Algebraic Analysis of Solvable Lattice Models
Algebraic Analysis of Solvable Lattice Models

Author: Michio Jimbo

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 180

ISBN-13: 0821803204

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Based on the NSF-CBMS Regional Conference lectures presented by Miwa in June 1993, this book surveys recent developments in the interplay between solvable lattice models in statistical mechanics and representation theory of quantum affine algebras. Because results in this subject were scattered in the literature, this book fills the need for a systematic account, focusing attention on fundamentals without assuming prior knowledge about lattice models or representation theory. After a brief account of basic principles in statistical mechanics, the authors discuss the standard subjects concerning solvable lattice models in statistical mechanics, the main examples being the spin 1/2 XXZ chain and the six-vertex model. The book goes on to introduce the main objects of study, the corner transfer matrices and the vertex operators, and discusses some of their aspects from the viewpoint of physics. Once the physical motivations are in place, the authors return to the mathematics, covering the Frenkel-Jing bosonization of a certain module, formulas for the vertex operators using bosons, the role of representation theory, and correlation functions and form factors. The limit of the XXX model is briefly discussed, and the book closes with a discussion of other types of models and related works.

Yang-Baxter Equation in Integrable Systems
Yang-Baxter Equation in Integrable Systems

Author: Michio Jimbo

Publisher: World Scientific

Published: 1990

Total Pages: 740

ISBN-13: 9789810201203

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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.

Theta Functions, Bowdoin 1987
Theta Functions, Bowdoin 1987

Author: Leon Ehrenpreis

Publisher: American Mathematical Soc.

Published: 1989

Total Pages: 730

ISBN-13: 0821814834

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During his long and productive career, Salomon Bochner worked in a variety of different areas of mathematics. This four part set brings together his collected papers, illustrating the range and depth of his mathematical interests. The books are available either individually or as a set.

Braid Group, Knot Theory, and Statistical Mechanics II
Braid Group, Knot Theory, and Statistical Mechanics II

Author: Chen Ning Yang

Publisher: World Scientific

Published: 1994

Total Pages: 496

ISBN-13: 9789810215248

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The present volume is an updated version of the book edited by C N Yang and M L Ge on the topics of braid groups and knot theory, which are related to statistical mechanics. This book is based on the 1989 volume but has new material included and new contributors.

A Guide to Quantum Groups
A Guide to Quantum Groups

Author: Vyjayanthi Chari

Publisher: Cambridge University Press

Published: 1995-07-27

Total Pages: 672

ISBN-13: 9780521558846

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Since they first arose in the 1970s and early 1980s, quantum groups have proved to be of great interest to mathematicians and theoretical physicists. The theory of quantum groups is now well established as a fascinating chapter of representation theory, and has thrown new light on many different topics, notably low-dimensional topology and conformal field theory. The goal of this book is to give a comprehensive view of quantum groups and their applications. The authors build on a self-contained account of the foundations of the subject and go on to treat the more advanced aspects concisely and with detailed references to the literature. Thus this book can serve both as an introduction for the newcomer, and as a guide for the more experienced reader. All who have an interest in the subject will welcome this unique treatment of quantum groups.

Topological Field Theory, Primitive Forms and Related Topics
Topological Field Theory, Primitive Forms and Related Topics

Author: Masaki Kashiwara

Publisher: Springer Science & Business Media

Published: 1998-12

Total Pages: 512

ISBN-13: 9780817639754

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As the interaction of mathematics and theoretical physics continues to intensify, the theories developed in mathematics are being applied to physics, and conversely. This book centers around the theory of primitive forms which currently plays an active and key role in topological field theory (theoretical physics), but was originally developed as a mathematical notion to define a "good period mapping" for a family of analytic structures. The invited papers in this volume are expository in nature by participants of the Taniguchi Symposium on "Topological Field Theory, Primitive Forms and Related Topics" and the RIMS Symposium bearing the same title, both held in Kyoto. The papers reflect the broad research of some of the world's leading mathematical physicists, and should serve as an excellent resource for researchers as well as graduate students of both disciplines.

Topological Field Theory, Primitive Forms and Related Topics
Topological Field Theory, Primitive Forms and Related Topics

Author: A. Kashiwara

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 492

ISBN-13: 1461207053

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As the interaction of mathematics and theoretical physics continues to intensify, the theories developed in mathematics are being applied to physics, and conversely. This book centers around the theory of primitive forms which currently plays an active and key role in topological field theory (theoretical physics), but was originally developed as a mathematical notion to define a "good period mapping" for a family of analytic structures. The invited papers in this volume are expository in nature by participants of the Taniguchi Symposium on "Topological Field Theory, Primitive Forms and Related Topics" and the RIMS Symposium bearing the same title, both held in Kyoto. The papers reflect the broad research of some of the world's leading mathematical physicists, and should serve as an excellent resource for researchers as well as graduate students of both disciplines.

Representations of Quantum Algebras and Combinatorics of Young Tableaux
Representations of Quantum Algebras and Combinatorics of Young Tableaux

Author: Susumu Ariki

Publisher: American Mathematical Soc.

Published: 2002

Total Pages: 169

ISBN-13: 0821832328

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This book contains most of the nonstandard material necessary to get acquainted with this new rapidly developing area. It can be used as a good entry point into the study of representations of quantum groups. Among several tools used in studying representations of quantum groups (or quantum algebras) are the notions of Kashiwara's crystal bases and Lusztig's canonical bases. Mixing both approaches allows us to use a combinatorial approach to representations of quantum groups and toapply the theory to representations of Hecke algebras. The primary goal of this book is to introduce the representation theory of quantum groups using quantum groups of type $A {r-1 {(1) $ as a main example. The corresponding combinatorics, developed by Misra and Miwa, turns out to be thecombinatorics of Young tableaux. The second goal of this book is to explain the proof of the (generalized) Leclerc-Lascoux-Thibon conjecture. This conjecture, which is now a theorem, is an important breakthrough in the modular representation theory of the Hecke algebras of classical type. The book is suitable for graduate students and research mathematicians interested in representation theory of algebraic groups and quantum groups, the theory of Hecke algebras, algebraic combinatorics, andrelated fields.

Infinite-Dimensional Lie Algebras
Infinite-Dimensional Lie Algebras

Author: Victor G. Kac

Publisher: Cambridge University Press

Published: 1990

Total Pages: 428

ISBN-13: 9780521466936

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The third, substantially revised edition of a monograph concerned with Kac-Moody algebras, a particular class of infinite-dimensional Lie albegras, and their representations, based on courses given over a number of years at MIT and in Paris.