Actions of skew braces and set-theoretic solutions of the reflection equation

TL;DR

This paper introduces a new concept of action for skew braces, demonstrating how it can generate solutions to the reflection equation, thus advancing the understanding of algebraic structures related to quantum integrable systems.

Contribution

It defines a notion of action for skew braces and shows its application in constructing solutions to the reflection equation, linking algebraic actions to quantum algebra.

Findings

Defined a new action concept for skew braces

Connected skew brace actions to solutions of the reflection equation

Provided a framework for constructing algebraic solutions

Abstract

A skew brace, as introduced by L. Guarnieri and L. Vendramin, is a set with two group structures interacting in a particular way. When one of the group structures is abelian, one gets back the notion of brace as introduced by W. Rump. Skew braces can be used to construct solutions of the quantum Yang-Baxter equation. In this article, we introduce a notion of action of a skew brace, and show how it leads to solutions of the closely associated reflection equation.