On semisimple Hopf algebras of dimension $2^m$, II

Abstract

In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension $2^{2n+1}$ for $n\ge 2$ over an algebraically closed field of characteristic $0$ with the group of group-like elements isomorphic to ${\mathbb{Z}}_{2^n}\times {\mathbb{Z}}_{2^n}$. Moreover we classify all such nonisomorphic Hopf algebras of dimension $32$ and show that they are not twist-equivalent to each other. More generally, given an abelian group of order $2^{m-1}$ we give an upper bound for the number of nonisomorphic nontrivial Hopf algebras of dimension $2^m$ which have this group as their group of group-like elements.