Combinatorial solutions to the reflection equation

TL;DR

This paper introduces algebraic methods, including ring theory and skew braces, to generate solutions to the reflection equation, advancing the understanding of its set-theoretic and parameter-dependent forms.

Contribution

It combines ring-theoretic and skew brace techniques to systematically construct solutions to the reflection equation, including parameter-dependent cases.

Findings

Constructed new set-theoretic solutions to the reflection equation.

Developed methods to produce solutions for parameter-dependent reflection equations.

Enhanced algebraic framework for solving the reflection equation.

Abstract

We use ring-theoretic methods and methods from the theory of skew braces to produce set-theoretic solutions to the reflection equation. We also use set-theoretic solutions to construct solutions to the parameter-dependent reflection equation.