Hopf algebras and finite tensor categories in conformal field theory

Understand

In conformal field theory the understanding of correlation functions can be divided into two distinct conceptual levels: The analytic properties of the correlators endow the representation categories of the underlying chiral symmetry algebras with additional structure, which in suitable cases is the one of a finite tensor category.

• The problem of specifying the correlators can then be encoded in algebraic structure internal to those categories.
• After reviewing results for conformal field theories for which these representation categories are semisimple, we explain what is known about representation categories of chiral symmetry algebras that are not semisimple.
• We focus on generalizations of the Verlinde formula, for which certain finite-dimensional complex Hopf algebras are used as a tool, and on the structural importance of the presence of a Hopf algebra internal to finite tensor categories.