Summer 24 - V5A5 - Advanced Topics in Representation Theory - Kac-Moody Algebras and Flag Varieties

This is the course website for the lecture course Advanced Topics in Representation Theory in Summer 2024.

Content

Kac-Moody algebras and groups are (mostly infinite-dimensional) generalisations of semisimple Lie algebras and groups. We recall the classification and Serre relations of semisimple Lie algebras via their root system as well as their Verma modules and highest weight representations. We then define Kac-Moody Lie algebras via their generalised Cartan matrix and discuss their structure and representation theory in analogy to the semisimple Lie algebras. Lastly, we will discuss the construction of Kac-Moody groups and flag varieties. We will discuss the geometry of their Schubert varieties, with a focus on the affine Grassmannian.

Prerequisites

Lecture Notes

Preliminary lecture notes will be found here.

Sources

Some sources that the course is based on:

Schedule

Lectures

There are two lectures per week:

Wednesday 12-14, Room: Endenicher Allee 60, Seminarraum 1.008,
Friday 10-12, Room: Endenicher Allee 60, Seminarraum 0.006.

Homework

There will be exercise problems posed in the lectures.

Exams

There will be an oral exam on Friday, the 26th of July. More information on the time will follow.