Existence of integral Hopf orders in twists of group algebras

Abstract

We find a group-theoretical condition under which a twist of a group algebra, in Movshev's way, admits an integral Hopf order. Let $K$ be a (large enough) number field with ring of integers $R$. Let $G$ be a finite group and $M$ an abelian subgroup of $G$ of central type. Consider the twist $J$ for $K\hspace{-0.8pt}G$ afforded by a non-degenerate $2$-cocycle on the character group $\widehat{M}$. We show that if there is a Lagrangian decomposition $\widehat{M} \simeq L \times \widehat{L}$ such that $L$ is contained in a normal abelian subgroup $N$ of $G$, then the twisted group algebra $(K\hspace{-0.8pt}G)_J$ admits a Hopf order $X$ over $R$. The Hopf order $X$ is constructed as the $R$-submodule generated by the primitive idempotents of $K\hspace{-1.1pt}N$ and the elements of $G$. It is indeed a Hopf order of $K\hspace{-0.8pt}G$ such that $J^{\pm 1} \in X \otimes_R X$. Furthermore, we give some criteria for this Hopf order to be unique. We illustrate this construction with several families of examples. As an application, we provide a further example of simple and semisimple complex Hopf algebra that does not admit integral Hopf orders.