A Combinatorial Approach to the Set-theoretic Solutions of the Yang-baxter Equation

A bijective map r : X −→ X, where X = {x1, · · · , xn} is a finite set, is called a set-theoretic solution of the Yang-Baxter equation (YBE) if the braid relation r12r23r12 = r23r12r23 holds in X. A non-degenerate involutive solution (X, r) satisfying r(xx) = xx, for all x ∈ X, is called squarefree solution. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions (X, r) and the associated Yang-Baxter algebraic structures — the semigroup S(X, r), the group G(X, r) and the kalgebra A(k, X, r) over a field k, generated by X and with quadratic defining relations naturally arising and uniquely determined by r. We study the properties of the associated Yang-Baxter structures and prove a conjecture of the present author that the three notions: a squarefree solution of (set-theoretic) YBE, a semigroup of I type, and a semigroup of skew-polynomial type, are equivalent. This implies that the Yang-Baxter algebra A(k, X, r) is Poincaré-Birkhoff-Witt type algebra, with respect to some appropriate ordering of X. We conjecture that every square-free solution of YBE is retractable, in the sense of Etingof-Schedler.