A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation

A bijective map $r:X^2⟶X^2$, where $X=\{x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n\}$ is a finite set, is called a \emph{set-theoretic solution of the Yang-Baxter equation} (YBE) if the braid relation $r_{12}{r}_{23}{r}_{12}=r_{23}{r}_{12}{r}_{23}$ holds in $X^3\mathrm{.}$ A non-degenerate involutive solution $(X,r)$ satisfying $r(xx)=xx$, for all $x\in X$, is called \emph{square-free solution}. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions $(X,r)$ and the associated Yang-Baxter algebraic structures – the semigroup $S(X,r)$, the group $G(X,r)$ and the $k$- algebra $A(k,X,r)$ over a field $k$, generated by $X$ and with quadratic defining relations naturally arising and uniquely determined by $r$. We study the properties of the associated Yang-Baxter structures and prove a conjecture of the present author that the three notions: a square-free solution of (set-theoretic) YBE, a semigroup of I type, and a semigroup of skew-polynomial type, are equivalent. This implies that the Yang-Baxter algebra $A(k,X,r)$ is Poincar'{e}-Birkhoff-Witt type algebra, with respect to some appropriate ordering of $X$. We conjecture that every square-free solution of YBE is retractable, in the sense of Etingof-Schedler.