# Quantum groups, braidings and crystals

*Algebra/Geometry/Topology Seminar*

*by Peter Tingley*

*Institution:*Melbourne

*Date: Mon 9th March 2009*

*Time: 2:15 PM*

*Location: David Caro Building, Room 211*

*Abstract*: I will introduce quantized universal enveloping algebras and their representations. I will then discuss two beautiful facts about this theory. 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. I will then discuss how these things are related. In particular, there is a structure on the combinatorial category of crystals which is analogous to the braiding, except that the braid group is replaced by the so called cactus group. Like the braid group, the cactus group is the fundamental group of a nice space, so I will end with some topology. This talk is largely expository, and will be presented via examples.