Complex Semisimple Lie Algebras: (Springer Monographs in Mathematics)

These notes are a record of a course given in Algiers from 10th to 21st May, 1965. Their contents are as follows. The first two chapters are a summary, without proofs, of the general properties of nilpotent, solvable, and semisimple Lie algebras. These are well-known results, for which the reader can refer to, for example, Chapter I of Bourbaki or my Harvard notes. The theory of complex semisimple algebras occupies Chapters III and IV. The proofs of the main theorems are essentially complete; however, I have also found it useful to mention some complementary results without proof. These are indicated by an asterisk, and the proofs can be found in Bourbaki, Groupes et Algebres de Lie, Paris, Hermann, 1960-1975, Chapters IV-VIII. A final chapter shows, without proof, how to pass from Lie algebras to Lie groups (complex-and also compact). It is just an introduction, aimed at guiding the reader towards the topology of Lie groups and the theory of algebraic groups. I am happy to thank MM. Pierre Gigord and Daniel Lehmann, who wrote up a first draft of these notes, and also Mlle. Franl(oise Pecha who was responsible for the typing of the manuscript.

Table of Contents:
I Nilpotent Lie Algebras and Solvable Lie Algebras.- 1. Lower Central Series.- 2. Definition of Nilpotent Lie Algebras.- 3. An Example of a Nilpotent Algebra.- 4. Engel’s Theorems.- 5. Derived Series.- 6. Definition of Solvable Lie Algebras.- 7. Lie’s Theorem.- 8. Cartan’s Criterion.- II Semisimple Lie Algebras (General Theorems).- 1. Radical and Semisimpiicity.- 2. The Cartan-Killing Criterion.- 3. Decomposition of Semisimple Lie Algebras.- 4. Derivations of Semisimple Lie Algebras.- 5. Semisimple Elements and Nilpotent Elements.- 6. Complete Reducibility Theorem.- 7. Complex Simple Lie Algebras.- 8. The Passage from Real to Complex.- III Cartan Subalgebras.- 1. Definition of Cartan Subalgebras.- 2. Regular Elements: Rank.- 3. The Cartan Subalgebra Associated with a Regular Element.- 4. Conjugacy of Cartan Subalgebras.- 5. The Semisimple Case.- 6. Real Lie Algebras.- IV The Algebra SI2 and Its Representations.- 1. The Lie Algebra sl2.- 2. Modules, Weights, Primitive Elements.- 3. Structure of the Submodule Generated by a Primitive Element.- 4. The Modules Wm.- 5. Structure of the Finite-Dimensional g-Modules.- 6. Topological Properties of the Group SL2.- V Root Systems.- 1. Symmetries.- 2. Definition of Root Systems.- 3. First Examples.- 4. The Weyl Group.- 5. Invariant Quadratic Forms.- 6. Inverse Systems.- 7. Relative Position of Two Roots.- 8. Bases.- 9. Some Properties of Bases.- 10. Relations with the Weyl Group.- 11. The Cartan Matrix.- 12. The Coxeter Graph.- 13. Irreducible Root Systems.- 14. Classification of Connected Coxeter Graphs.- 15. Dynkin Diagrams.- 16. Construction of Irreducible Root Systems.- 17. Complex Root Systems.- VI Structure of Semisimple Lie Algebras.- 1. Decomposition of g.- 2. Proof of Theorem 2.- 3. Borei Subalgebras.- 4. WeylBases.- 5. Existence and Uniqueness Theorems.- 6. Chevalley’s Normalization.- Appendix. Construction of Semisimple Lie Algebras by Generators and Relations.- VII Linear Representations of Semisimple Lie Algebras.- 1. Weights.- 2. Primitive Elements.- 3. Irreducible Modules with a Highest Weight.- 4. Finite-Dimensional Modules.- 5. An Application to the Weyl Group.- 6. Example: sl n+1.- 7. Characters.- 8. H. Weyl’s formula.- VIII Complex Groups and Compact Groups.- 1. Cartan Subgroups.- 2. Characters.- 3. Relations with Representations.- 4. Berel Subgroups.- 5. Construction of Irreducible Representations from Boret Subgroups.- 6. Relations with Algebraic Groups.- 7. Relations with Compact Groups.

Review :
From the reviews of the French edition: "...the book is intended for those who have an acquaintance with the basic parts of the theory, namely, with those general theorems on Lie algebras which do not depend on the notion of Cartan subalgebra. The author begins with a summary of these general theorems and then discusses in detail the structure and representation theory of complex semisimple Lie algebras. One recognizes here a skillful ordering of the material, many simplifications of classical arguments and a new theorem describing fundamental relations between canonical generators of semisimple Lie algebras. The classical theory being thus introduced in such modern form, the reader can quickly reach the essence of the theory through the present book." (Mathematical Reviews)