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 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 regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k \{0}, and z, t ∈ X is satisfied in A. We prove the equivalence of the notions a binomial skew polynomial ring and a binomial solution of YBE. This implies that the Yang-Baxter algebra of such a solution is of Poincaré-Birkhoff-Witt type, and possesses a number of other nice properties such as being Koszul, Noetherian, and an Artin-Schelter regular domain.