Strictification of weakly equivariant Hopf algebras

TL;DR

This paper demonstrates that weakly equivariant Hopf algebras can be replaced by Morita equivalent weak Hopf algebras with strict group actions, preserving their representation categories, but with limitations on coproduct unitality.

Contribution

It introduces a method to strictify weakly equivariant Hopf algebras to weak Hopf algebras with strict group actions, maintaining tensor equivalence of categories.

Findings

Weakly equivariant Hopf algebras can be Morita replaced by strictified weak Hopf algebras.

The strictification preserves the tensor category of representations.

The coproduct in the strictified algebra cannot generally be unital.

Abstract

A weakly equivariant Hopf algebra is a Hopf algebra A with an action of a finite group G up to inner automorphisms. We show that each weakly equivariant Hopf algebra can be replaced by a Morita equivalent algebra B with a strict action of G and with a coalgebra structure that leads to a tensor equivalent representation category. However, the coproduct of this strictification cannot, in general, be chosen to be unital, so that a strictification of the G-action can only be found on a weak Hopf algebra B.