Enumeration of set-theoretic solutions to the Yang-Baxter equation

We use Constraint Satisfaction methods to enumerate and construct set-theoretic solutions to the Yang-Baxter equation of small size. We show that there are 321931 involutive solutions of size nine, 4895272 involutive solutions of size ten and 422449480 non-involutive solution of size eight. Our method is then used to enumerate non-involutive biquandles.Comment: 11 pages, 8 table

Enumeration of set-theoretic solutions to the Yang-Baxter equation

Funding: The second author is partially supported by PICT 2018-3511 and is also a Junior Associate of the ICTP. The third author acknowledges support of NYU-ECNU Institute of Mathematical Sciences at NYU–Shanghai and he is supported in part by PICT 2016-2481 and UBACyT 20020170100256BA.We use Constraint Satisfaction methods to enumerate and construct set-theoretic solutions to the Yang-Baxter equation of small size. We show that there are 321931 involutive solutions of size nine, 4895272 involutive solutions of size ten and 422449480 non-involutive solution of size eight. Our method is then used to enumerate non-involutive biquandles.PostprintPeer reviewe

On structure groups of set-theoretic solutions to the Yang-Baxter equation

This paper explores the structure groups $G_{(X,r)}$ of finite non-degenerate set-theoretic solutions $(X,r)$ to the Yang-Baxter equation. Namely, we construct a finite quotient $\overline{G}_{(X,r)}$ of $G_{(X,r)}$, generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if $X$ injects into $G_{(X,r)}$, then it also injects into $\overline{G}_{(X,r)}$. We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of $G_{(X,r)}$. We show that multipermutation solutions are the only involutive solutions with diffuse structure group; that only free abelian structure groups are biorderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: biorderable, left-orderable, abelian, free abelian, torsion free.Comment: 32 pages. Final version. Accepted for publication in Proc. Edinburgh Math. So

Cycle matrices: A combinatorial approach to the set-theoretic solutions of the Quantum Yang-Baxter Equation

An $n\times n$ matrix $M=[m_{ij}]$ with $m_{ij}\in U_n=\{1,2,\ldots,n\}$ will be called a cycle matrix if $(U_n,\cdot)$ is a cycle set, where $i\cdot j=m_{ij}$. We study these matrices in this article. Using these matrices, we give some recipes to construct solutions, which include the multipermutation level $2$ solutions. As an application of these, we construct a multi-permutation solution of level $r$ for all $r\geq 1$. Our method gives alternate proof that the class of permutation groups of solutions contains all finite abelian groups.Comment: Minor changes after comments by Prof. G. Militaru and Prof. L. Vendramin; No changes in contents; Comments are welcom

On Skew Braces (with an appendix by N. Byott and L. Vendramin)

Braces are generalizations of radical rings, introduced by Rump to study involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation (YBE). Skew braces were also recently introduced as a tool to study not necessarily involutive solutions. Roughly speaking, skew braces provide group-theoretical and ring-theoretical methods to understand solutions of the YBE. It turns out that skew braces appear in many different contexts, such as near-rings, matched pairs of groups, triply factorized groups, bijective 1-cocycles and Hopf-Galois extensions. These connections and some of their consequences are explored in this paper. We produce several new families of solutions related in many different ways with rings, near-rings and groups. We also study the solutions of the YBE that skew braces naturally produce. We prove, for example, that the order of the canonical solution associated with a finite skew brace is even: it is two times the exponent of the additive group modulo its center.Comment: 37 pages. Final versio

A combinatorial approach to noninvolutive set-theoretic solutions of the Yang-Baxter equation

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(k, X, r) over a field 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 ≥ 3 which satisfy the minimality condition, that is, dimk A2 = 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 \∗ Y , then its braided monoid SZ is a generalized strong twisted union SZ = SX \ SY of the braided monoids SX and SY . We propose a construction of a generalized strong twisted union Z = X \ Y of braided sets (X, rX) and (Y, rY ), where the map r has a high, explicitly prescribed order