Fixed-point free endomorphisms and Hopf Galois structures

Abstract

Let L | K L|K be a Galois extension of fields with finite Galois group G G . Greither and Pareigis showed that there is a bijection between Hopf Galois structures on L | K L|K and regular subgroups of P e r m ( G ) Perm(G) normalized by G G , and Byott translated the problem into that of finding equivalence classes of embeddings of G G in the holomorph of groups N N of the same cardinality as G G . In 2007 we showed, using Byott’s translation, that fixed point free endomorphisms of G G yield Hopf Galois structures on L | K 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 H λ H_{\lambda } , the K K -Hopf algebra that arises from the left regular representation of G G in P e r m ( G ) Perm(G) . The paper concludes with various old and new examples of abelian fixed point free endomorphisms.