Simple Hopf algebras and deformations of finite groups

TL;DR

This paper investigates the simplicity and semisolvability of Hopf algebras derived from specific group deformations, revealing that these properties are not solely determined by their representation categories.

Contribution

It demonstrates that certain twisting deformations of supersolvable and symmetric groups produce simple Hopf algebras, challenging existing assumptions about their classification.

Findings

Certain supersolvable group deformations are simple Hopf algebras.

Twisting deformations of the symmetric group can be simple.

Deformations of nilpotent groups are semisolvable.

Abstract

We show that certain twisting deformations of a family of supersolvable groups are simple as Hopf algebras. These groups are direct products of two generalized dihedral groups. Examples of this construction arise in dimensions 60 and p^2q^2, for prime numbers p, q with q dividing p-1. We also show that certain twisting deformation of the symmetric group is simple as a Hopf algebra. On the other hand, we prove that every twisting deformation of a nilpotent group is semisolvable. We conclude that the notions of simplicity and (semi)solvability of a semisimple Hopf algebra are not determined by its tensor category of representations.