Structure constants related to symmetric Hopf algebras

Let k be an algebraically closed field of characteristic 0. In this paper we continue our study of structure constants for semisimple Hopf algebras H whose character algebra is commutative, and for non-semisimple factorizable ribbon Hopf algebras. This is done from the point of view of symmetric algebras, such as group algebras. In particular we consider general fusion rules which are structure constants associated to products of irreducible characters and structure constants associated to generalizations of class sums and conjugacy classes. Our methods are reminiscent on one hand of the methods employed when analyzing tilting modules for certain quantum groups and on the other hand of "splitting modules" for certain Drinfeld doubles. The family of irreducible characters is divided in two according to the vanishing of their quantum dimension. The fusion rules on the character algebra are computed with respect to this division.