Reflection equation as a tool for studying solutions to the Yang-Baxter equation

TL;DR

This paper introduces a novel approach using reflection equations to generate and analyze solutions to the Yang-Baxter equation, revealing new connections and methods for constructing solutions.

Contribution

It constructs a family of YBE solutions from reflections, relates their structure monoids via cocycle maps, and simplifies the reflection equation for involutive solutions.

Findings

Constructed a family of solutions $r^{(k)}$ from reflections.

Established isomorphisms between actions of braid groups on these solutions.

Provided systematic methods for constructing new reflections.

Abstract

Given a right-non-degenerate set-theoretic solution $(X,r)$ to the Yang-Baxter equation, we construct a whole family of YBE solutions $r^{(k)}$ on $X$ indexed by its reflections $k$ (i.e., solutions to the reflection equation for $r$). This family includes the original solution and the classical derived solution. All these solutions induce isomorphic actions of the braid group/monoid on $X^n$. The structure monoids of $r$ and $r^{(k)}$ are related by an explicit bijective $1$-cocycle-like map. We thus turn reflections into a tool for studying YBE solutions, rather than a side object of study. In a different direction, we study the reflection equation for non-degenerate involutive YBE solutions, show it to be equivalent to (any of the) three simpler relations, and deduce from the latter systematic ways of constructing new reflections.