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 KG afforded by a non-degenerate 2-cocycle on the character group M . We show that if there is a Lagrangian decomposition M ≃ L × L such that L is contained in a normal abelian subgroup N of G, then the twisted group algebra (KG)J admits a Hopf order X over R. The Hopf order X is constructed as the R-submodule generated by the primitive idempotents of KN and the elements of G. It is indeed a Hopf order of KG such that J ±1 ∈ X ⊗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.

A modest proposal: explicit support for foundational pluralism

“…See e.g., the note accompanying Theorem 7.2 here[12], or the note preceding Lemma 43 here[25] 3. As in the Agda standard library, for example: https://agda.github.io/agda-stdlib/v1.1/ Relation.Nullary.Negation.html#1727.…”