Primitive set-theoretic solutions of the Yang-Baxter equation

To every involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation on a finite set $X$ there is a naturally associated finite solvable permutation group ${mathcal G}(X,r)$ acting on $X$. We prove that every primitive permutation group of this type is of prime order $p$. Moreover, $(X,r)$ is then a so called permutation solution determined by a cycle of length $p$. This solves a problem recently asked by A. Ballester-Bolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive non-degenerate set-theoretic solutions.

Every finite abelian group is a subgroup of the additive group of a finite simple left brace

Left braces, introduced by Rump, have turned out to provide an important tool in the study of set theoretic solutions of the quantum Yang-Baxter equation. In particular, they have allowed to construct several new families of solutions. A left brace $(B,+,cdot )$ is a structure determined by two group structures on a set $B$: an abelian group $(B,+)$ and a group $(B,cdot)$, satisfying certain compatibility conditions. The main result of this paper shows that every finite abelian group $A$ is a subgroup of the additive group of a finite simple left brace $B$ with metabelian multiplicative group with abelian Sylow subgroups. This result complements earlier unexpected results of the authors on an abundance of finite simple left braces.

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

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 $A=A(K,X,r)=Klangle Xmid xy=uv mbox{ whenever }r(x,y)=(u,v)rangle$. Note that $A=oplus_{ngeq 0} A_n$ is a graded algebra, where $A_n$ is the linear span of all the elements $x_1cdots x_n$, for $x_1,dots ,x_nin X$. 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.