A crash course in Lie algebras

Skoruppa N.-P.

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.

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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