  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.
