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

TL;DR

This paper investigates the minimal possible dimensions of the quadratic algebra associated with finite set-theoretic solutions of the Yang-Baxter equation, providing classifications and bounds related to derived solutions, racks, and quandles.

Contribution

It determines lower bounds for the algebra dimension and classifies solutions achieving these bounds, extending understanding of the algebraic structures linked to the Yang-Baxter equation.

Findings

Lower bounds for the dimension of A_2 are established.

Complete classification of solutions attaining minimal dimension bounds.

Results apply to both general and square-free solutions.

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 $A=A(K,X,r)=K\langle X\mid xy=uv \mbox{ whenever }r(x,y)=(u,v)\rangle$. Note that $A=\oplus_{n\geq 0} A_n$ 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 \cite{GI2018}, 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 in \cite{GI2018} are solved.