Hopf Powers and Orders for Some Bismash Products

The Hopf order of an element $h$ of a Hopf algebra $H$ is the least $n$ such that the $n$-th Hopf power of $h$ is trivial. For some bismash product Hopf algebras obtained from factorizable groups (including Drinfeld doubles of some groups) we compute the dimensions of the spaces of elements with trivial $n$-th Hopf powers, and determine which Hopf orders of elements are possible. We discuss the behavior under twists and duality.