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.
Chapter VII - Cartan Subalgebras and Regular Elements: Primary Decomposition of Linear Representations.- Cartan Subalgebras and Regular Elements of a Lie Algebra.- Conjugacy Theorems.- Regular Elements of a Lie Group.- Decomposable Linear Lie Algebras.- Chapter VIII - Split Semi-Simple Lie Algebras: Root System of a Split Semi-Simple Lie Algebra.- Subalgebras of Split Semi-Simple Lie Algebras.- Automorphisms of a Semi-Simple Lie Algebra.- Modules over a Split Semi-Simple Lie Algebra.- Finite Dimensional Modules over a Split Semi-simple Lie.- Symmetric Invariants.- The Formula of Hermann Weyl.- Maximal Subalgebras.- Chevalley Orders.- Classical Splittable Simple Lie Algebras.- Chapter IX - Compact Real Lie Groups: Compact Lie Algebras.- Maximal Tori of Compact Lie Groups.- Compact Forms of Complex Semi-Simple Lie Algebras.- Root System Associatedto a Compact Group. Conjugacy Classes. Integration on Compact Lie Groups.- Irreducible Representations of connected compact Lie Groups.- Fourier Transform.- Compact Lie Groups Operating on Manifolds.- Appendices.