Bi-skew braces and Hopf Galois structures

TL;DR

This paper introduces bi-skew braces, a mathematical structure with two group operations, and explores their connection to Hopf Galois structures, providing new examples and classifications related to field extensions.

Contribution

The paper defines bi-skew braces and demonstrates their correspondence to Hopf Galois structures, including classifications and new examples from radical rings and semidirect products.

Findings

Bi-skew braces correspond to Hopf Galois structures of two types.

Radical rings with $A^3=0$ yield bi-skew braces.

Semidirect products produce numerous non-abelian bi-skew braces.

Abstract

We define a bi-skew brace to be a set $G$ with two group operations $\star$ and $\circ$ so that $(G, \circ, \star)$ is a skew brace with additive group $(G, \star)$ and also with additive group $(G, \circ)$. If $G$ is a skew brace, then $G$ corresponds to a Hopf Galois structure of type $(G, \star)$ on any Galois extension of fields with Galois group isomorphic to $(G, \circ)$. If $G$ is a bi-skew brace, then $G$ also corresponds to a Hopf Galois structure of type $(G, \circ)$ on a Galois extension of fields with Galois group isomorphic to $(G, \star)$. Many non-trivial examples exist. One source is radical rings $A$ with $A^3 = 0$, where one of the groups is abelian and the other need not be. The left braces of degree $p^3$ classified by Bachiller are bi-skew braces if and only they are radical rings. A different source of bi-skew braces is semidirect products of arbitrary finite groups, which yield many examples where both groups are non-abelian, and a skew brace proof of a result of Crespo, Rio and Vela that if $G = H\rtimes J$ is a semidirect product of finite groups, then a Galois extension of fields with Galois group $G$ has a Hopf Galois structure of type $H \times J$.