TY  - BOOK
AU  - Serre,Jean Pierre
TI  - Complex semisimple Lie algebras
T2  - Springer monographs in mathematics,
SN  - 3540678271
AV  - QA252.3 .S48 1987
PY  - 2001///]
CY  - Berlin, New York
PB  - Springer
KW  - Lie algebras
N1  - "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.
ER  -