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