UAlberta-PIMS Mathematics and Statistics Colloquium: Jinwei Yang

The tensor category of vertex operator algebras arises from two dimensional conformal field theories and has many applications such as representation theory, topological quantum field theory and quantum computations. A major problem of tensor category theory of vertex operator algebras is the existence of the tensor category structure, especially when the vertex operator algebra is neither rational nor C_2-cofinite. We develop a few general methods to establish the tensor category structure on representation categories of vertex operator algebras, based on Huang-Lepowsky-Zhang's tensor category theory. In particular, we obtain the tensor structures on representation categories of affine Lie algebras at various positive rational levels and of the Virasoro algebra at all central charges. As applications, we are able to study representations and tensor categories of many less-well-understood vertex operator algebras including affine Lie superalgebras, singlet algebras, and certain W-algebras.