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Theory of Lie groups. I. 15th printing. (English) Zbl 0946.22001

Princeton Landmarks in Mathematics. Princeton Mathematical Series. 8. Princeton, NJ: Princeton University Press. viii, 217 p. (1999).
The present monograph is the 15th printing of Chevalley’s classical book on Lie groups which appeared for the first time in 1946 [Zbl 0063.00842]. It was the first systematic treatment of Lie groups from a global point of view, treating a Lie group as an object carrying three structures: it is a group, a topological space and an analytic manifold. The contents of the book is as follows.
In the first section the classical groups are presented, using systematically the polar decomposition as a tool to reduce topological questions to compact groups. Chapter II deals with topological groups with an emphasis on covering theory, where the approach to coverings is more by the universal property of a universal covering than by direct constructions using paths. Here quotient groups and homogeneous spaces of topological groups are discussed, and the universal coverings of the orthogonal groups, the spin groups, are constructed using Clifford algebras. In Chapter III the author introduces analytic manifolds, as we would nowadays say, from a sheaf theoretic point of view, by specifying certain properties of locally defined functions.
In Chapter IV, Lie groups are considered as analytic manifolds, where the existence of analytic subgroups corresponding to Lie subalgebras is derived by an analytic version of Frobenius’ Theorem on the existence of maximal submanifolds for involutive distributions. Here one also finds the interesting sufficient condition that a locally compact group is a Lie group if it has a continuous injection into a Lie group. Since Hilbert’s 5th problem was not yet solved at the time of the first printing, this was a very useful tool.
In Chapter V differential forms on manifolds and in particular Maurer-Cartan forms on Lie groups and their applications are discussed. This chapter also describes the basic techniques of integrating differential forms which leads to a direct construction of Haar measure on a Lie group, which is needed in Chapter VI for compact groups.
The last Chapter VI develops the representation theory of compact Lie groups. An interesting point is the discussion of the complex algebraic group \(M(G)\) defined by the representation ring of a compact Lie group \(G\). Nowadays we would call \(M(G)\) the universal complexification \(G_\mathbb C\) of \(G\). It is shown that this group has a polar decomposition, hence is a topological product of \(G\) and a vector space. The passage from a compact group \(G\) to an algebraic group \(M(G)\) was one of the main initial motivations to start a systematic study of algebraic groups in their own right.
Although the book is self-contained, it would have been interesting (nowadays more for historic reasons) to have a bibliography, which is completely missing.
It is interesting to compare Chevalley’s book with the modern developments in Lie theory concerned with concepts of infinite-dimensional Lie groups. There are several approaches to infinite-dimensional Lie theory. One is due to J. Milnor [Remarks on infinite-dimensional Lie groups, Proc. Summer School on Quantum Gravity, B. DeWitt ed., Les Houches (1984; Zbl 0594.22009)]: An infinite-dimensional Lie group is a manifold modeled over a sequentially complete locally convex space for which the group operations are smooth. This is still very much in the spirit of Chevalley’s approach because these Lie groups are still topological groups, but it turns out that it is not natural to ask for an analytic structure because it does not exist for the important examples of diffeomorphism groups of compact manifolds. A slightly different approach is due to A. Kriegl and P. Michor [The convenient setting of global analysis, Math.Surveys and Monographs 53, Am. Math. Soc. (1997; Zbl 0889.58001)]. Their Lie groups are defined in a setting for calculus which is broader than Milnor’s but has the disadvantage that smooth mappings are no longer continuous. Therefore their Lie groups are still manifolds, but no longer topological groups. There are further more recent approaches (in particular one due to H.Omori), which permit to consider closed subgroups and quotient groups by closed normal subgroups still as some kind of Lie groups which do no longer have any manifold structure.

MSC:

22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups
22E15 General properties and structure of real Lie groups
22C05 Compact groups
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