Abelian maps, bi-skew braces, and opposite pairs of {H}opf-{G}alois structures

TL;DR

This paper explores how certain homomorphisms from nonabelian groups induce multiple Hopf-Galois structures and skew braces, leading to new solutions to the Yang-Baxter equation and insights into Galois extensions.

Contribution

It introduces a novel method to construct multiple Hopf-Galois structures and associated bi-skew braces from homomorphisms with abelian images, extending previous work.

Findings

Constructs two Hopf-Galois structures from a single homomorphism.

Identifies a bi-skew brace among the constructed skew braces.

Derives four set-theoretic solutions to the Yang-Baxter equation.

Abstract

Let $G$ be a finite nonabelian group, and let $\psi:G\to G$ be a homomorphism with abelian image. We show how $\psi$ gives rise to two Hopf-Galois structures on a Galois extension $L/K$ with Galois group (isomorphic to) $G$; one of these structures generalizes the construction given by a ``fixed point free abelian endomorphism'' introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equation as well as a pair of Hopf-Galois structures on a (potentially) different finite Galois extension.