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 braided monoid $S(X,r)$, the quadratic Yang–Baxter algebra $A= A(\mathbf{k}, X, r)$ over a field $\mathbf{k}$, and its Koszul dual $A^{!}$. More generally, we continue our systematic study of nondegenerate quadratic sets $(X,r)$ and their associated algebraic objects. Next we investigate the class of (noninvolutive) square-free solutions $(X,r)$. This contains the 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, that is, $\dim_{\mathbf{k}} A_2 =2n-1$. 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 nondegenerate $2$-cancellative braided set splitting as a generalized strong twisted union of $r$-invariant subsets $Z = X\mathbin{\natural}^{\ast} Y$, then its braided monoid $S_Z$ is a generalized strong twisted union $S_Z= S_X\mathbin{\natural}^{\ast} S_Y$ of the braided monoids $S_X$ and $S_Y$. We propose a construction of a generalized strong twisted union $Z = X\mathbin{\natural}^{\ast} Y$ of braided sets $(X,r_X)$ and $(Y,r_Y)$, where the map $r$ has a high, explicitly prescribed order.