## Infinite-Dimensional Lie AlgebrasThis is the third, substantially revised edition of this important monograph. The book is concerned with Kac-Moody algebras, a particular class of infinite-dimensional Lie algebras, and their representations. It is based on courses given over a number of years at MIT and in Paris, and is sufficiently self-contained and detailed to be used for graduate courses. Each chapter begins with a motivating discussion and ends with a collection of exercises, with hints to the more challenging problems. |

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### Contents

Chapter 1 Basic Definitions | 1 |

Chapter 2 The Invariant Bilinear Form and the Generalized Casimir Operator | 16 |

Chapter 3 Integrable Representations of KacMoody Algebras and the Weyl Group | 30 |

Chapter 4 A Classification of Generalized Cartan Matrices | 47 |

Chapter 5 Real and Imaginary Roots | 59 |

the Normalized Invariant Form the Root System and the Weyl Group | 79 |

Chapter 7 Affine Algebras as Central Extensions of Loop Algebras | 96 |

Chapter 8 Twisted Affine Algebras and Finite Order Automorphisms | 125 |

the Character Formula | 171 |

the Weight System and the Unitarizability | 190 |

Chapter 12 Integrable HighestWeight Modules over Affine Algebras Application to rjFunction Identities Sugawara Operators and Branching Functions | 216 |

Chapter 13 Affine Algebras Theta Functions and Modular Forms | 248 |

Chapter 14 The Principal and Homogeneous Vertex Operator Constructions of the Basic Representation BosonFermion Correspondence Application t... | 292 |

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### Common terms and phrases

adjoint affine algebra affine Lie algebra affine type algebra of type associated basic representation basis branching functions called Cartan matrix Cartan subalgebra central extension Chapter Chevalley commutation Corollary corresponding Coxeter number defined denote dimensional direct sum dual Dynkin diagram element equation exists finite number finite type finite-dimensional Lie algebra follows from Proposition formula given hence Hermitian form highest-weight vector Hirota holomorphic hyperbolic identity imaginary roots implies infinite invariant bilinear form irreducible isomorphic Kac-Moody algebra Kac-Peterson 1984 KP hierarchy Lemma linear modular forms multiplicity nondegenerate nonzero normalized invariant form Note obtain polynomial principal gradation Proof prove real roots resp root lattice root system Show simple finite-dimensional Lie simple roots subgroup submodule subspace symmetric symmetrizable Table Aff Theorem theta functions unique unitarizable vector space Verma module vertex operator construction Vir-module Virasoro algebra Weyl group