## University of Sydney Algebra Seminar

# Peter Tingley (University of Melbourne)

## Friday 12th June, 12.05-12.55pm, Carslaw 375

### Quantum groups, braidings and crystals

We will discuss two beautiful facts about the representation theory of quantized universal enveloping algebras. The first is that the category of representations is braided, which is a key ingredient in the celebrated quantum group knot invariants. The second is the existence of crystal bases for the representations. These are extremely nice bases which, among other things, describe much of the structure of the representations in a purely combinatorial way. We will then discuss how these things are related. Much of this talk is expository, but I will end with a precise statement of this relationship which was developed in joint work with Joel Kamnitzer.