Title: Exploring representations of Kac-Moody algebras using combinatorics, algebra and geometry
Abstract: Finite dimensional modules of complex simple Lie algebras have been studied for a long time. The theory involves understanding roots, root spaces, and the action of the Weyl group. All this structure exists for the larger class of (symmetric) Kac-Moody algebras. A natural generalization of finite dimensional modules is integrable highest weight modules. We consider these in some detail, using the naïve approach that to understand a module one should understand the action in some explicit basis. We mostly focus on the affine Kac-Moody algebras. The story involves a combinatorial aspect (crystals, and MV polytopes, which index the basis) an algebraic approach (PBW bases), and a geometric approach (Quiver varieties). Much of this is expository, but it is related to a 2011 paper with Baumann and Kamnitzer. At the end I will mention some interesting observations outside of affine type, which is the topic of some current work with Patrick Chan.