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Lie algebras with triangular decompositions. (English) Zbl 0874.17026
Canadian Mathematical Society Series of Monographs and Advanced Texts. New York, NY: John Wiley & Sons. xx, 685 p. (1995).
The purpose of this book is stated in the Introduction: “Our objective is to present a self-contained development of the algebraic theory of the Kac-Moody algebras, their representations, and their close relatives, the Virasoro and Heisenberg algebras.”
The notion that allows the authors to give a unified treatment of these Lie algebras is that of a triangular decomposition: this is a decomposition of a Lie algebra $${\mathfrak g}$$ into a vector space direct sum of three subalgebras $${\mathfrak g}_\pm$$ and $${\mathfrak h}$$ such that $${\mathfrak h}$$ is abelian and $$[{\mathfrak h},{\mathfrak g}_\pm]\subseteq{\mathfrak g}_\pm$$ (together with certain other conditions that will be familiar to readers conversant with the theory of semisimple Lie algebras). After beginning, in Chapter 1, with a standard introduction to the general theory of Lie algebras, triangular decompositions are studied in Chapter 2. In this general context, the authors are able to define such notions as highest weight modules, category $${\mathcal O}$$, the Shapovalov form, Jantzen filtrations, BGG duality, etc.
Chapter 3 turns to root systems associated to Lie algebras with triangular decompositions. First, we are given a fairly standard treatment of the classification of finite root systems via Dynkin diagrams. Then follows a novel treatment of the finite-affine-indefinite type classification of generalized Cartan matrices using the Perron-Frobenius theorem. The chapter concludes with the construction of Lie algebras starting from (root) lattices.
The rest of the book is devoted to contragredient Lie algebras in which, roughly speaking, the triangular decomposition is generated by copies of $$sl_2$$. Chapter 4 begins by introducing the Weyl group, studying special properties of the root systems of contragredient Lie algebras and determining the condition (symmetrizability) for the existence of an invariant bilinear form. The chapter concludes with a proof of the Gabber-Kac theorem.
Chapter 5 is devoted to a detailed study of the geometry of Weyl groups and abstract root systems, including a proof that Weyl groups are Coxeter groups, a discussion of the Bruhat ordering, the length functions, etc.
Chapter 6 studies category $${\mathcal O}$$ for Kac-Moody algebras. Included are the Weyl-Kac character formula (and its generalization due to Kac-Wakimoto, whose proof requires the use of translation functors), the Shapovalov determinant formula, proof of complete reducibility, the BGG theorem and more.
The aim of the final Chapter 7 is to generalize the well known conjugacy of Cartan subalgebras from the finite-dimensional case to the case of symmetrizable Kac-Moody algebras. This is essentially the only part of the book which is not self-contained, as it makes use of several results from the theory of algebraic groups, in particular the Borel fixed point theorem.
The book is very clearly written and all the results are proved in complete detail (with the exceptions just noted). Each chapter ends with an interesting collection of exercises. The book would be very suitable for independent study by a well motivated graduate student.

##### MSC:
 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras 17-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras