Complex Semisimple Lie Algebras

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.

Nilpotent Lie Algebras and Solvable Lie Algebras

1

(5)

Lower Central Series

1

(1)

Definition of Nilpotent Lie Algebras

1

(1)

An Example of a Nilpotent Algebra

2

(1)

Engel's Theorems

2

(1)

Derived Series

3

(1)

Definition of Solvable Lie Algebras

3

(1)

Lie's Theorem

4

(1)

Cartan's Criterion

4

(1)

Semisimple Lie Algebras (General Theorems)

5

(5)

Radical and Semisimplicity

5

(1)

The Cartan-Killing Criterion

6

(1)

Decomposition of Semisimple Lie Algebras

6

(1)

Derivations of Semisimple Lie Algebras

7

(1)

Semisimple Elements and Nilpotent Elements

7

(1)

Complete Reducibility Theorem

8

(1)

Complex Simple Lie Algebras

8

(1)

The Passage from Real to Complex

9

(1)

Cartan Subalgebras

10

(7)

Definition of Cartan Subalgebras

10

(1)

Regular Elements: Rank

10

(1)

The Cartan Subalgebra Associated with a Regular Element

11

(1)

Conjugacy of Cartan Subalgebras

12

(3)

The Semisimple Case

15

(1)

Real Lie Algebras

16

(1)

The Algebra sl2 and Its Representations

17

(7)

The Lie Algebra sl2

17

(1)

Modules, Weights, Primitive Elements

17

(1)

Structure of the Submodule Generated by a Primitive Element

18

(1)

The Modules Wm

19

(1)

Structure of the Finite-Dimensional g-Modules

20

(1)

Topological Properties of the Group SL2

21

(1)

Applications

22

(2)

Root Systems

24

(19)

Symmetries

24

(1)

Definition of Root Systems

25

(1)

First Examples

26

(1)

The Weyl Group

27

(1)

Invariant Quadratic Forms

27

(1)

Inverse Systems

28

(1)

Relative Position of Two Roots

29

(1)

Bases

30

(1)

Some Properties of Bases

31

(2)

Relations with the Weyl Group

33

(1)

The Cartan Matrix

34

(1)

The Coxeter Graph

35

(1)

Irreducible Root Systems

36

(1)

Classification of Connected Coxeter Graphs

37

(1)

Dynkin Diagrams

38

(1)

Construction of Irreducible Root Systems

39

(2)

Complex Root Systems

41

(2)

Structure of Semisimple Lie Algebras

43

(13)

Decomposition of g

43

(2)

Proof of Theorem 2

45

(2)

Borel Subalgebras

47

(1)

Weyl Bases

48

(2)

Existence and Uniqueness Theorems

50

(1)

Chevalley's Normalization

51

(5)

Appendix. Construction of Semisimple Lie Algebras by Generators and Relations

52

(4)

Linear Representations of Semisimple Lie Algebras

56

(10)

Weights

56

(1)

Primitive Elements

57

(1)

Irreducible Modules with a Highest Weight

58

(2)

Finite-Dimensional Modules

60

(2)

An Application to the Weyl Group

62

(1)

Example: sln+1

62

(1)

Characters

63

(1)

H. Weyl's formula

64

(2)

Complex Groups and Compact Groups

66

(6)

Cartan Subgroups

66

(1)

Characters

67

(1)

Relations with Representations

68

(1)

Borel Subgroups

68

(1)

Construction of Irreducible Representations from Borel Subgroups

69

(1)

Relations with Algebraic Groups

70

(1)

Relations with Compact Groups

70

(2)

Bibliography

72

(1)

Index

73

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Complex Semisimple Lie Algebras

Summary

Table of Contents

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