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Lie groups and algebraic groups. Translated from the Russian by D. A. Leites. (English) Zbl 0722.22004
Springer Series in Soviet Mathematics. Berlin etc.: Springer-Verlag. xvii, 328 p. (1990).
[For a review of the Russian original (1988) cf. Zbl 0648.22009.]
The book under review is based on notes of the authors’ seminar on algebraic and Lie groups held at Moscow in the late sixties. The central theme is the structure theory of semisimple Lie groups which the authors approach via the theory of algebraic groups. The necessary prerequisites from algebraic geometry are provided in Chapter 2 which is an independent introduction to affine and projective algebraic varieties and also covers some dimension theory. Chapter 3 deals with the basics of algebraic groups such as algebraic tori, Jordan decomposition, Borel groups and the tangent algebra. The last section of this chapter is devoted to compact linear groups and complete reducibility. Chapter 4 is the heart of the book. Here one finds all the standard theory of complex semisimple Lie groups up to the classification via root systems and Dynkin diagrams. Moreover there is a classification of those automorphisms of a complex simple Lie algebra whose spectrum is contained in the unit circle.
The part of the book dealing with real Lie groups has been added later to the seminar notes. Chapter 1 is a general introduction to Lie groups in terms of differentiable manifolds. The results of this chapter are only very occasionally used throughout the text up to Chapter 5 which is devoted to a classification of the real semisimple Lie groups. On the way the authors prove the existence of compact real forms for complex semisimple groups, the Cartan decomposition and the standard conjugacy theorems on maximal tori and maximal compact subgroups. In addition there is a section on restricted root systems and the Iwasawa decomposition. Finally in Chapter 6 Levi’s and Malcev’s theorems are proven and used to show the existence of a Lie group for any given finite dimensional Lie algebra.
The book ends with a reference chapter with a lot of useful formulae and tables concerning root systems and weight lattices for certain representations.
As one can see from the above description this book covers far more material than one would expect in a book of roughly 290 pages (without the reference chapter). The way to achieve this is to formulate almost every statement as a problem, i.e. as an exercise for the reader. The reviewer is convinced that a student who has worked his way through this book has learnt a lot about the subject, but it takes a very energetic and persevering student to do that.
The book considerably underscores the interplay between Lie groups and Lie algebras choosing on various occasions methods of proof from the theory of algebraic groups rather than Lie theoretic ones. As long as one is primarily concerned with semisimple groups this is certainly just a matter of taste and the authors make quite clear where they see the importance of Lie theory: “Confirming ourselves to algebraic Lie groups and their algebraic actions we may get rid of various nuisances without substantially impoverishing the Lie group theory” (p. 104).

22Exx Lie groups
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
20Gxx Linear algebraic groups and related topics
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
17B20 Simple, semisimple, reductive (super)algebras
22E46 Semisimple Lie groups and their representations
14L35 Classical groups (algebro-geometric aspects)
17B45 Lie algebras of linear algebraic groups