Publisher’s description: This is the English translation of Bourbaki’s text “Groupes et Algèbres de Lie, Chapters 7 to 9”. It completes the previously published translations of Chapters 1 to 3 (1998; Zbl 0904.17001) and 4 to 6 (2002; Zbl 0983.17001) by covering the structure and representation theory of semi-simple Lie algebras and compact Lie groups. Chapter 7 deals with Cartan subalgebras of Lie algebras, regular elements and conjugacy theorems. Chapter 8 begins with the structure of split semisimple Lie algebras and their root systems. It goes on to describe the finite-dimensional modules for such algebras, including the character formula of Hermann Weyl. It concludes with the theory of Chevalley orders. Chapter 9 is devoted to the theory of compact Lie groups, beginning with a discussion of their maximal tori, root systems and Weyl groups. It goes on to describe the representation theory of compact Lie groups, including the application of integration to establish Weyl’s formula in this context. The chapter concludes with a discussion of the actions of compact Lie groups on manifolds.
The nine chapters together form the most comprehensive text available on the theory of Lie groups and Lie algebras.
For reviews of the original editions see Ch. 1: Zbl 0199.35203 (2nd ed. Zbl 0213.04103); Ch. 2–3: Zbl 0244.22007; Ch. 4–6: Zbl 0186.33001; Ch. 7–8: Zbl 0329.17002; Ch. 9: Zbl 0505.22006.