QUADRATIC PBW-ALGEBRAS, YANG-BAXTER EQUATION AND ARTIN-SCHELTER REGULARITY

We study quadratic algebras over a field k. We show that an n-generated PBW-algebra A has finite global dimension and polynomial growth iff its Hilbert series is HA(z) = 1/(1 − z)n. A surprising amount can be said when the algebra A has quantum binomial relations, that is the defining relations are binomials xy − cxyzt, cxy ∈ k×, which are square-free and nondegenerate. We prove that in this case various good algebraic and homological properties are closely related. The main result shows that for an n-generated quantum binomial algebra A the following conditions are equivalent: (i) A is a PBW-algebra with finite global dimension; (ii) A is PBW and has polynomial growth; (iii) A is an Artin-Schelter regular PBW-algebra; (iv) A is a Yang-Baxter algebra; (v) HA(z) = 1/(1−z)n; (vi) The dual A! is a quantum Grassman algebra; (vii) A is a binomial skew polynomial ring. This implies that the problem of classification of Artin-Schelter regular PBW-algebras of global dimension n is equivalent to the classification of square-free set-theoretic solutions of the Yang-Baxter equation (X, r), on sets X of order n. MIRAMARE – TRIEST