A crash course in Lie algebras

✍ Scribed by Skoruppa N.-P.

• Year: 2003
• Tongue: English
• Leaves: 61
• Edition: version 1.2, web draft
• Category: Library

✦ Synopsis

These are the notes of a course on Lie algebras which I gave at the university of Bordeaux in spring 1997. The course was a so-called "Cours PostDEA", and as such had to be held within 12 hours. Even more challenging, no previous knowledge about Lie algebras should be assumed. Nevertheless, I had the goal to reach as peak of the course the character formula for Kac-Moody algebras, and, at the same time, to give complete proofs as far as possible.

✦ Table of Contents

1.1 Definitions......Page 7
1.2 The universal enveloping algebra......Page 11
2.1 Cartan, Killing......Page 13
2.2 Casimir, Weyl......Page 15
2.3 Jordan......Page 17
2.4 Representations of sl 2......Page 19
2.5 Appendix: Engel, Lie......Page 21
3.1 The root space decomposition......Page 23
3.1.1 Appendix: Cartan subalgebras......Page 29
3.2 Root systems......Page 31
3.3 Serre's Theorem and consequences......Page 34
4.1 Definitions......Page 37
4.2 Remarks on a classification......Page 43
4.3 Affine Kac-Moody algebras......Page 44
4.4 Highest weight representations......Page 47
4.5 Character Formula and consequences......Page 49
4.6 Proof of the Character Formula......Page 51
A Dynkin Diagrams......Page 59