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

Abstract

A bijective map r: X2→X2, 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 X3. A nondegenerate involutive solution (X,r) satisfying r(xx)=xx, for all x∈X, is called square-free 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 k-algebra 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 square-free 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 a 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–Solovyev.