书籍详情
卡茨：穆迪代数及其表示（英文版）
作者：Xiaoping Xu 著
出版社：科学出版社
出版时间：2007-01-01
ISBN：9787030186089
定价：¥49.00
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内容简介
In order to study infinite-dimensional Lie algebras with root space decomposition as finite-dimensional simple Lie algebras，Victor Kac and Robert Moody independently introduced Lie algebras associated with generalized Caftan matrices，So-called”Kac-Moody algebras”in later 1960s。In 1ast near forty years，these algebras have played important roles in the other mathematical fields such as combinatorics．number theory，topology，integrable systems，operator theory，quantum stochastic process，and in quantum field theory of physics，This book gives a systematic exposition on the structure of Kac-Moody algebras，their representation theory and some connections with combinatorics，number theory，integrable systems and quantum field theory in physics．In particular，we give many details that Kac’S book lacks，correct some mistakes and reorganize the materials．This book may serve as a text book for graduate students in mathematics and physics，and a reference book for researchers。
作者简介
Xiaoping Xu，a professor at Institute of Mathematics，Academy of Mathematics and System Sciences，Chinese Academy of Sciences．
目录
Preface
Notational Conventions
Introduction
Chapter 1 Structure of Kac-Moody Algebras
1.1 Lie Algebra Associated with a Matrix
1.2 Invariant Bilinear Form
1.3 Generalized Casimir Operators
1.4 Weyl Groups
1.5 Classification of Generalized Cartan Matrices
1.6 Real and Imaginary Roots
Chapter 2 Affine Kac-Moody Algebras
2.1 Affine Roots and Weyl Groups
2.2 Realizations of Untwisted Affine Algebras
2.3 Realizations of Twisted Affine Algebras
Chapter 3 Representation Theory
3.1 Highest-Weight Modules
3.2 Defining Relations of Kac-Moody Algebras
3.3 Character Formula
3.4 Weights
3.5 Unit arizability
3.6 Action of Imaginary Root Vectors
3.7 Implications of the Denominator Identity
Chapter 4 Representations of Afllne and Virasoro Algebras.
4.1 Macdonald Identities
4.2 Affine Weights
4.3 Virasoro Algebra
4.4 Sugawara Construction
4.5 Coset Construction
Chapter 5 Related Modular Forms
5.1 Theta Functions
5.2 Modular Transformations
5.3 Modular Forms
5.4 Applications to Affine Algebras
Chapter 6 Realizations of Modules
6.1 Generating Functions
6.2 Untwisted Vertex Operator Representations
6.3 Twisted Vertex Operator Representations
6.4 Free Fermionic Field Realizations
6.5 Boson-Fermion C0rreSpondenCe
Bibliography
Index
Notational Conventions
Introduction
Chapter 1 Structure of Kac-Moody Algebras
1.1 Lie Algebra Associated with a Matrix
1.2 Invariant Bilinear Form
1.3 Generalized Casimir Operators
1.4 Weyl Groups
1.5 Classification of Generalized Cartan Matrices
1.6 Real and Imaginary Roots
Chapter 2 Affine Kac-Moody Algebras
2.1 Affine Roots and Weyl Groups
2.2 Realizations of Untwisted Affine Algebras
2.3 Realizations of Twisted Affine Algebras
Chapter 3 Representation Theory
3.1 Highest-Weight Modules
3.2 Defining Relations of Kac-Moody Algebras
3.3 Character Formula
3.4 Weights
3.5 Unit arizability
3.6 Action of Imaginary Root Vectors
3.7 Implications of the Denominator Identity
Chapter 4 Representations of Afllne and Virasoro Algebras.
4.1 Macdonald Identities
4.2 Affine Weights
4.3 Virasoro Algebra
4.4 Sugawara Construction
4.5 Coset Construction
Chapter 5 Related Modular Forms
5.1 Theta Functions
5.2 Modular Transformations
5.3 Modular Forms
5.4 Applications to Affine Algebras
Chapter 6 Realizations of Modules
6.1 Generating Functions
6.2 Untwisted Vertex Operator Representations
6.3 Twisted Vertex Operator Representations
6.4 Free Fermionic Field Realizations
6.5 Boson-Fermion C0rreSpondenCe
Bibliography
Index
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