  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.
