Infinite dimensional Lie algebras. An introduction
type: Item
instance of: scholarly article
title: Infinite dimensional Lie algebras. An introduction
zbMATH Open document ID: 0537.17001
published in: Progress in Mathematics
publication date: 1983-01-01T00:00:00Z
review text: This book is a remarkable contribution to the ''hot'' area of Kac-Moody Lie algebras. These algebras appear in many problems of, as Kac points out, mathematical physics, combinatorics, modular forms, etc. Their most spectacular applications are in the areas of Hamiltonian systems and combinatorial identities. Besides being natural and useful they led to the re-examining of many results and proofs in the theory of finite dimensional Lie algebras thereby gaining often new insights into the structure of the latter; this work was done mostly by V. Kac himself, often in collaboration with D. Peterson. The most remarkable are, probably, the recent results of V. Kac on cohomology of compact Lie groups (and related results about Chow rings etc.). The book concentrates mostly on the areas where V. Kac contributed original work (and these are almost all areas). Some of the areas least covered in the book are applications to PDE, touched upon in the concluding chapter of the book, and cohomology of Kac-Moody Lie algebras, introduced by Garland and Lepowsky. The book is a very good introduction into the theory. It leads the reader directly through and to main results, ideas, and methods of the theory. In bibliographical notes at the end of each chapter the author gives an additional guidance to the literature. Some results, side-lines and problems are put forward as exercises so that the book does not get bogged down with unnecessary ''completeness'' but the reader gets clearly an idea of the state of affairs and receives a thorough orientation. The Kac-Moody Lie algebras were defined almost simultaneously by V. Kac and R. Moody. They are graded Lie algebras \(G=\oplus_{i\in\mathbb Z}G_i\) (i.e. \([G_i,G_j]\subseteq G_{i+j})\) such that \(G_1\) has basis \(e_1,\ldots,e_n\), \(G_ 0\) has basis \(h_1,\ldots,h_n\), \(G_{-1}\) has basis \(f_ 1,\ldots,f_n\) with (1) \(G_0\) commutative, (2) \([h_i,e_j]=a_{ij}e_j\), \([h_i,f_j]=- a_{ij}f_j\), (3)\ \([e_i,f_j]=\delta_{ij}h_ i\), (4) \…
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17-01
Mathematics Subject Classification ID: 17-02
Mathematics Subject Classification ID: 22E67
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 22-01
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 11P81
Mathematics Subject Classification ID: 11F22
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37K30
zbMATH DE Number: 3853316
zbMATH Keywords: symmetrizable Cartan matrix
zbMATH Keywords: Korteweg-de Vries hierarchies
zbMATH Keywords: Kac-Moody Lie algebras
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: combinatorial identities
zbMATH Keywords: highest weight modules
zbMATH Keywords: affine Kac-Moody Lie algebras
zbMATH Keywords: realizations
zbMATH Keywords: integrable highest weight modules
zbMATH Keywords: character formula
zbMATH Keywords: modular forms
zbMATH Keywords: \(\theta\)-functions
author: Victor G. Kac
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3853316