TY - UNPB

T1 - Existence of integral Hopf orders in twists of group algebras

AU - Cuadra, Juan

AU - Meir, Ehud

PY - 2022/10/31

Y1 - 2022/10/31

N2 - 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,J^-1∈X⊗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.

AB - 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,J^-1∈X⊗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.

U2 - 10.48550/arXiv.2211.00097

DO - 10.48550/arXiv.2211.00097

M3 - Preprint

BT - Existence of integral Hopf orders in twists of group algebras

PB - ArXiv

ER -