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| CHAPTER 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 CHAPTER 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 CHAPTER Ⅲ Cartan Subalgebras 1. Definition of Cartan Subalgebras 2. Regular Elements" Rank 3. The Cartan Subalgebra Associated with a Regular Element 4. Conjugacy ofCartan Subalgebras 5. The Semisimple Case 6. Real Lie Algebras CHAPTER Ⅳ The Algebra sl2 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 7. Applications CHAPTER 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 Caftan 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 CHAPTER 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 |
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