Table Of ContentYANG-BAXTER EQUATION ANO
QUANTUM ENVELOPING ALGEBRAS
ADVANCED SERIES ON THEORETICAL PHYSICAL SCIENCE
A Collaboration between World Scientific and Institute of
Theoretical Physics
Series Editors: Dai Yuan-Ben, Hao Bai-Lin, Su Zhao-Bin
(Institute of Theoretical Physics, Academia Sinica)
Vol. 1: Yang-Baxter Equation and Quantum Enveloping Algebras
(Zhong-Qi Ma)
Forthcoming:
Geometric Methods in Physics (Wu Yong-Shi)
Experimental Foundation of Special Relativity (Zhang Yuan-Zhong)
Liquid Crystal Models of Biomembranes (Ouyang Zhong-Can)
1
Advanced Series on Theoretical Physical Science
Volume
YANG-BAXTER
EQUATION AND
QUANTUM
ENVELOPING
ALGEBRAS
Zhong-Qi Ma
Institute of High Energy Physics
Academia Sinica
Beijing, China
World Scientific Institute of Theoretical Physics
Singapore • New Jersey London • Hong Kong Beijing
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Fairer Road, Singapore 9128
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 73 Lynton Mead, Totteridge, London N20 8DH
YANG-BAXTER EQUATION AND QUANTUM ENVELOPING
ALGEBRAS
Copyright © 1993 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book or parts thereof, may not be reproduced in any form
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Printed in Singapore.
PREFACE
The exact solutions to some simple but fundamental dynamical systems have
played crucial roles in the development of physics. Although at the early stage
of quantum mechanics there were several disputes concerning some fundamental
concepts and explanations, the precise coincidence between experimental results
and the theoretical predictions from the exact solutions on the hydrogen atom and
the linear harmonic oscillator was essential to delineate the fundamental concepts
and interpretations of quantum mechanics, and thus laid the foundation for it. The
exact solution on the many body problem in one dimension with repulsive delta-
function interaction presented by Professor Chen Ning Yang in 1967 and the exact
solution for the eight vertex statistical model presented by Professor Baxter in 1972
are brilliant achievements in the many body statistical theory. A key nonlinear
equation, called Yang-Baxter equation, is involved in solving both models. In terms
of its solutions, both models were exactly solved.
The Yang-Baxter equation describes the translation invariance of oriented lines.
It is why the Yang-Baxter equation plays the important roles in such diverse areas
of physics and mathematics like the completely integrable statistical models, the
conformal field theories, the topological field theories, the theory of braid groups,
the theory of knots and links, etc. In the past twenty years, mathematicians and
theoretical physicists have been working diligently on the Yang-Baxter equation,
and made a breakthrough in the mid-eighties, namely, the establishment of the
quantum enveloping algebras. The first motivation of presentation of the quantum
enveloping algebras is for solving the Yang-Baxter equation. However, the ele
gant theory of the quantum enveloping algebras itself appeared to be a significant
achievement in mathematics. In 1990 Drinfel'd and Jones won the Fields Award
for their outstanding contributions in this field.
Just before this subject became one of the main interests of theoretical physi
cists and mathematicians, in 1987 at Nankai Institute of Mathematics, Professor
Chen Ning Yang suggested again Chinese theoretical physicists and mathematicians
to keep up with and to contribute to the development of this field. He personally
invited some famous scientists who made important contributions in this field, such
as Faddeev, Friedan, Kohno, Jimbo, Kulish, Sklyanin, etc, to give systematic lec
tures in Tianjin. Under his encouragement and mentorship, Chinese scientists were
able to keep abreast with new trends in this field.
This book is intended to serve as a textbook at the graduate level and also
as a reference book. Originating from author's research experience, it attempts
to reformulate the results in this rapidly developing field and make the materials
more accessible. It explains the historical origin of the Yang-Baxter equation from
vi pre/ace
statistical models, and explores systematically the meaning of the equation and
methods for solving it. In order to make this book self-contained, it begins with the
mathematical preliminaries. From the viewpoint of theoretical physicists this book
intends to develop an intuitive understanding of the Hopf algebras, quantization of
Lie bialgebras, and the quantum enveloping algebras, and to place emphasis on the
introduction of the calculation skill in terms of the physical language. No attempt
is made to achieve mathematical rigour. The reference list at the end of this book
is representative rather than exhaustive.
This book is arranged as follows. Chapter 1 introduces the fundamental mathe
matical tools, mainly the knowledge on the group theory. The representations of the
permutation groups are treated by the Young operators. This method is generalized
in studying the braid groups. Some properties of simple Lie algebras and loop al
gebras are also listed. The formulas of q calculation are proved both for the generic
q and for q being a root of unity. Chapter 2 explains in detail the computations in
volved in exactly solving the many body Schrodinger system in one dimension with
repulsive delta-function interaction, and sketches the similar computations on the
six vertex model. The purpose is to provide readers with some physical background
of the Yang-Baxter equation. The Yang-Baxter equation is a nonlinear equation
and hard to solve generally. There are two kinds of parameters in the Yang-Baxter
equation: the quantum parameter and the spectral parameter. As the first step,
Chapter 3 and Chapter 6 study the solutions in two special cases where one of two
parameters is taken its limit value, respectively. The recent development of the
theory of knots and links is also discussed in Chapter 6.
The Yang-Baxter equation without the spectral parameter is called the simple
Yang-Baxter equation. The theory of the quantum enveloping algebras is an effec
tive tool for solving the simple Yang-Baxter equation. Chapter 4 introduces the
theory of the quantum enveloping algebras. It is an important progress of mathe
matics in eighties. This book explains the profound of the theory in simple terms
from the viewpoint of theoretical physicists. Chapter 5 expounds the irreducible
representations of the quantum enveloping algebras, the quantum Clebsch-Gordan
coefficients, and the Racah coefficients in very detail, and treats them as the quan
tum analogues of the theory of angular momemtrum and the theory of Lie algebras.
Because of the adoption of the physical notations, theoretical physicists may find
it easier to imagine and understand. Chapter 7 summarizes three methods for
computing trigonometric solutions of the Yang-Baxter equation, and discusses their
merits and shortcomings. Chapter 7 also gives a standard procedure to obtain the
rational solutions of the Yang-Baxter equation from the trigonometric ones. When
the quantum parameter q is a root of unity, some new phenomena appear. The gen
eral theory of representations for this case is in progress. Chapter 8 only discusses
the representations of UA\, and studies a physical model with the symmetry of the
t
quantum enveloping algebras UA.
i l
preface vii
This book originates from lectures given by the author in Shanghai Jiaotong Uni
versity, Fudan University, Institute of Physics (Beijing), Peking University, Lanzhou
University, Xiamen University and so on. Due to the rapid development in this field,
and the limited exposure of the author, this book cannot include all the progress
made in this field. I sincerely welcome any suggestion from the readers.
In conclusion, I must express my cordial gratitude to Professor Chen Ning Yang
for his guidance, under which I entered this field. I am also indebted to Professor
Bao-Heng Zhao for bis permission to use some materials in Chapter 2 from his lec
tures [185] given in Nankai Institute of Mathematics. I would like to thank Professor
Bo-Yu Hou and Professor Bo-Yuan Hou for the effective and fruitful cooperations
in research works. This book includes some results from the collaborations.
Inititute of High Energy Phyiici Zhong-Qi Ma
Beijing, China
May, 1993
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CONTENTS
CHAPTER 1. MATHEMATICAL PRELIMINARIES 1
1.1. Permutation Groups 1
1.2. Braid Groups 15
1.3. Simple Lie Algebras 28
1.4. Loop Algebras 43
1.5. Formulas lor q Calculations 48
CHAPTER 2. HISTORICAL ORIGIN OF THE YANG-BAXTER
EQUATION 57
2.1. The Many Body Problem in One Dimension with Repulsive Delta-
Function Interaction 57
2.2. The Six Vertex Model 70
2.3. Expressions for the Yang-Baxter Equation 76
CHAPTER 3. CLASSICAL YANG-BAXTER EQUATION 80
3.1. The Classical Yang-Baxter Equation 80
3.2. Rational Solutions 80
3.3. Trigonometric Solutions 82
3.4. Reformation of the Baxter Solution 86
3.5. Classification of Solutions of the Classical Yang-Baxter Equation 89
CHAPTER 4. QUANTUM ENVELOPING ALGEBRAS 91
4.1. Hopf Algebras 92
4.2. Quantization of Lie Bialgebras 101
4.3. Quantum Enveloping Algebras 109
4.4. Irreducible Representations of UAy 111
t
4.5. Universal K Matrix 116
CHAPTER 5. QUANTUM CLEBSCH-GORDAN COEFFICIENTS 124
5.1. Highest Weight Representations of Quantum Enveloping Algebras 124
5.2. Quantum Clebsch-Gordan Coefficients 135
5.3. Quantum CG Coefficients of UAi Algebra 158
t
5.4. Quantum Racah Coefficients 165
Contents x
CHAPTER 6. SIMPLE YANG-BAXTER EQUATION 181
6.1. Spectral Expansions of Solutions 181
6.2. Jones Polynomials 190
6.3. New Link Polynomials 196
6.4. Braiding Matrices 200
CHAPTER 7. TRIGONOMETRIC AND RATIONAL SOLUTIONS 207
7.1. Yang-Baxterization 207
7.2. Expansion Formulas 220
7.3. The Fusion Procedure 249
7.4. Rational Solutions 254
CHAPTER 8. NON-GENERIC q VALUES 258
8.1. The Lusztig Picture 258
8.2. Two Types of Representations 260
8.3. Quantum CG Coefficients for Non-Generic q Values 267
8.4. The XXZ Spin Chain Model 279
8.5. Cyclic Representations 297
REFERENCES 302
INDEX 315
