This is the softcover reprint of 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
(3-540-64242-0) and 4 to 6 (978-3-540-69171-6) 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.
