Bi-skew braces and Hopf Galois structures

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 $G=H⋊J$ is a semidirect product of finite groups, then a Galois extension of fields with Galois group $G$ has a Hopf Galois structure of type $H\times J$.

17 pages, submitted to New York Journal of Mathematics

References in corpus (2)

• Solubility Criteria for Hopf-Galois Structures
• Classification of braces of order $p^3$