The Representation Theory of Co-triangular Semisimple Hopf Algebras

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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∈ C[G] 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.