Set-theoretical reflection equation: Classification of reflection 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.

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Cited by in corpus (17)

• Set theoretic Yang-Baxter & reflection equations and quantum group symmetries
• Combinatorial solutions to the reflection equation
• Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency
• Soliton interactions and Yang-Baxter maps for the complex coupled short-pulse equation
• Two-component Yang-Baxter maps and star-triangle relations
• 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
• Boundary from bulk integrability in three dimensions: 3D reflection maps from tetrahedron maps
• Integrable boundary conditions for quad equations, open boundary reductions and integrable mappings
• Integrable multi-component difference systems of equations
• Breather interactions in the integrable discrete Manakov system and trigonometric Yang-Baxter maps
• Reflections to set-theoretic solutions of the Yang-Baxter equation
• Reflection Maps Associated with Involutions and Factorization Problems, and Their Poisson Geometry
• New Solutions to the Reflection Equation with Braces
• Classification of integrable boundary equations for integrable quad-graph systems
• Algebraic structures in set-theoretic Yang-Baxter & reflection equations
• Integrable extensions of Adler's map via Grassmann algebras