From endomorphisms to bi-skew braces, regular subgroups, the Yang--Baxter equation, and Hopf--Galois structures

TL;DR

This paper characterizes endomorphisms of groups that generate bi-skew braces via Koch's construction, linking set-theoretic solutions of the Yang--Baxter equation with Hopf--Galois structures.

Contribution

It provides a characterization of endomorphisms leading to bi-skew braces and extends Koch's construction to a more general setting, connecting multiple algebraic structures.

Findings

Characterization of endomorphisms producing bi-skew braces.

Extension of Koch's construction to broader cases.

Connection between set-theoretic solutions and Hopf--Galois structures.

Abstract

The interplay between set-theoretic solutions of the Yang--Baxter equation of Mathematical Physics, skew braces, regular subgroups, and Hopf--Galois structures has spawned a considerable body of literature in recent years. In a recent paper, Alan Koch generalised a construction of Lindsay N.~Childs, showing how one can obtain bi-skew braces $(G, \cdot, \circ)$ from an endomorphism of a group $(G, \cdot)$ whose image is abelian. In this paper, we characterise the endomorphisms of a group $(G, \cdot)$ for which Koch's construction, and a variation on it, yield (bi-)skew braces. We show how the set-theoretic solutions of the Yang--Baxter equation derived by Koch's construction carry over to our more general situation, and discuss the related Hopf--Galois structures.