Fixed-point free endomorphisms and Hopf Galois structures

Abstract:

Let $L|K$ be a Galois extension of fields with finite Galois group $G$. Greither and Pareigis showed that there is a bijection between Hopf Galois structures on $L|K$ and regular subgroups of $Perm(G)$ normalized by $G$, and Byott 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 2007 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 $H_{\lambda }$, 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.