The first week of the course will focus on the category O for finite-dimensional simple Lie algebras, and will cover the material found in the first few chapters of Humphrey’s book Representations of semisimple Lie algebras in the BGG category O. The second week will deal with the case of affine Lie algebras. The focus will be on discussing integrable weight representations. The connections between positive level representations of affine Lie algebras and the finite-dimensional representations of the maximal parabolic of the affine Lie algebra will be made. Finally, the course will discuss how the methods and problems used and studied in O can be formulated and studied to understand the category of finite-dimensional representations of the affine Lie algebra.

We will introduce vertex algebras and operator product expansion from first prin- ciples, as well as touch on their applications to Lie algebras, representation theory, and modular forms. A number of examples will be featured, including vertex Lie algebras, lattice vertex algebras, and W-algebras. If time permits, we will discuss some of the geometry around vertex algebras in the context of conformal blocks.