Homomorphisms and Rigid Isomorphisms of Twisted Group Doubles

Abstract

We prove several results concerning quasi-bialgebra morphisms ${\mathcal {D}^{\omega }(G)\to \mathcal {D}^{\eta }(H)}$ of twisted group doubles. We take a particular focus on the isomorphisms which are simultaneously isomorphisms ${\mathcal {D}(G)\to \mathcal {D}(H)}$ and completely determine them. Whenever ω ∈ Z3(G/Z(G), U(1)) this suffices to completely describe ${\text {Aut}(\mathcal {D}^{\omega }(G))}$, the group of quasi-Hopf algebra isomorphisms of ${\mathcal {D}^{\omega }(G)}$, and so generalizes existing descriptions for the case where ω is trivial.