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.

给定Yang-Baxter方程的有限非退化集理论解（X，  r）和场K，（X，  r）的结构K-代数为\（A = A（K，X，r） = K \ langle X \ mid xy = uv \ text {每当} r（x，y）=（u，v）\ rangle \）。请注意，\（A = \ oplus _ {n \ ge 0} A_n \）是渐变的代数，其中\（A_n \）是所有元素\（x_1 \ cdots x_n \）对于\（x_1 ，\ dots，x_n \ in X \）。已知结果之一断言\（\ dim（A_2）\）的最大可能值对应于对合解，并暗示A（K，  X，  r）的几个深层重要属性。遵循Gateva-Ivanova的最新思想（一种针对Yang-Baxter方程的非对合集理论解的组合方法。arXiv：1808.03938v3 [math.QA]，2018年），我们着眼于\（ A_2 \）。我们确定下界并完全分类解（X，  r），在一般情况下以及在无平方情况下都可以达到这些限制。这是由Soloviev引入的，与机架和分位数密切相关的所谓派生解决方案完成的。解决了Gateva-Ivanova（2018）提出的几个问题。