Constructing finite simple solutions of the Yang-Baxter equation (Q2232717)

In the main results of the present paper, the authors provide various methods to construct simple, involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation. As a by-product, they produce examples of such kind of solutions with cardinality \(p^2\) and \(p_1^{m_1}\cdots p_n^{m_n}\) where \(p,p_1, \ldots, p_n\) are primes and \(m_1+\cdots+ m_n>n\). For the benefit of the reader, let us give the definition of ``simple'', ``involutive'' and ``non-degenerate''. Let \(r:X\times X\longrightarrow X\times X\) be a set-theoretic solution of the Yang-Baxter equation given by \(r(x,y)=(\sigma_x(y),\gamma_y(x))\). It is involutive if \(r^2=\operatorname{id}\). It is non-degenerate if the functions \(\sigma_x,\gamma_y:X\rightarrow X\) are bijective for all \(x,y\in X\). It is simple if there is no non-trivial epimorphism from \(X\) to other solution. Other properties of the simple solutions are explored. For instance, the authors show that finite simple solutions are indecomposable and irretractable (if their order is not a prime). They finish the manuscript by proposing some open questions on the matter. This work is a contribution to the aim of classifying all the set-theoretic solutions of the Yang-Baxter equation as these can be build up from the simple ones. In the Introduction, the authors make a review of the state of the art of this problem; we can add [\textit{N. Andruskiewitsch} and \textit{M. Graña}, Adv. Math. 178, No. 2, 177--243 (2003; Zbl 1032.16028)] at the references there.