Structure monoids of set-theoretic solutions of the Yang-Baxter equation

Given a set-theoretic solution $(X,r)$ of the Yang--Baxter equation, we denote by $M=M(X,r)$ the structure monoid and by $A=A(X,r)$, respectively $A=A(X,r)$, the left, respectively right, derived structure monoid of $(X,r)$. It is shown that there exist a left action of $M$ on $A$ and a right action of $M$ on $A$ and 1-cocycles $pi$ and $pi$ of $M$ with coefficients in $A$ and in $A$ with respect to these actions respectively. We investigate when the 1-cocycles are injective, surjective or bijective. In case $X$ is finite, it turns out that $pi$ is bijective if and only if $(X,r)$ is left non-degenerate, and $pi$ is bijective if and only if $(X,r)$ is right non-degenerate. In case $(X,r) $ is left non-degenerate, in particular $pi$ is bijective, we define a semi-truss structure on $M(X,r)$ and then we show that this naturally induces a set-theoretic solution $(bar M, bar r)$ on the least cancellative image $bar M= M(X,r)/eta$ of $M(X,r)$. In case $X$ is naturally embedded in $M(X,r)/eta$, for example when $(X,r)$ is irretractable, then $bar r$ is an extension of $r$. It also is shown that non-degenerate irretractable solutions necessarily are bijective.

Braces and the Yang-Baxter equation

Several aspects of relations between braces and non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation are discussed and many consequences are derived. In particular, for each positive integer $n$ a finite square-free multipermutation solution of the Yang-Baxter equation with multipermutation level $n$ and an abelian involutive Yang-Baxter group is constructed. This answers a problem of Gateva-Ivanova and Cameron. It is also proved that finite non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation whose associated involutive Yang-Baxter group is abelian are retractable in the sense of Etingof, Schedler and Soloviev. Earlier the authors proved this with the additional square-free hypothesis on the solutions. Retractability of solutions is also proved for finite square-free non-degenerate involutive set-theoretic solutions associated to a left brace.