Non-existence of Hopf orders for a twist of the alternating and symmetric groups

TL;DR

This paper proves that certain complex semisimple Hopf algebras, constructed as twists of group algebras of alternating and symmetric groups, cannot be realized over number rings, highlighting limitations in their algebraic structure.

Contribution

It demonstrates the non-existence of Hopf orders over number rings for specific twisted group algebra constructions involving alternating and symmetric groups.

Findings

Hopf algebras from twists of $A_n$ and $S_{2m}$ lack Hopf orders over number rings.

These Hopf algebras are simple and provide new examples of such limitations.

The twists are derived from specific 2-cocycles on subgroups.

Abstract

We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfel'd twists of group algebras for the following groups: $A_n$, the alternating group on $n$ elements, with $n \geq 5$; and $S_{2m}$, the symmetric group on $2m$ elements, with $m \geq 4$ even. The twist for $A_n$ arises from a $2$-cocycle on the Klein four-group contained in $A_4$. The twist for $S_{2m}$ arises from a $2$-cocycle on a subgroup generated by certain transpositions which is isomorphic to $\mathbb{Z}_2^m$. This provides more examples of complex semisimple Hopf algebras that can not be defined over number rings. As in the previous family known, these Hopf algebras are simple.