What is categorification?

If we de-categorify Vector-Spaces, we recover the ring of integers as follows. We first replace a vector space with its dimension; that is to say, we only keep track of isomorphism classes of objects. In so doing, we also replace direct sum with addition of those integers, and replace tensor product with multiplication. To categorify is to undo this process. For instance, one might start with the ring of symmetric functions and realize it has replaced the representation theory of the symmetric group.

In this talk, I will discuss how Khovanov-Lauda-Rouquier (KLR) algebras categorify quantum groups. I will discuss their simple modules, and in particular that they carry the combinatorial structure of a crystal graph. This is joint work with Aaron Lauda.