Non-existence of Hopf orders for a twist of the alternating and symmetric groups

Abstract

We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfel'd twists of group algebras for the following groups: (Formula presented.), the alternating group on (Formula presented.) elements, with (Formula presented.), and (Formula presented.), the symmetric group on (Formula presented.) elements, with (Formula presented.) even. The twist for (Formula presented.) arises from a 2-cocycle on the Klein four-group contained in (Formula presented.). The twist for (Formula presented.) arises from a 2-cocycle on a subgroup generated by certain transpositions, which is isomorphic to (Formula presented.). This provides more examples of complex semisimple Hopf algebras that cannot be defined over number rings. As in the previous family known, these Hopf algebras are simple.