On Semisimple Hopf Algebras of Dimension 2m, II

In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension 22n+1 for n ≥ 2 over an algebraically closed field of characteristic 0 with the group of group-like elements isomorphic to ℤ2n×ℤ2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {Z}_{2^{n}}\times \mathbb {Z}_{2^{n}}$\end{document}. 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 2m−1 we give an upper bound for the number of nonisomorphic nontrivial Hopf algebras of dimension 2m which have this group as their group of group-like elements.