Quadratic algebras, Yang–Baxter equation, and Artin–Schelter regularity

Abstract

We study two classes of quadratic algebras over a field k: the class Cn of all n-generated PBW algebras with polynomial growth and finite global dimension, and the class of quantum binomial algebras. We show that a PBW algebra A is in Cniff its Hilbert series is HA(z)=1/(1−z)n. Furthermore, the class Cn contains a unique (up to isomorphism) monomial algebra, A=k〈x1,…,xn〉/(xjxi∣1≤i<j≤n). A surprising amount can be said when A is a quantum binomial algebra, that is its defining relations are nondegenerate square-free binomials xy−cxyzt, cxy∈k×. Our main result shows that for an n-generated quantum binomial algebra A the following conditions are equivalent: (i) A is an Artin–Schelter regular PBW algebra. (ii) A is a Yang–Baxter algebra, that is the set of relations ℜ defines canonically a solution of the Yang–Baxter equation. (iii) A is a binomial skew polynomial ring, with respect to an enumeration of X. (iv) The Koszul dual A! is a quantum Grassmann algebra.