目錄
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.Engei'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 Semisimplv Lie Algebras
4.Derivations of Scmisimple 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 Caftan 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
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 9-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.lnvariant 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
CHAPTER VI Structure of Semisimple Lie Algebras
1.Decomposition ofg
2.Proof of Theorem 2
3.Borei Subalgebras
4.Weyl Bases
5.Existence and Uniqueness Theorems
6.Chevalley's Normalization
Appendix.Construction of Semisimple Lie Algebras by Generators and Relations
CHAPTER 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: sln+l
7.Characters
8.H.Weyl's formula
CHAPTER VIII Complex Groups and Compact Groups
1.Cartan Subgroups
2.Characters
3.Relations with Representations
4.Borel Subgroups
5.Construction of Irreducible Representations from Borei Subgroups
6.Relations with Algebraic Groups
7.Relations with Compact Groups
Bibliography
Index
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