New Solutions to the Reflection Equation with Braces

TL;DR

This paper introduces new solutions to the reflection equation using brace structures, expanding the set of known solutions and highlighting the role of braces and near-rings in integrable systems.

Contribution

It demonstrates that every brace produces multiple solutions and identifies a natural class of solutions as near-rings, advancing the understanding of the reflection equation.

Findings

Every brace yields several solutions.

A natural class of solutions is a near-ring.

More solutions are found for factorizable rings.

Abstract

The reflection equation of Cherednik is a counterpart to the celebrated Yang-Baxter equation, with importance in the theory of integrable systems. We obtain several new solutions of the reflection equation using braces building on the work of Smoktunowicz, Vendramin and Weston. In particular, we show that every brace yields several simple solutions and that a natural class of solutions is a near-ring. We find more solutions for factorizable rings. Some of our solutions apply to the original parameter-dependent equation.