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.
Bibliographical noteJuan Cuadra was supported by grant MTM2017-86987-P from MICINN and FEDER and by the research group FQM0211 from Junta de Andalucía. Ehud Meir was supported by the RTG 1670 “Mathematics inspired by String theory and Quantum Field Theory”.
The authors would like to thank Sonia Natale for her comments on a first version
of this paper and, specially, for drawing their attention to . The authors are
indebted to the attentive referee for his careful revision and apt comments, which
helped to improve the original manuscript.
- 16T05 (primary)
- 16G30 (secondary)