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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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A CRASH COURSE IN LIE ALGEBRAS Nils-Peter Skoruppa
Universit´ e Bordeaux 1 — UFR Math´ ematiques et Informatique — Ecole doctorale 1998 —
Version: Id: lie.tex,v 1.2 2003/11/20 22:49:00 fenrir Exp
c Nils-Peter Skoruppa, 1998 - 2003 ° www.countnumber.de
Preface 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 KacMoody algebras, and, at the same time, to give complete proofs as far as possible. Accordingly, the present notes are a somewhat streamlined approach to Kac-Moody algebras, and some absolutely crucial topics (like root systems and Weyl groups) are only very briefly touched. Proofs are almost always completely given, with a few exceptions: some facts about root systems and Weyl groups, and the classical basic theorems of Lie, Engel and Cartan (criterion for solvability) are explained, but not proved. However, the missing proofs can be easily complemented from the literature. The material is taken from various sources, in particular from courses of Serre, the book of Humphreys and the basic monograph of Kac on KacMoody algebras. Probably there is not a single theorem i