On the algebra structure of some bismash products

TL;DR

This paper investigates the algebraic structure of bismash product Hopf algebras derived from specific finite groups, revealing that most are not group algebras except in special cases, while others are.

Contribution

It demonstrates that certain bismash products from PGL_2(q) are not group algebras, unlike those from Frobenius groups, providing new insights into their algebraic structures.

Findings

Bismash products from PGL_2(q) are not group algebras for most q.

Bismash products from Frobenius groups are isomorphic to group algebras.

Most constructed Hopf algebras are non-trivial and not group algebras.

Abstract

We study several families of semisimple Hopf algebras, arising as bismash products, which are constructed from finite groups with a certain specified factorization. First we associate a bismash product $H_q$ of dimension $q(q-1)(q+1)$ to each of the finite groups $PGL_2(q)$ and show that these $H_q$ do not have the structure (as algebras) of group algebras (except when $q =2,3$). As a corollary, all Hopf algebras constructed from them by a comultiplication twist also have this property and are thus non-trivial. We also show that bismash products constructed from Frobenius groups do have the structure (as algebras) of group algebras.