Strictification of weakly equivariant Hopf algebras

Abstract

A weakly equivariant Hopf algebra is a Hopf algebra $A$ with an action of a finite group $G$ up to inner automorphisms of $A$. We show that each weakly equivariant Hopf algebra can be replaced by a Morita equivalent algebra $A^{str}$ 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 \emph{weak} Hopf algebra $A^{str}$.