Gateva-Ivanova, T. (2021). A combinatorial approach to noninvolutive set-theoretic solutions of the Yang–Baxter equation. Publicacions Matematiques, 65, 747-808. doi:10.5565/PUBLMAT6522113
Abstract: 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.