## Advanced Algebra I: Lie algebras and their representations

Thorsten Heidersdorf

Universität Bonn WS 2020/21

Tuesday, 8.15 - 10.00 am, Thursday 10.15 - 12.00 am

First lecture: 27.10.2020

First tutorials: 02.11.2020

**Exam**: 1st try: 22.02.2021, 2nd try: 31.03.2021

**Assistant**: Edgar Assing

**Zoom login**: If you would like to attend the course, please send me an email.

**eCampus**: The eCampus system is running now. You can find the lecture notes there as well as a forum for all kinds of questions about the course. You need to register there to join the tutorials and submit your exercise sheets.

**Tutorials**: Monday, 10.00 - 12.00 (Zbigniew Wojciechowski), Wednesday 12.00 - 14.00 (Lukas Bonfert) and Wednesday 16.00 - 18.00 (Siu Hang Man). Registration will be done via eCampus on a first come, first serve basis. Remember to switch on your video for the tutorials.

**The registration opens Wednesday, 21.00pm**

**Exercises**: Exercises will be published on eCampus each Friday by 4.00pm. They need to be submitted the following week on Friday 4.00pm. You need to get 50% of the possible points for the exam admission.

**Content**: Lie algebras and their representations are ubiquitious in many branches of mathematics. They arise for example as tangent spaces of the identity element of a Lie group or algebraic group. As such they control major parts of the structure and representation theory of Lie groups and algebraic groups. They play a pivotal role not only in representation theory, but also in algebraic and differential geometry and theoretical physics.

This course will prove the standard results on the classification and (finite dimensional) representation theory of finite dimensional semisimple Lie algebras in characteristic 0.

### Main topics

- Basics on Lie algebras, Poincare-Birkhoff-Witt theorem
- Levi-Malcev decomposition for Lie algebras
- Weyl's complete reducibility for semisimple Lie algebras
- Root space decomposition for semisimple Lie algebras
- Classification of semisimple Lie algebras
- Finite dimensional representations of semisimple Lie algebras, Verma modules, highest weight theory
- Weyl character formula
- Schur-Weyl duality, Irreducible representations in type ABCD

**Prerequisites**: Good knowledge of linear algebra. Some background on R-modules (e.g. tensor products, symmetric algebra, tensor algebra), algebras and basics of the language of categories and functors.

Not needed: Knowledge of "F4A1 Foundations in Representation Theory" (Schroer SS 20).