Binomial skew polynomial rings, Artin-Schelter regularity, and binomial solutions of the Yang-Baxter equation

Abstract

Let $k$ be a field and $X$ be a set of $n$ elements. We introduce and study a class of quadratic $k$-algebras called \emph{quantum binomial algebras}. Our main result shows that such an algebra $A$ defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual $A^{!}$ is Frobenius of dimension $n,$ with a \emph{regular socle} and for each $x,y \in X $ an equality of the type $xyy=\alpha zzt,$ where $\alpha \in k \setminus\{0\},$ and $z,t \in X$ is satisfied in $A$. We prove the equivalence of the notions \emph{a binomial skew polynomial ring} and \emph{a binomial solution of YBE}. This implies that the Yang-Baxter algebra of such a solution is of Poincar\'{e}-Birkhoff-Witt type, and possesses a number of other nice properties such as being Koszul, Noetherian, and an Artin-Schelter regular domain.