This is the English translation of Bourbaki's text Groupes et Algebres de Lie,

Chapters 7 to 9. It completes the previously published translations of Chapters

1 to 3 (3-540-50218-1) and 4 to 6 (3-540-42650-7) 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 semi-simple 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.