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

We study two classes of quadratic algebras over a field k: the class CnCn of all nn-generated PBW algebras with polynomial growth and finite global dimension, and the class of quantum binomial algebras. We show that a PBW algebra AA is in CnCniff   its Hilbert series is HA(z)=1/(1−z)nHA(z)=1/(1−z)n. Furthermore, the class CnCn contains a unique (up to isomorphism) monomial algebra, A=k〈x1,…,xn〉/(xjxi∣1≤i