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

TL;DR

This paper classifies semisimple Hopf algebras of certain dimensions with specific group-like structures, providing explicit classifications for dimension 32 and bounds for higher dimensions, advancing understanding of their algebraic structures.

Contribution

It offers a complete classification of semisimple Hopf algebras of dimension 2^{2n+1} with a specific group of group-like elements and establishes bounds for their enumeration in higher dimensions.

Findings

Classified all such Hopf algebras of dimension 32

Proved they are not twist-equivalent

Provided upper bounds for nonisomorphic Hopf algebras in higher dimensions

Abstract

In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension $2^{2n+1}$ for $n\geq 2$ 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.