Complex semisimple Lie algebras

✍ Scribed by Serre J.-P.

• Publisher: Springer
• Year: 2000
• Tongue: English
• Leaves: 86
• Category: Library

✦ Synopsis

These short notes, already well-known in their original French edition, present the basic theory of semisimple Lie algebras over the complex numbers. 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 linear representations. The last chapter discusses the connection between Lie algebras, complex groups and compact groups. The book is intended to guide the reader towards further study.

✦ Table of Contents

Cover......Page 1
Contents......Page 9
2. Definition of Nilpotent Lie Algebras......Page 12
4.Engel's Theorems......Page 13
6. Definition of Solvable Lie Algebras......Page 14
8. Cartan's Criterion......Page 15
1. Radical and Semisimplicity......Page 16
3. Decomposition of Semisimple Lie Algebras......Page 17
5. Semisimple Elements and Nilpotent Elements......Page 18
7. Complex Simple Lie Algebras......Page 19
8. The Passage from Real to Complex......Page 20
2. Regular Elements: Rank......Page 21
3. The Cartan Subalgebra Associated with a Regular Element......Page 22
4. Conjugacy of Cartan Subalgebras......Page 23
5. The Semisimple Case......Page 26
6. Real Lie Algebras......Page 27
2. Modules, Weights, Primitive Elements......Page 28
3. Structure of the Submodule Generated by a Primitive Element......Page 29
4. The Modules W_m......Page 30
5. Structure of the Finite-Dimensional g-Modules......Page 31
6. Topological Properties of the Group SL_2......Page 32
7. Applications......Page 33
1. Symmetries......Page 35
2. Definition of Root Systems......Page 36
3. First Examples......Page 37
5. Invariant Quadratic Forms......Page 38
6. Inverse Systems......Page 39
7. Relative Position of Two Roots......Page 40
8. Bases......Page 41
9. Some Properties of Bases......Page 42
10. Relations with the Weyl Group......Page 44
11. The Cartan Matrix......Page 45
12. The Coxeter Graph......Page 46
13. Irreducible Root Systems......Page 47
14. Classification of Connected Coxeter Graphs......Page 48
15. Dynkin Diagrams......Page 49
16. Construction of Irreducible Root Systems......Page 50
17. Complex Root Systems......Page 52
1. Decomposition of g......Page 54
2. Proof of Theorem 2......Page 56
3. Borel Subalgebras......Page 58
4. Weyl Bases......Page 59
5. Existence and Uniqueness Theorems......Page 61
6. Chevalley's Normalization......Page 62
Appendix. Construction of Semisimple Lie Algebras by Generators and Relations......Page 63
1. Weights......Page 67
2. Primitive Elements......Page 68
Uniqueness......Page 69
Existence......Page 70
4. Finite-Dimensional Modules......Page 71
6. Example: sl_{n+1}......Page 73
7. Characters......Page 74
8. H.Weyl's Formula......Page 75
1. Cartan Subgroups......Page 77
2. Characters......Page 78
4. Borel Subgroups......Page 79
5. Construction of Irreducible Representations from Borel Subgroups......Page 80
7. Relations with Compact Groups......Page 81
Bibliography......Page 83
Index......Page 84
Back cover......Page 86