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

Juan Cuadra (Corresponding Author), Ehud Meir (Corresponding Author)

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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: 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.
Original languageEnglish
Pages (from-to)137-158
Number of pages22
JournalJournal of the London Mathematical Society
Volume100
Issue number1
Early online date3 Jan 2019
DOIs
Publication statusPublished - Aug 2019

Keywords

  • 16T05 (primary)
  • 16G30 (secondary)

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