Let V be an affine vertex algebra of some simple Lie algebra 𝔤 and some level. Let KL be the category of V-modules whose conformal weight spaces are integrable 𝔤-modules. A famous result of Kazhdan and Lusztig tells us that for almost all levels KL is a braided tensor category and as such equivalent to a category of weight modules of the quantum group Uq(𝔤) of 𝔤 for suitable q.
It is desired to have similar results for suitable categories of W-algebras and superalgebras. In particular one wants to understand tensor structure and equivalences to quantum supergroups.
I will outline how to prove such statements and illustrate this in some examples.
This video was produced by the International Centre for Mathematical Sciences, as part of the workshop Geometric Representation Theory and W-algebras.
