Complex semisimple Lie algebras / Jean-Pierre Serre ; translated from the French by G.A. Jones

"Reprint of the 1987 edition."

Includes bibliographical references (pages [72]) and index

Contents Ch. 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. Ch. II. Semisimple Lie Algebras (General Theorems). 1. Radical and Semisimplicity. 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. Ch. 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. Ch. IV. The Algebra sl[subscript 2] and Its Representations. 1. The Lie Algebra sl[subscript 2]. 2. Modules, Weights, Primitive Elements. 3. Structure of the Submodule Generated by a Primitive Elements. 4. The Modules W. 5. Structure of the Finite-Dimensional g-Modules. 6. Topological Properties of the Group SL[subscript 2]. 7. Applications. Ch. 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. Ch. VI. Structure of Semisimple Lie Algebras. 1. Decomposition of g. 2. Proof of Theorem 2. 3. Borel Subalgebras. 4. Weyl Bases. 5. Existence and Uniqueness Theorems. 6. Chevalley's Normalization. Appendix. Construction of Semisimple Lie Algebras by Generators and Relations. Ch. 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. Examples: sl[subscript n+1]. 7. Characters. 8. H. Weyl's formula. Ch. VIII. Complex Group and Compact Groups. 1. Cartan Subgroups. 2. Characters. 3. Relations with Representations. 4. Borel Subgroups. 5. Construction of Irreducible Representations from Borel Subgroups. 6. Relations with Algebraic Groups. 7. Relations with Compact Groups. Bibliography. Index.