On the existence of orders in semisimple Hopf algebras. (Q2787980)

The famous Frobenius theorem on finite groups \(G\) states that the degree of any irreducible complex representation of \(G\) divides \(|G|\). Kaplanski conjectured that this theorem carries over to semisimple Hopf algebras. Larson proved that Kaplanski's conjecture holds for Hopf algebras which admit an integral form over a ring of algebraic numbers. The authors show that such integral forms do not exist for a class of Hopf algebras obtained via Drinfeld twists. However, these examples all satisfy Kaplanski's conjecture. On the other hand, it is shown that the Frobenius theorem holds for a complex semisimple Hopf algebra if and only if it admits a weak integral form in the sense of Rumynin and Lorenz, or equivalently, if the Casimir element satisfies a monic integral polynomial.