北京雁栖湖应用数学研究院

2023年02月27日 至 05月23日

You should be familiar with fundamental notions in algebra and representation theory (basic ring theory, associative algebras, Lie algebras and their universal enveloping algebras, highest weight theory of finite-dimensional Lie algebras). This will be a useful background for the whole course. For the second part of the course, it will be good if you have followed a course on Hopf algebras, braided tensor categories and quantum groups (but I will spend a few lectures on recalling this material).

This course roughly splits up into two parts. In the first part we focus on infinite-dimensional Lie algebras such as the oscillator and Virasoro algebras and their representations. We will then look at Kac-Moody Lie algebras, especially those of affine type. In the second part we will explore q-deformed enveloping algebras of Kac-Moody Lie algebras, focusing on the affine sl2 example. We will have two online lectures per week, about 90 minutes per lecture, with plenty of time for questions and discussions. In order to fully appreciate the material, you should spend at least the same amount of time per week working on homework exercises, which I will set in the lectures.

Bombay Lectures on Highest Weight Representations Representations of Infinite-Dimensional Lie Algebras, V. Kac and A. Raina, Advanced Series in Mathematical Physics: Volume 2.
Lie Algebras of Finite and Affine Type, R. Carter, Cambridge studies in advanced mathematics 96, Cambridge University Press, 2005.
A Guide to Quantum Groups, V. Chari and A. Pressley, Cambridge University Press, 1994.

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