Non-existence of integral Hopf orders for twists of several simple groups of Lie type

Abstract

Let p be a prime number and q = p m , with m ≥ 1 if p = 2, 3 and m > 1 otherwise. Let Ω be any non-trivial twist for the complex group algebra of PSL2(q) arising from a 2-cocycle on an abelian subgroup of PSL2(q). We show that the twisted Hopf algebra (CPSL2(q))Ω does not admit a Hopf order over any number ring. The same conclusion is proved for the Suzuki group 2 B2(q) and SL3(p) when the twist stems from an abelian p-subgroup. This supplies new families of complex semisimple (and simple) Hopf algebras that do not admit a Hopf order over any number ring. The strategy of the proof is formulated in a general framework that includes the finite simple groups of Lie type.As an application, we combine our results with two theorems of Thompson and Barry and Ward on minimal simple groups to establish that for any finite non-abelian simple group G there is a twist Ω for CG, arising from a 2-cocycle on an abelian subgroup of G, such that (CG)Ω does not admit a Hopf order over any number ring. This partially answers in the negative a question posed by Meir and the second author.

Existence of integral Hopf orders in twists of group algebras

“…The results obtained so far in [4] and [8,Theorem 3.3] support a negative answer. In this final section, we provide one more instance of partial negative answer through PSL 2n+1 (q).…”

“…The preliminary material necessary for this paper is the same as that of [6], [8], and [4]. For convenience, we briefly collect here the indispensable content and refer the reader to there for further information.…”

“…In this final section, we provide one more instance of partial negative answer through PSL 2n+1 (q). The strategy of proof deployed here is different to that in [4] and [8,Theorem 3.3]. Here, we embed H q,n in a twist of KPSL 2n+1 (q) and exploit the concrete form of the unique Hopf order of H q,n .…”

“…Our first step was taken in [6], where we found that, unlike group algebras, complex semisimple Hopf algebras may not admit Hopf orders over number rings. This striking phenomenon was further explored in [8] and [4]. The semisimple Hopf algebras analyzed in these works are twisted group algebras of non-abelian groups; i.e., group algebras in which the coalgebra structure is modified in Movshev's way.…”