Complex semisimple lie algebras复半单李代数
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作者： Jean-Pierre Serre 著
出 版 社： 广东教育出版社
出版时间： 2001-1-1字数：版次： 1页数： 67印刷时间： 2001/01/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783540678274包装： 精装内容简介
These notes, already well known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers including the basic classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and representation theory. The theory is illustrated by using the example of sln; in particular, the representation theory of sl2 is completely worked out. The last chapter discusses the connection between Lie algebras and Lie groups, and is intended to guide the reader towards further study.
目录
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