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

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 non-degenerate 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, dim(k) A(2) = 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 (sic)* Y, then its braided monoid S-Z is a generalized strong twisted union S-Z = S-X (sic)* S-Y of the braided monoids S-X and S-Y. We propose a construction of a generalized strong twisted union Z = X (sic)* Y of braided sets (X, r(X)) and (Y, r(Y)), where the map r has a high, explicitly prescribed order.