Generators and Relations Talk 1

From ScottWiki

I'll describe some of Kuperberg's work on `spiders', which provide a diagrammatic approach to the representation theory of $\mathfrak{sl}_3$.

I'll then tell you about my work on generalising this to $U_q(\mathfrak{sl}_n})$. Firstly I'll describe a category of diagrams (with no relations), and a full functor into the representation theory. The main task then is to calculate the kernel.

To do this, I'll use Gelfand-Tsetlin bases. It will turn out that (with some rejiggering of the categories involved) the forgetful functor $Rep(U_q(\mathfrak{sl}_{n+1})) \Rightarrow Rep(U_q(\mathfrak{sl}_n))$ can be lifted to an easy to understand functor between the categories of diagrams. Eventually, this leads to an inductive method of finding relations, and (hopefully!) to show that we've found all of them.