Set-theoretical reflection equation: Classification of reflection maps

TL;DR

This paper introduces the set-theoretical reflection equation and classifies its solutions, called reflection maps, especially in relation to quadrirational Yang-Baxter maps, providing a systematic construction method.

Contribution

It presents a general framework for reflection maps and offers a complete classification for solutions linked to quadrirational Yang-Baxter maps.

Findings

Classification of reflection maps for all quadrirational Yang-Baxter maps

A new method for constructing reflection maps

Connection established between reflection maps and Yang-Baxter maps

Abstract

The set-theoretical reflection equation and its solutions, the reflection maps, recently introduced by two of the authors, is presented in general and then applied in the context of quadrirational Yang-Baxter maps. We provide a method for constructing reflection maps and we obtain a classification of solutions associated to all the families of quadrirational Yang-Baxter maps that have been classified recently.