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

Author: L.A. Lambe

Publisher: Springer Science & Business Media

Published: 2013-11-22

Total Pages: 314

ISBN-13: 1461541093

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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.

Yang-Baxter Equation and Quantum Enveloping Algebras
Yang-Baxter Equation and Quantum Enveloping Algebras

Author: Zhongqi Ma

Publisher: World Scientific

Published: 1993

Total Pages: 336

ISBN-13: 9789810213831

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This is the first-ever textbook on the Yang-Baxter equation. A key nonlinear equation for solving two important models in many-body statistical theory - the many-body problem in one dimension with repulsive delta-function interaction presented by Professor Baxter in 1972 - it has become one of the main concerns of physicists and mathematicians in the last ten years. A textbook on this subject which also serves as a reference book is vital for an equation which plays important roles in diverse areas of physics and mathematics like the completely integrable statistical models, conformal field theories, topological field theories, the theory of braid groups, the theory of knots and links, etc. This book arose from lectures given by the author in an attempt to reformulate the results of the rapidly developing research and make the material more accessible. It explains the presentation of the Yang-Baxter equation from statistical models, and expound systematically the meaning and methods of solving for this equation. From the viewpoint of theoretical physics it aims to develop an intuitive understanding of the fundamental knowledge of the Hopf algebras, quantization of Lie bialgebras, and the quantum enveloping algebras, and places emphasis on the introduction of the calculation skill in terms of the physical language.

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.

Yang-baxter Equation And Quantum Enveloping Algebras
Yang-baxter Equation And Quantum Enveloping Algebras

Author: Zhong-qi Ma

Publisher: World Scientific

Published: 1993-12-30

Total Pages: 331

ISBN-13: 9814504262

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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.

Quantum Groups
Quantum Groups

Author: Christian Kassel

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 540

ISBN-13: 1461207835

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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.

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.

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.

Fifty Years of Mathematical Physics
Fifty Years of Mathematical Physics

Author: Molin Ge

Publisher: World Scientific Publishing Company

Published: 2016-02-16

Total Pages: 596

ISBN-13: 9814340960

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This unique volume summarizes with a historical perspective several of the major scientific achievements of Ludwig Faddeev, with a foreword by Nobel Laureate C N Yang. The volume that spans over fifty years of Faddeev's career begins where he started his own scientific research, in the subject of scattering theory and the three-body problem. It then continues to describe Faddeev's contributions to automorphic functions, followed by an extensive account of his many fundamental contributions to quantum field theory including his original article on ghosts with Popov. Faddeev's contributions to soliton theory and integrable models are then described, followed by a survey of his work on quantum groups. The final scientific section is devoted to Faddeev's contemporary research including articles on his long-term interest in constructing knotted solitons and understanding confinement. The volume concludes with his personal view on science and mathematical physics in particular.