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: An, the alternating group on n elements, with n ≥ 5; and S2m, the symmetric group on 2m elements, with m ≥ 4 even. The twist for An arises from a 2-cocycle on the Klein four-group contained in A4. The twist for S2m arises from a 2-cocycle on a subgroup generated by certain transpositions which is isomorphic to Zm2 . This provides more examples of complex semisimple Hopf algebras that can not be defined over number rings. As in the previous family known, these Hopf algebras are simple.
- 16T05 (primary)
- 16G30 (secondary)