Reflections to set-theoretic solutions of the Yang-Baxter equation

Abstract

The main aim of this paper is to determine reflections to bijective and non-degenerate solutions of the Yang-Baxter equation, by exploring their connections with their derived solutions. This is motivated by a recent description of left non-degenerate solutions in terms of a family of automorphisms of their associated left rack. In some cases, we show that the study of reflections for bijective and non-degenerate solutions can be reduced to those of derived type. Moreover, we extend some results obtained in the literature for reflections of involutive non-degenerate solutions to more arbitrary solutions. Besides, we provide ways for defining reflections for solutions obtained by employing some classical construction techniques of solutions. Finally, we gather some numerical data on reflections for bijective non-degenerate solutions associated with skew braces of small order.

Postprint version, 33 pages

References in corpus (13)

• Skew braces and the Yang-Baxter equation
• On structure groups of set-theoretic solutions to the Yang-Baxter equation
• Automorphism groups of Quandles
• Hecke algebraic approach to the reflection equation for spin chains
• Reflection equations and q-Minkowski space algebras
• Set theoretic Yang-Baxter & reflection equations and quantum group symmetries
• Covariance Properties of Reflection Equation Algebras
• Set-theoretical reflection equation: Classification of reflection maps
• Set-theoretic solutions to the Yang-Baxter equation and generalized semi-braces
• Combinatorial solutions to the reflection equation
• Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices
• Set-theoretical solutions to the reflection equation associated to the quantum affine algebra of type $A_{n-1}^{(1)}$
• Reflection equation as a tool for studying solutions to the Yang-Baxter equation