Set-theoretic solutions of the Yang–Baxter equation, associated quadratic algebras and the minimality condition

Abstract

Given a finite non-degenerate set-theoretic solution (X, r) of the Yang–Baxter equation and a field K, the structure K-algebra of (X, r) is $A=A(K,X,r)=K\langle X\mid xy=uv \text{ whenever } r(x,y)=(u,v)\rangle$. Note that $A=\oplus _{n\ge 0} A_n$ is a graded algebra, where $A_n$ is the linear span of all the elements $x_1\cdots x_n$, for $x_1,\dots ,x_n\in X$. One of the known results asserts that the maximal possible value of $\dim (A_2)$ corresponds to involutive solutions and implies several deep and important properties of A(K, X, r). Following recent ideas of Gateva-Ivanova (A combinatorial approach to noninvolutive set-theoretic solutions of the Yang–Baxter equation. arXiv:1808.03938v3 [math.QA], 2018), we focus on the minimal possible values of the dimension of $A_2$. We determine lower bounds and completely classify solutions (X, r) for which these bounds are attained in the general case and also in the square-free case. This is done in terms of the so called derived solution, introduced by Soloviev and closely related with racks and quandles. Several problems posed by Gateva-Ivanova (2018) are solved.