Algebraic Structures in Set-Theoretic Yang-Baxter and Reflection Equations

We present resent results regarding invertible, non-degenerate solutions of the set-theoretic Yang-Baxter and reflection equations. We recall the notion of braces and we present and prove various fundamental properties required for the solutions of the set theoretic Yang-Baxter equation. We then restrict our attention on involutive solutions and consider λ parametric set-theoretic solutions of the Yang-Baxter equation and we extract the associated quantum algebra. We also discuss the notion of the Drinfeld twist for involutive solutions and their relation to the Yangian. We next focus on reflections and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the symmetric group. We show that there exists a “reflection׳׳ finite sub-algebra for some special choice of reflection maps.