Triviality theorems for Yetter–Drinfel'd Hopf algebras

Abstract

Under suitable assumptions on the base field, we prove that a commutative semisimple Yetter–Drinfel'd Hopf algebra over a finite abelian group is trivial, i.e., is an ordinary Hopf algebra, if its dimension is relatively prime to the order of the finite abelian group. Furthermore, we prove that a finite-dimensional cocommutative cosemisimple Yetter–Drinfel'd Hopf algebra contains a trivial Yetter–Drinfel'd Hopf subalgebra of dimension greater than one, at least if the Yetter–Drinfel'd Hopf algebra itself has dimension greater than one.