# On skew braces (with an appendix by N. Byott and L. Vendramin)

## TL;DR

This paper explores skew braces, their connections to various algebraic structures, and their role in generating solutions to the Yang-Baxter equation, including new families of solutions and properties of canonical solutions.

## Contribution

It introduces new families of solutions to the YBE related to rings, near-rings, and groups, and studies properties of solutions derived from skew braces.

## Key findings

- The order of the canonical solution for a finite skew brace is even.
- Skew braces connect to near-rings, matched pairs, and Hopf-Galois extensions.
- New solutions to the YBE are constructed from skew braces.

## Abstract

Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation (YBE). Skew braces were also recently introduced as a tool to study not necessarily involutive solutions. Roughly speaking, skew braces provide group-theoretical and ring-theoretical methods to understand solutions of the YBE. It turns out that skew braces appear in many different contexts, such as near-rings, matched pairs of groups, triply factorized groups, bijective 1-cocycles and Hopf-Galois extensions. These connections and some of their consequences are explored in this paper. We produce several new families of solutions related in many different ways with rings, near-rings and groups. We also study the solutions of the YBE that skew braces naturally produce. We prove, for example, that the order of the canonical solution associated with a finite skew brace is even: it is two times the exponent of the additive group modulo its center.