Left semi-braces and solutions of the Yang-Baxter equation

Let r : X2 → X2 be a set-theoretic solution of the Yang-Baxter equation on a finite set X. It was proven by Gateva-Ivanova and Van den Bergh that if r is non-degenerate and involutive then the algebra K⟨x ∈ X | xy = uv if r(x, y) = (u, v)⟩ shares many properties with commutative polynomial algebras in finitely many variables; in particular this algebra is Noetherian, satisfies a polynomial identity and has Gelfand-Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non- degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions rB that are associated to a left semi-brace B; such solutions can be degenerate or can even be idempotent. In order to do so we first describe such semi-braces and we prove some decompositions results extending results of Catino, Colazzo, and Stefanelli.

Publication year:2019

Keywords:Yang-Baxter equation, groups, monoid algebras, monoids, semi-brace

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