# Introduction to Lie algebras

**Dozent:** Prof. Dr. Nicolas Perrin

**Zeit:** Mittwoch 10-12 und Freitag 10-12

Übung: Freitag 14-16

**Raum:** Lectures in LWK 0.008, Exercices in LWK 0.006 and 0.007.

Noten: here

Lecture Notes:

PART I: General properties of Lie algebras

Chapter 1, general definitions on algebras: PDF

Chapter 2, first basic definitions on Lie algebras: PDF

Chapter 3, envelopping algebras: PDF

Chapter 4, representations, first definitions and properties: PDF

Chapter 5, nilpotent Lie algebras: PDF

Chapter 6, Jordan-Chevalley decomposition: PDF

Chapter 7, solvable Lie algebras: PDF

Chapter 8, semisimple Lie algebras: PDF

PART II: Classification of complex semisimple Lie algebras.

Chapter 9, Cartan subalgebras: PDF

Chapter 10, the lie algebra sl2: PDF

Chapter 11, root systems: PDF

Chapter 12, classification of connected Coxeter graphs: PDF

Chapter 13, classification of complex semisimple Lie algebras: PDF

Chapter 14, representations of semisimple Lie algebras: PDF (Weyl's character formula is stated without proof).

Chapter 15, Poicaré-Birkhoff-Witt theorem: PDF (only a statement of the result, no proof).

Chapter 16, Groups: PDF (an overview of the classification of complex semisimple Lie groups).

Exercice sheets:

Sheet 1, to be hand in on Wednesday 21.10.2009 and corrected on Friday 23.10.2009: PDF

Sheet 2, to be hand in on Wednesday 28.10.2009 and corrected on Friday 30.10.2009: PDF

Sheet 3, to be hand in on Wednesday 04.11.2009 and corrected on Friday 06.11.2009: PDF

Sheet 4, to be hand in on Wednesday 11.11.2009 and corrected on Friday 13.11.2009: PDF

Sheet 5, to be hand in on Wednesday 18.11.2009 and corrected on Friday 20.11.2009: PDF

Sheet 6, to be hand in on Wednesday 25.11.2009 and corrected on Friday 27.11.2009: PDF

Sheet 7, to be hand in on Wednesday 02.12.2009 and corrected on Friday 04.12.2009: PDF

Sheet 8, to be hand in on Wednesday 09.12.2009 and corrected on Friday 11.12.2009: PDF

Sheet 9, to be hand in on Wednesday 16.12.2009 and corrected on Friday 18.12.2009: PDF

Sheet 10, to be hand in on Wednesday 06.01.2010 and corrected on Friday 08.01.2010: PDF

Sheet 11, to be hand in on Wednesday 20.01.2010 and corrected on Friday 22.01.2010: PDF

Sheet 12, to be hand in on Wednesday 27.01.2010 and corrected on Friday 29.01.2010: PDF

Sheet 13, to be hand in on Wednesday 03.03.2010 and corrected on Friday 05.01.2010: PDF

Abstract:

Lie algebras appear almost everywhere in mathematics. Furthermore, the theory of Lie algebras is build on very simple arguments of linear algebra and the classification of semisimple Lie algebras over an algebraically closed field of characteristic zero is very simple. All this make the theory of Lie algberas very attractive.

In these lectures we will start from the beginning the theory of Lie algebras and their representations. There will be mainly two topics in the lecture:

- The first part will be devoted to general theory of Lie algebras.
- Thesecond part will be devoted to classification of complex semisimple Lie algebras in terms of Dynkin diagrams and root system.
- The (short) third part will be devoted to the proof Weyl's character formula.

If time permits we may study Lie algebras over the field of real numbers or look at Jordan algebras.

**References:**

- Bourbaki, Nicolas. Groupes et algèbres de Lie 1. Hermann 1954.
- Humphreys, James E. Introduction to Lie algebras and representation theory. Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York-Berlin, 1978.
- Jacobson, Nathan. Lie algebras. Interscience Tracts in Pure and Applied Mathematics, No. 10 Interscience Publishers, New York-London 1962.
- Serre, Jean-Pierre. Algèbres de Lie semi-simples complexes. W. A. Benjamin, inc., New York-Amsterdam 1966.