On the connection between Hopf--Galois structures and skew braces

TL;DR

This paper introduces a new perspective on the relationship between Hopf--Galois structures and skew braces, extending known results and revealing new insights, especially regarding the surjectivity of the Hopf--Galois correspondence and the role of bi-skew braces.

Contribution

It presents a novel approach to connect Hopf--Galois structures with skew braces, leading to new theoretical results and deeper understanding of the surjectivity problem.

Findings

Known results extend to the new perspective

New results on the surjectivity of the Hopf--Galois correspondence

Enhanced understanding of bi-skew braces in Hopf--Galois theory

Abstract

We present a different version of the well-known connection between Hopf--Galois structures and skew braces, building on a recent paper of A. Koch and P. J. Truman. We show that the known results that involve this connection easily carry over to this new perspective, and that new ones naturally appear. As an application, we present new insights on the study of the surjectivity of the Hopf--Galois correspondence, explaining in more detail the role of bi-skew braces in Hopf--Galois theory.