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: 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 Z m 2 . 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.Similarly to the previous family, they are constructed as Drinfel'd twists of certain group algebras. The twist is in turn constructed using the following method due to Movshev [16].Let M be an abelian subgroup of G. Let K be a number field. Suppose that K is large enough so that the group algebra KM splits. Denote by M the character group of M . For φ ∈ M , let e φ be the associated idempotent in KM . If ω : M × M → K × is a normalized 2-cocycle, thenis a twist for KG. The twisting procedure alters the coalgebra structure and the antipode of KG but leaves unchanged the algebra structure. So, as algebras, these Hopf algebras are group algebras.Here we deal with the following two families of groups and twists:(1) The alternating group A n on n elements, with n 5. Consider the double transpositions d 1 = (12)(34) and d 2 = (13)(24). The subgroup M is generated by them. The character group M is generated by ϕ i , for i = 1, 2, given by ϕ i (d j ) = (−1) δij . The 2-cocycle ω is the bicharacter:This family was introduced by Nikshych in [17] and provided the first non-trivial examples of simple and semisimple Hopf algebras.(2) The symmetric group S 2m on 2m elements, with m 4 even.This family was introduced by Bichon in [2] and further discussed by Galindo and Natale in [10]. These examples are also simple.Our main results, Theorems 3.3 and 4.5, are abridged in the following statement:Theorem. Let K be a number field with ring of integers O K . Let R ⊂ K be a Dedekind domain such that O K ⊆ R. Let G be one of the above groups and J the twist arising from the corresponding cocycle. If the twisted Hopf algebra (KG) J admits a Hopf order over R, then 1 2 ∈ R.As a consequence, (KG) J does not admit a Hopf order over any number ring.This theorem implies that the complexified Hopf algebra (CG) J does not admit a Hopf order over any number ring.The strategy of the proof can be outlined as follows. Hopf orders are inherited to Hopf subalgebras and quotient Hopf algebras (Proposition 1.1). For our purposes, this fact allows us to focus immediately on A 5 in the case of A n . The case of S 2m needs several reductions to subgroups and quotient groups to focus ultimately on S 8 . Assume now that H is a Hopf algebra over K and X is a Hopf order of H over R. Consider the dual Hopf algebra H * . The dual order X consists of those ϕ ∈ H * such that ϕ(X) ⊆ R. Th...