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Hardcover ISBN:  9780821828748 
Product Code:  GSM/42 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470420932 
Product Code:  GSM/42.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9780821828748 
eBook ISBN:  9781470420932 
Product Code:  GSM/42.B 
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MAA Member Price:  $165.60 $127.35 
AMS Member Price:  $147.20 $113.20 

Book DetailsGraduate Studies in MathematicsVolume: 42; 2002; 307 ppMSC: Primary 17; Secondary 81; 82
The notion of a “quantum group” was introduced by V.G. Dinfel&dacute; and M. Jimbo, independently, in their study of the quantum YangBaxter equation arising from 2dimensional solvable lattice models. Quantum groups are certain families of Hopf algebras that are deformations of universal enveloping algebras of KacMoody 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 KacMoody 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.
ReadershipGraduate students and research mathematicians interested in nonassociative rings and algebras.

Table of Contents

Chapters

Chapter 1. Lie algebras and Hopf algebras

Chapter 2. KacMoody algebras

Chapter 3. Quantum groups

Chapter 4. Crystal bases

Chapter 5. Existence and uniqueness of crystal bases

Chapter 6. Global bases

Chapter 7. Young tableaux and crystals

Chapter 8. Crystal graphs for classical Lie algebras

Chapter 9. Solvable lattice models

Chapter 10. Perfect crystals

Chapter 11. Combinatorics of Young walls


Reviews

Book by Hong and Kang is the first expository text which gives a detailed account on the relationship between crystal bases and combinatorics. This book provides an accessible and “crystal clear” introduction and overview of the relatively new subject of quantum groups and crystal bases, ... It will be an indispensable companion to the research papers.
Mathematical Reviews


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The notion of a “quantum group” was introduced by V.G. Dinfel&dacute; and M. Jimbo, independently, in their study of the quantum YangBaxter equation arising from 2dimensional solvable lattice models. Quantum groups are certain families of Hopf algebras that are deformations of universal enveloping algebras of KacMoody 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 KacMoody 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.
Graduate students and research mathematicians interested in nonassociative rings and algebras.

Chapters

Chapter 1. Lie algebras and Hopf algebras

Chapter 2. KacMoody algebras

Chapter 3. Quantum groups

Chapter 4. Crystal bases

Chapter 5. Existence and uniqueness of crystal bases

Chapter 6. Global bases

Chapter 7. Young tableaux and crystals

Chapter 8. Crystal graphs for classical Lie algebras

Chapter 9. Solvable lattice models

Chapter 10. Perfect crystals

Chapter 11. Combinatorics of Young walls

Book by Hong and Kang is the first expository text which gives a detailed account on the relationship between crystal bases and combinatorics. This book provides an accessible and “crystal clear” introduction and overview of the relatively new subject of quantum groups and crystal bases, ... It will be an indispensable companion to the research papers.
Mathematical Reviews