From endomorphisms to bi-skew braces, regular subgroups, the Yang--Baxter equation, 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.

v1 of this manuscript has been split in two. The first part has appeared in J. Algebra 587 (2021), 462-487, and carries a slightly revised title - it is v2 here (19 pages). The second part is now arXiv:2110.11028