The Representation Theory of Co-triangular Semisimple Hopf Algebras

In a previous paper we prove that any semisimple triangular Hopf algebra A over an algebraically closed field of characteristic 0 (say the field of complex numbers C) is obtained from a finite group after twisting the ordinary comultiplication of its group algebra in the sense of Drinfeld; that is A=C[G]^J for some finite group G and a twist J\in C[G]\ot C[G]. In this paper we explicitly describe the representation theory of co-triangular semisimple Hopf algebras A^=(C[G]^J)^ in terms of representations of some associated groups. As a corollary we prove that Kaplansky’s 6th conjecture from 1975 holds for A^; that is that the dimension of any irreducible representation of A^ divides the dimension of A.