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

Abstract

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 $A^{\prime }=A^{\prime }(X,r)$, 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 $A^{\prime }$ and 1-cocycles $\pi$ and ${\pi }^{\prime }$ of $M$ with coefficients in $A$ and in $A^{\prime }$ 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 ${\pi }^{\prime }$ 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.

21 pages. Some minor changes have been implemented. To appear in Publicacions Matemàtiques Some additional minor changes were implemented