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

Abstract

We study noninvolutive set-theoretic solutions $(X,r)$ of the Yang-Baxter equations in terms of the properties of the canonically associated algebraic objects-the braided monoid $S(X,r)$, the quadratic Yang-Baxter algebra $A= A(\textbf{k}, X, r)$ over a field $\textbf{k}$ and its Koszul dual, $A^{!}$. More generally, we continue our systematic study of nondegenerate quadratic sets $(X,r)$ and the associated algebraic objects. Next we investigate the class of (noninvolutive) square-free solutions $(X,r)$. It contains the special class of self distributive solutions (quandles). We make a detailed characterization in terms of various algebraic and combinatorial properties each of which shows the contrast between involutive and noninvolutive square-free solutions. We introduce and study a class of finite square-free braided sets $(X,r)$ of order $n\geq 3$ which satisfy "the minimality condition \textbf{M}", that is $\dim_{\textbf{k}} A_2 =2n-1$. Examples are some simple racks of prime order $p$. Finally, we discuss general extensions of solutions and introduce the notion of "a generalized strong twisted union of braided sets". We prove that if $(Z,r)$ is a non-degenerate 2-cancellative braided set splitting as $Z = X\natural^{\ast} Y$, then its braided monoid $S_Z$ is a generalized strong twisted union $S_Z= S_X\natural^{\ast} S_Y$ of the braided monoids $S_X$ and $S_Y$. Moreover, if $(Z,r)$ is injective then its braided group $G_Z=G(Z,r)$ also splits as $G_Z= G_X\natural^{\ast} G_Y$ of the associated braided groups of $X$ and $Y$. We propose a construction of a generalized strong twisted union $Z = X\natural^{\ast} Y$ of braided sets $(X,r_X)$, and $(Y, r_Y)$, where the map $r$ has high, explicitly prescribed order.