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

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.

• Semisimple Lie algebras and their finite dimensional representations
• basic knowledge in algebraic geometry, for example, of projective complex varieties

• Infinite-Dimensional Lie Algebras, Victor G. Kac,
• Infinite-Dimensional Lie Algebras (An Introduction), Victor G. Kac,
• Infinite-dimensional Lie algebras, Minori Wakimoto,
• Kac-Moody Groups, their Flag Varieties and Representation Theory, Shrawan Kumar.

Lectures

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

There will be exercise problems posed in the lectures.

Exams

There will be an oral exam at the end of the semester. More information on the dates will follow.

Links

The BASIS entry for the lecture.