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

Abstract

A bijective map $r:X^2⟶X^2$, where $X=\{x_1,...,x_n\}$ is a finite set, is called a \emph{set-theoretic solution of the Yang-Baxter equation} (YBE) if the braid relation $r_{12}{r}_{23}{r}_{12}=r_{23}{r}_{12}{r}_{23}$ holds in $X^3.$ A non-degenerate involutive solution $(X,r)$ satisfying $r(xx)=xx$, for all $x\in X$, is called \emph{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 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.

34 pages

References in corpus (1)

• Permutation-type solutions to the Yang-Baxter and other n-simplex equations

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• Decomposition theorems for involutive solutions to the Yang-Baxter equation
• Retractability of solutions to the Yang-Baxter equation and $p$-nilpotency of skew braces
• Nilpotency in left semi-braces
• Inverse semi-braces and the Yang-Baxter equation
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• Two-component Yang-Baxter maps and star-triangle relations
• Skew braces: a brief survey
• Near braces and p-deformed braided groups
• Segre products and Segre morphisms in a class of Yang-Baxter algebras
• The Yang-Baxter equation, Quantum computing and Quantum entanglement
• Multinomial expansion and Nichols algebras associated to non-degenerate involutive solutions of the Yang-Baxter equation