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 $X^n$. 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.

18 pages, 12 figures. Final version

References in corpus (5)

• Set theoretic Yang-Baxter & reflection equations and quantum group symmetries
• Set-theoretical reflection equation: Classification of reflection maps
• Radical and weight of skew braces and their applications to structure groups of solutions of the Yang-Baxter equation
• Primitive set-theoretic solutions of the Yang-Baxter equation
• Set-theoretic solutions of the Yang--Baxter equation, associated quadratic algebras and the minimality condition

Cited by in corpus (4)

• Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices
• Reflections to set-theoretic solutions of the Yang-Baxter equation
• Reflection Maps Associated with Involutions and Factorization Problems, and Their Poisson Geometry
• Self-distributive structures, braces & the Yang-Baxter equation