Equivalences of Categories of Modules Over Quantum Groups and Vertex Algebras

Vertex operator algebras are the symmetry algebras of two dimensional conformal field theory. In a famous series of papers, Kazhdan and Lusztig proved an equivalence between particular semisimple categories of modules over affine Lie algebras and quantum groups, the former of which can also be realized as modules over a corresponding vertex operator algebra. Such equivalences between representation categories of vertex operator algebras and quantum groups are now broadly referred to as the Kazhdan-Lusztig correspondence. There has been substantial research interest over the last two decades in understanding the Kazhdan-Lusztig correspondence for vertex operator algebras with non-semisimple representation theory. In this talk I will present an overview of this research area and discuss recent results and future directions.