Fixed-point Free Endomorphisms and Hopf Galois Structures

Let L|K be a Galois extension of fields with finite Galois group G. Greither and Pareigis [GP87] showed that there is a bijection between Hopf Galois structures on L|K and regular subgroups of Perm(G) normalized by G, and Byott [By96] translated the problem into that of finding equivalence classes of embeddings of G in the holomorph of groups N of the same cardinality as G. In [CCo06] we showed, using Byott’s translation, that fixed point free endomorphisms of G yield Hopf Galois structures on L|K. Here we show how abelian fixed point free endomorphisms yield Hopf Galois structures directly, using the Greither-Pareigis approach, and, in some cases, also via the Byott translation. The Hopf Galois structures that arise are “twistings” of the Hopf Galois structure by Hλ, the K-Hopf algebra that arises from the left regular representation of G in Perm(G). The paper concludes with various old and new examples of abelian fixed point free endomorphisms. Hopf Galois structures We first review the Greither-Pareigis approach to Hopf Galois structures. Let G be a finite group. The left (resp. right) regular representation λ (resp. ρ) of G in Perm(G) is the map from G to Perm(G) given by