Elements of mathematics. Lie groups and Lie algebras. Chapter 9. Compact real Lie groups. Reprint of the 1982 original. (Éléments de mathématique. Groupes et algèbres de Lie. Chapitre 9: Groupes de Lie réels compacts.) (French) Zbl 1123.22005

Berlin: Springer (ISBN 978-3-540-34392-9/pbk). 138 p. (2007).
Within Nicolas Bourbaki’s monumental treatise “Éléments de Mathématique”, Book 9 is devoted to a systematic development of the modern abstract theory of Lie algebras and Lie groups. This book so far consists of nine chapters, successively published between 1960 and 1982.
The book under review is the unaltered reprint of the French original edition of the so far last chapter (Chapter 9) of Book 9 which first appeared in 1982 (Zbl 0505.22006). An English translation of Chapters 7, 8 and 9 has been made available, in one volume, as late as in 2005 by Springer Verlag (Zbl 1139.17002).
Referring to the review of the French original of this particular volume (Zbl 0505.22006), which remains entirely applicable and up-to-date, we just recall here that Chapter 9 is exclusively devoted to the structure theory of compact real Lie groups and their associated Lie algebras. The nine sections of this chapter treat the following topics:
1. Compact Lie algebras, compact Lie groups, and Lie groups with compact Lie algebras.
2. Maximal tori in compact Lie groups and Cartan subalgebras of compact Lie algebras.
3. Compact forms of complex semi-simple Lie algebras, Chevalley systems, and real forms associated to them.
4. Root systems of Lie algebras of compact Lie groups and automorphisms of compact connected Lie groups.
5. Conjugacy classes, Weyl chambers, regular elements, and their behaviour under automorphisms of Lie groups.
6. Integration theory on Lie groups, Lie algebras and vector bundles, H. Weyl’s integral formula, and invariant differential forms.
7. Irreducible representations of compact connected Lie groups, characters, maximal weights, and Casimir elements.
8. Fourier transforms of integrable functions on compact Lie groups.
9. Actions of compact Lie groups on manifolds, embedding theorems, the slice theorem, and orbit types.
There are two appendices to the main text. Appendix I briefly discusses additional properties of compact Lie groups, including embeddings into products of Lie groups, projective limits of Lie groups, and a proof of the fact that solvable compact connected Lie groups are always commutative. Appendix II provides some supplementary material on representations of real, complex, or quaternion algebras.
As usual for most Bourbaki volumes, there is a tremendously rich amount of complementing and further-leading exercises enhancing each single chapter, many of which provide alternative proofs of occurring theorems, specific examples, or additional theorems. As it is noteworthy for most of the later chapters of Bourbaki’s books, also this volume does (unfortunately) not contain the otherwise usual, Bourbaki-typical extra section of historical annotations to the main text.


22E15 General properties and structure of real Lie groups
22C05 Compact groups
22E60 Lie algebras of Lie groups
22E46 Semisimple Lie groups and their representations
22E30 Analysis on real and complex Lie groups
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