Conjugacy Classes, Class Sums and Character Tables for Hopf Algebras

We extend the notion of conjugacy classes and class sums from finite groups to semisimple Hopf algebras and show that the conjugacy classes are obtained from the factorization of H as irreducible left D(H)-modules. For quasitriangular semisimple Hopf algebras H, we prove that the product of two class sums is an integral combination of the class sums up to d ^-2 where d = dim H. We show also that in this case the character table is obtained from the S-matrix associated to D(H). Finally, we calculate explicitly the generalized character table of D(kS ₃, which is not a character table for any group. It moreover provides an example of a product of two class sums which is not an integral combination of class sums.