Some Properties of Finite-Dimensional Semisimple Hopf Algebras

Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH). It was proved by Montgomery and Witherspoon that the conjecture is true for H of dimension p^n, p prime, and by Nichols and Richmond that if H has a 2-dimensional representation then dimH is even. In this paper we first prove that if V is an irreducible representation of D(H), the Drinfeld double of any finite-dimensional semisimple Hopf algebra H over k, then dimV divides dimH (not just dimD(H)=(dimH)^2). In doing this we use the theory of modular tensor categories (in particular Verlinde formula). We then use this statement to prove that Kaplansky's conjecture is true for finite-dimensional semisimple quasitriangular Hopf algebras over k. As a result we prove easily the result of Zhu that Kaplansky's conjecture on prime dimensional Hopf algebras over k is true, by passing to their Drinfeld doubles. Second, we use a theorem of Deligne on characterization of tannakian categories to prove that triangular semisimple Hopf algebras over k are equivalent to group algebras as quasi-Hopf algebras.