Extensions and Well's type exact sequence of skew braces

TL;DR

This paper explores the structure of split exact sequences in left skew braces, introduces a cohomology group action, and generalizes Well's exact sequence for trivial skew brace extensions.

Contribution

It provides a new description of split exact sequences, defines a cohomology group action, and extends Well's exact sequence to skew braces.

Findings

Describes split exact sequences of left skew braces

Defines a cohomology group action on extensions

Generalizes Well's type exact sequence

Abstract

In this article, we give a description of the split exact sequences of left skew braces. We define a free action of the second cohomology group of a left skew brace $H$ by $Ann(I)$ on $Ext_{\alpha}(H, I)$ and show that this action becomes transitive if $I$ is a trivial skew brace. We also generalize the Well's type exact sequence for extensions by the trivial skew brace.