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　　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 Cartan matrices,so-called ＂Kac-Moody algebras＂in later 1960s. In last 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.

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• 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 Unitarizability
  3.6 Action of Imaginary Root Vectors
  3.7 Implications of the Denominator Identity
  Chapter 4 Representations of Affine 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 Correspondence
  Bibliography
  Index