Yang-Baxter Equation and Quantum Enveloping Algebras (Advanced Series on Theoretical Physical Science, Vol 1)

This is a textbook examining the Yang-Baxter equation. The Yang-Baxter equation was presented a quarter of a century ago and became one of the main concerns of physicists and mathematicians in recent years. This book arose from lectures given by the author in an attempt to reformulate the results in the rapidly developing research works and make the materials more accessible. It explains the presentation of the Yang-Baxter equation from statistical models, and expounds systematically the meaning and solving methods for this equation. From the viewpoint of theoretical physicists it intends to develop an intuitive understanding of the fundamental knowledge on the Hopf algebras, quantization of Lie bialgebras, and the quantum enveloping algebras, but placing emphasis on the introduction of the calculation skill in terms of the physical language. 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. 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. 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 m The Yang-Baxter equation has become one of the main concerns of physicists and mathematicians in recent years. This book arose from lectures given by the author in an attempt to reformulate the results in the rapidly developing research works, and to make the materials more accessible. A permutation of N things, represented by numbers 1, 2, . . ., N, is a transformation changing the order of things.