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

## TL;DR

This paper introduces quantum binomial algebras and establishes their connection to solutions of the classical Yang-Baxter equation, showing these algebras are Koszul, Noetherian, and Artin-Schelter regular domains.

## Contribution

It characterizes quantum binomial algebras that yield YBE solutions and proves their equivalence to binomial skew polynomial rings, revealing their algebraic properties.

## Key findings

- Quantum binomial algebras define solutions to the classical YBE.
- Such algebras are Koszul, Noetherian, and Artin-Schelter regular.
- Equivalence between binomial skew polynomial rings and YBE solutions.

## 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.