Kac, Victor G. Infinite dimensional Lie algebras. 2nd ed. (English) Zbl 0574.17010 Cambridge etc.: Cambridge University Press. XVII, 280 p. £25.00; $ 24.95 (1985). The publication of a second edition just two years after the first one (Birkhäuser 1983, Progr. Math. 44) proves the author’s presentation to be very successful and also shows the rapid development of this theory. For the contents of this monograph see B. Weisfeiler’s extensive review in Zbl 0537.17001. Preface to the second edition. ”The most important additions reflect recent developments in the theory of infinite-dimensional groups (some key facts, like Proposition 3.8 and Exercise 5.19 are among them) and in the soliton theory (like Exercises 14.37-14.40 which uncover the role of the Virasoro algebra). The most important correction concerns the proof of the complete reducibility Proposition 9.10. The previous proof used Lemma 9.10 b) of the first edition which is false, as Exercise 9.15 shows. A correct version of Lemma 9.10 b) is the new Proposition 10.4 which gives a characterization of integrable highest weight modules. In addition to correcting misprints and errors and adding a few dozen of new exercises, I have brought to date the list of references and related bibliographical comments.” Reviewer: O.Ninnemann Cited in 15 ReviewsCited in 156 Documents MSC: 17B65 Infinite-dimensional Lie (super)algebras 22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups 17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras 05A19 Combinatorial identities, bijective combinatorics 11P81 Elementary theory of partitions 11F11 Holomorphic modular forms of integral weight 17-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties 35Q99 Partial differential equations of mathematical physics and other areas of application 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems Keywords:symmetrizable Cartan matrix; Korteweg-de Vries hierarchies; Kac-Moody Lie algebras; Hamiltonian systems; combinatorial identities; highest weight modules; affine Kac-Moody Lie algebras; realizations; integrable highest weight modules; character formula; modular forms; theta-functions; infinite-dimensional groups; soliton theory; Virasoro algebra Citations:Zbl 0537.17001 PDFBibTeX XML