is to him we dedicate this volume. The organizers wish to extend their thanks to Irving Kaplansky, Director of MSRI, and the staff for all of their efforts in making this conference such a success. Susan Montgomery Lance Small vii NONCOMMUTATIVE RINGS TABLE OF CONTENTS PREFACE. . . . . . . . . . . . . . . . . . . . . . .. . . . Vll . . . . . . . . . . K. A. Brown THE REPRESENTATION THEORY OF NOETHERIAN RINGS 1 A. Joseph SOME RING THEORETIC TECHNIQUES AND OPEN PROBLEMS IN ENVELOPING ALGEBRAS. . . . . . . . . . . 27 . . . M. Lorenz CROSSED PRODUCTS: CHARACTERS, CYCLIC HOMOLOGY, AND GROTHENDIECK GROUPS . . . . . . . . . . . . . . . 69 . . . . . .


Algèbre Locale, Multiplicités

Algèbre Locale, Multiplicités [Pdf/ePub] eBook Author: Jean-Pierre Serre
Editor: Springer
ISBN-10: 3540371230
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Algèbre Locale, Multiplicités by Jean-Pierre Serre Book Summary:

Chapitre I. 1DIAUX PIlEMIEIS IT LOCALISATION I I. Wotationa et definitions I 2. Lemme de Bakay. . . . 2 3. Localisation • • • 4. Anneaux et 80dules noethiriens 2 5. Spectre•••••• 3 4 6. Le cas noetherien. 4 7. Ideaux pre. iers associe. Chapitre 11. OUTILS IT SOUTES A) Filtr·ations et graduations. 8 I. Anneaux et modules filtres • 8 2. Topologie definie par UDe filtration 9 10 3. Coapletion des modules filtres • • • II 4. Anneaux et modules graduis • • • • • 5. au tout redevient noethirien; filtrations ~-adiques. 15 20 6. Modules differentiels filtres•••••••••••• B) Polynoaes de Hilbert-SamueL ••••••••••• 26 I. Rappel sur les polynOmes Ii valeurs entieres•••• 26 27 2. Fonctions additives sur les categories de modules. 29 3. Le polynOme caractiristique de Hilbert 32 4. Les invariants de Hilbert-Samuel Chapitre 111. T1I£ORlE DE LA DDlE!ISION A) Dimension des extensions. entieres. 38 I. Definitions. • • • • • • • • • • • • 38 2. Le premier theore- de Cohen-Seidenberg. 39 3. Le second theoreme de Cohen-Seidenberg • 4I B) Dimension dans les anneaux noetheriens. 43 I. Dimension d'un module. • • • 43 2. Le cas semi-local noetherien 44 3. Syste. es de parametres 47 C) Anneaux normaux 48 I. caracterisation des anneaux normaux. 48 2. Proprietes des anneaux noraaux 51 3. Fermeture integrale. 53 D) Anneaux de polynomes. • • • • • 54 I.

Download or read Algèbre Locale, Multiplicités book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). Chapitre I. 1DIAUX PIlEMIEIS IT LOCALISATION I I. Wotationa et definitions I 2. Lemme de Bakay. . . . 2 3. Localisation • • • 4. Anneaux et 80dules noethiriens 2 5. Spectre•••••• 3 4 6. Le cas noetherien. 4 7. Ideaux pre. iers associe. Chapitre 11. OUTILS IT SOUTES A) Filtr·ations et graduations. 8 I. Anneaux et modules filtres • 8 2. Topologie definie par UDe filtration 9 10 3. Coapletion des modules filtres • • • II 4. Anneaux et modules graduis • • • • • 5. au tout redevient noethirien; filtrations ~-adiques. 15 20 6. Modules differentiels filtres•••••••••••• B) Polynoaes de Hilbert-SamueL ••••••••••• 26 I. Rappel sur les polynOmes Ii valeurs entieres•••• 26 27 2. Fonctions additives sur les categories de modules. 29 3. Le polynOme caractiristique de Hilbert 32 4. Les invariants de Hilbert-Samuel Chapitre 111. T1I£ORlE DE LA DDlE!ISION A) Dimension des extensions. entieres. 38 I. Definitions. • • • • • • • • • • • • 38 2. Le premier theore- de Cohen-Seidenberg. 39 3. Le second theoreme de Cohen-Seidenberg • 4I B) Dimension dans les anneaux noetheriens. 43 I. Dimension d'un module. • • • 43 2. Le cas semi-local noetherien 44 3. Syste. es de parametres 47 C) Anneaux normaux 48 I. caracterisation des anneaux normaux. 48 2. Proprietes des anneaux noraaux 51 3. Fermeture integrale. 53 D) Anneaux de polynomes. • • • • • 54 I.


Introduction to Quantum Groups and Crystal Bases

Introduction to Quantum Groups and Crystal Bases [Pdf/ePub] eBook Author: Jin Hong,Seok-Jin Kang
Editor: American Mathematical Soc.
ISBN-10: 0821828746
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Introduction to Quantum Groups and Crystal Bases by Jin Hong,Seok-Jin Kang Book Summary:

The notion of a ``quantum group'' was introduced by V.G. Dinfeld and M. Jimbo, independently, in their study of the quantum Yang-Baxter equation arising from 2-dimensional solvable lattice models. Quantum groups are certain families of Hopf algebras that are deformations of universal enveloping algebras of Kac-Moody algebras. And over the past 20 years, they have turned out to be the fundamental algebraic structure behind many branches of mathematics and mathematical physics, such as solvable lattice models in statistical mechanics, topological invariant theory of links and knots, representation theory of Kac-Moody algebras, representation theory of algebraic structures, topological quantum field theory, geometric representation theory, and $C^*$-algebras. In particular, the theory of ``crystal bases'' or ``canonical bases'' developed independently by M. Kashiwara and G. Lusztig provides a powerful combinatorial and geometric tool to study the representations of quantum groups. The purpose of this book is to provide an elementary introduction to the theory of quantum groups and crystal bases, focusing on the combinatorial aspects of the theory.

Download or read Introduction to Quantum Groups and Crystal Bases book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). The notion of a ``quantum group'' was introduced by V.G. Dinfeld and M. Jimbo, independently, in their study of the quantum Yang-Baxter equation arising from 2-dimensional solvable lattice models. Quantum groups are certain families of Hopf algebras that are deformations of universal enveloping algebras of Kac-Moody algebras. And over the past 20 years, they have turned out to be the fundamental algebraic structure behind many branches of mathematics and mathematical physics, such as solvable lattice models in statistical mechanics, topological invariant theory of links and knots, representation theory of Kac-Moody algebras, representation theory of algebraic structures, topological quantum field theory, geometric representation theory, and $C^*$-algebras. In particular, the theory of ``crystal bases'' or ``canonical bases'' developed independently by M. Kashiwara and G. Lusztig provides a powerful combinatorial and geometric tool to study the representations of quantum groups. The purpose of this book is to provide an elementary introduction to the theory of quantum groups and crystal bases, focusing on the combinatorial aspects of the theory.


Classical and Quantum Nonlinear Integrable Systems

Classical and Quantum Nonlinear Integrable Systems [Pdf/ePub] eBook Author: A Kundu
Editor: CRC Press
ISBN-10: 1420034618
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Classical and Quantum Nonlinear Integrable Systems by A Kundu Book Summary:

Covering both classical and quantum models, nonlinear integrable systems are of considerable theoretical and practical interest, with applications over a wide range of topics, including water waves, pin models, nonlinear optics, correlated electron systems, plasma physics, and reaction-diffusion processes. Comprising one part on classical theories

Download or read Classical and Quantum Nonlinear Integrable Systems book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). Covering both classical and quantum models, nonlinear integrable systems are of considerable theoretical and practical interest, with applications over a wide range of topics, including water waves, pin models, nonlinear optics, correlated electron systems, plasma physics, and reaction-diffusion processes. Comprising one part on classical theories


Algebraic Groups and Their Generalizations: Quantum and infinite dimensional methods

Algebraic Groups and Their Generalizations: Quantum and infinite dimensional methods [Pdf/ePub] eBook Author: William Joseph Haboush,Brian Parshall
Editor: American Mathematical Soc.
ISBN-10: 0821815415
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Algebraic Groups and Their Generalizations: Quantum and infinite dimensional methods by William Joseph Haboush,Brian Parshall Book Summary:

Proceedings of a research institute held at Pennsylvania State University, July 1991, focusing on quantum and infinite-dimensional methods of algebraic groups. Topics include perverse sheaves, finite Chevalley groups, the general theory of algebraic groups, representations, invariant theory, general

Download or read Algebraic Groups and Their Generalizations: Quantum and infinite dimensional methods book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). Proceedings of a research institute held at Pennsylvania State University, July 1991, focusing on quantum and infinite-dimensional methods of algebraic groups. Topics include perverse sheaves, finite Chevalley groups, the general theory of algebraic groups, representations, invariant theory, general


Quantum Group Symmetry and Q-Tensor Algebras

Quantum Group Symmetry and Q-Tensor Algebras [Pdf/ePub] eBook Author: L C Biedenharn,M A Lohe
Editor: World Scientific
ISBN-10: 9814500135
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Quantum Group Symmetry and Q-Tensor Algebras by L C Biedenharn,M A Lohe Book Summary:

Quantum groups are a generalization of the classical Lie groups and Lie algebras and provide a natural extension of the concept of symmetry fundamental to physics. This monograph is a survey of the major developments in quantum groups, using an original approach based on the fundamental concept of a tensor operator. Using this concept, properties of both the algebra and co-algebra are developed from a single uniform point of view, which is especially helpful for understanding the noncommuting co-ordinates of the quantum plane, which we interpret as elementary tensor operators. Representations of the q-deformed angular momentum group are discussed, including the case where q is a root of unity, and general results are obtained for all unitary quantum groups using the method of algebraic induction. Tensor operators are defined and discussed with examples, and a systematic treatment of the important (3j) series of operators is developed in detail. This book is a good reference for graduate students in physics and mathematics. Contents:Origins of Quantum GroupsRepresentations of Unitary Quantum GroupsTensor Operators in Quantum GroupsThe Dual Algebra and the Factor GroupQuantum Rotation MatricesQuantum Groups at Roots of UnityAlgebraic Induction of Quantum Group RepresentationsSpecial TopicsBibliographyIndex Readership: Physicists and mathematicians interested in symmetry techniques in physics. keywords:Quantum Groups;Quantum Algebras;Tensor Operators;Symmetries;Representations;q-Boson Operators;q-Clebsch-Gordan Coefficients;Vector Coherent States;Algebraic Induction;Weyl-Ordered Polynomials

Download or read Quantum Group Symmetry and Q-Tensor Algebras book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). Quantum groups are a generalization of the classical Lie groups and Lie algebras and provide a natural extension of the concept of symmetry fundamental to physics. This monograph is a survey of the major developments in quantum groups, using an original approach based on the fundamental concept of a tensor operator. Using this concept, properties of both the algebra and co-algebra are developed from a single uniform point of view, which is especially helpful for understanding the noncommuting co-ordinates of the quantum plane, which we interpret as elementary tensor operators. Representations of the q-deformed angular momentum group are discussed, including the case where q is a root of unity, and general results are obtained for all unitary quantum groups using the method of algebraic induction. Tensor operators are defined and discussed with examples, and a systematic treatment of the important (3j) series of operators is developed in detail. This book is a good reference for graduate students in physics and mathematics. Contents:Origins of Quantum GroupsRepresentations of Unitary Quantum GroupsTensor Operators in Quantum GroupsThe Dual Algebra and the Factor GroupQuantum Rotation MatricesQuantum Groups at Roots of UnityAlgebraic Induction of Quantum Group RepresentationsSpecial TopicsBibliographyIndex Readership: Physicists and mathematicians interested in symmetry techniques in physics. keywords:Quantum Groups;Quantum Algebras;Tensor Operators;Symmetries;Representations;q-Boson Operators;q-Clebsch-Gordan Coefficients;Vector Coherent States;Algebraic Induction;Weyl-Ordered Polynomials


Produits Tensoriels Topologiques et Espaces Nucleaires

Produits Tensoriels Topologiques et Espaces Nucleaires [Pdf/ePub] eBook Author: Alexandre Grothendieck
Editor: American Mathematical Soc.
ISBN-10: 0821812165
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Produits Tensoriels Topologiques et Espaces Nucleaires by Alexandre Grothendieck Book Summary:

Download or read Produits Tensoriels Topologiques et Espaces Nucleaires book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc).


Groupes et algèbres de Lie

Groupes et algèbres de Lie [Pdf/ePub] eBook Author: N. Bourbaki
Editor: Springer Science & Business Media
ISBN-10: 9783540344919
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Groupes et algèbres de Lie by N. Bourbaki Book Summary:

Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce troisième volume du Livre sur les Groupes et algèbres de Lie, neuvième Livre du traité, est consacré aux structures de systèmes de racines, de groupes de Coxeter et de systèmes de Tits, qui apparaissent naturellement dans l’étude des groupes de Lie analytique ou algébriques. Il comprend les chapitres: -Groupes de Coxeter et systèmes de Tits, -Groupes engenders par des reflexions, -Systèmes de racines. Ce volume contient également des planches décrivant les différents types de systèmes de raciness et des notes historiques. Ce volume est une réimpression de l’édition de 1968.

Download or read Groupes et algèbres de Lie book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce troisième volume du Livre sur les Groupes et algèbres de Lie, neuvième Livre du traité, est consacré aux structures de systèmes de racines, de groupes de Coxeter et de systèmes de Tits, qui apparaissent naturellement dans l’étude des groupes de Lie analytique ou algébriques. Il comprend les chapitres: -Groupes de Coxeter et systèmes de Tits, -Groupes engenders par des reflexions, -Systèmes de racines. Ce volume contient également des planches décrivant les différents types de systèmes de raciness et des notes historiques. Ce volume est une réimpression de l’édition de 1968.


Physics, Geometry and Topology

Physics, Geometry and Topology [Pdf/ePub] eBook Author: H.C. Lee
Editor: Springer Science & Business Media
ISBN-10: 1461538025
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Physics, Geometry and Topology by H.C. Lee Book Summary:

The Banff NATO Summer School was held August 14-25, 1989 at the Banff Cen tre, Banff, Albert, Canada. It was a combination of two venues: a summer school in the annual series of Summer School in Theoretical Physics spon sored by the Theoretical Physics Division, Canadian Association of Physi cists, and a NATO Advanced Study Institute. The Organizing Committee for the present school was composed of G. Kunstatter (University of Winnipeg), H.C. Lee (Chalk River Laboratories and University of Western Ontario), R. Kobes (University of Winnipeg), D.l. Toms (University of Newcastle Upon Tyne) and Y.S. Wu (University of Utah). Thanks to the group of lecturers (see Contents) and the timeliness of the courses given, the school, entitled PHYSICS, GEOMETRY AND TOPOLOGY, was popular from the very outset. The number of applications outstripped the 90 places of accommodation reserved at the Banff Centre soon after the school was announced. As the eventual total number of participants was increased to 170, it was still necessary to tum away many deserving applicants. In accordance with the spirit of the school, the geometrical and topologi cal properties in each of the wide ranging topics covered by the lectures were emphasized. A recurring theme in a number of the lectures is the Yang-Baxter relation which characterizes a very large class of integrable systems including: many state models, two-dimensional conformal field theory, quantum field theory and quantum gravity in 2 + I dimensions.

Download or read Physics, Geometry and Topology book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). The Banff NATO Summer School was held August 14-25, 1989 at the Banff Cen tre, Banff, Albert, Canada. It was a combination of two venues: a summer school in the annual series of Summer School in Theoretical Physics spon sored by the Theoretical Physics Division, Canadian Association of Physi cists, and a NATO Advanced Study Institute. The Organizing Committee for the present school was composed of G. Kunstatter (University of Winnipeg), H.C. Lee (Chalk River Laboratories and University of Western Ontario), R. Kobes (University of Winnipeg), D.l. Toms (University of Newcastle Upon Tyne) and Y.S. Wu (University of Utah). Thanks to the group of lecturers (see Contents) and the timeliness of the courses given, the school, entitled PHYSICS, GEOMETRY AND TOPOLOGY, was popular from the very outset. The number of applications outstripped the 90 places of accommodation reserved at the Banff Centre soon after the school was announced. As the eventual total number of participants was increased to 170, it was still necessary to tum away many deserving applicants. In accordance with the spirit of the school, the geometrical and topologi cal properties in each of the wide ranging topics covered by the lectures were emphasized. A recurring theme in a number of the lectures is the Yang-Baxter relation which characterizes a very large class of integrable systems including: many state models, two-dimensional conformal field theory, quantum field theory and quantum gravity in 2 + I dimensions.


Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups

Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups [Pdf/ePub] eBook Author: Paul Sally
Editor: American Mathematical Soc.
ISBN-10: 0821851861
Size: 1848 kb
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Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups by Paul Sally Book Summary:

This book contains papers presented by speakers at the AMS-IMS-SIAM Joint Summer Research Conference on Conformal Field Theory, Topological Field Theory and Quantum Groups, held at Mount Holyoke College in June 1992. One group of papers deals with one aspect of conformal field theory, namely, vertex operator algebras or superalgebras and their representations. Another group deals with various aspects of quantum groups. Other topics covered include the theory of knots in three-manifolds, symplectic geometry, and tensor products. This book provides an excellent view of some of the latest developments in this growing field of research.

Download or read Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). This book contains papers presented by speakers at the AMS-IMS-SIAM Joint Summer Research Conference on Conformal Field Theory, Topological Field Theory and Quantum Groups, held at Mount Holyoke College in June 1992. One group of papers deals with one aspect of conformal field theory, namely, vertex operator algebras or superalgebras and their representations. Another group deals with various aspects of quantum groups. Other topics covered include the theory of knots in three-manifolds, symplectic geometry, and tensor products. This book provides an excellent view of some of the latest developments in this growing field of research.


La géométrie et le quantique

La géométrie et le quantique [Pdf/ePub] eBook Author: Alain Connes
Editor: Cnrs
ISBN-10: 2271127726
Size: 1622 kb
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La géométrie et le quantique by Alain Connes Book Summary:

En 1637, Descartes révolutionne la manière que l'on a de faire de la géométrie : en associant à chaque point de l'espace trois coordonnées, il pose les bases de la géométrie algébrique. Cette géométrie est dite " commutative " : le produit de deux quantités ne dépend pas de l'ordre des termes, et A × B = B × A. Cette propriété est fondamentale, l'ensemble de l'édifice mathématique en dépend. Mais au début du XXe siècle, la découverte du monde quantique vient tout bouleverser. L'espace géométrique des états d'un système microscopique, un atome par exemple, s'enrichit de nouvelles propriétés, qui ne commutent plus. Il faut donc adapter l'ensemble des outils mathématiques. Cette nouvelle géométrie, dite " non commutative ", devenue essentielle à la recherche en physique, a été développée par Alain Connes. En un texte court, vif et fascinant, ce grand mathématicien nous introduit à la poésie de sa discipline.

Download or read La géométrie et le quantique book by clicking button below to visit the book download website. There are multiple format available for you to choose (Pdf, ePub, Doc). En 1637, Descartes révolutionne la manière que l'on a de faire de la géométrie : en associant à chaque point de l'espace trois coordonnées, il pose les bases de la géométrie algébrique. Cette géométrie est dite " commutative " : le produit de deux quantités ne dépend pas de l'ordre des termes, et A × B = B × A. Cette propriété est fondamentale, l'ensemble de l'édifice mathématique en dépend. Mais au début du XXe siècle, la découverte du monde quantique vient tout bouleverser. L'espace géométrique des états d'un système microscopique, un atome par exemple, s'enrichit de nouvelles propriétés, qui ne commutent plus. Il faut donc adapter l'ensemble des outils mathématiques. Cette nouvelle géométrie, dite " non commutative ", devenue essentielle à la recherche en physique, a été développée par Alain Connes. En un texte court, vif et fascinant, ce grand mathématicien nous introduit à la poésie de sa discipline.