Classification of quadratic and cubic PBW algebras on three generators

TL;DR

This paper provides a comprehensive classification of quadratic and cubic PBW algebras on three generators, including Koszul and Artin-Schelter regular algebras, extending previous foundational results.

Contribution

It offers a complete classification of quadratic and cubic PBW algebras with specific Hilbert series, including Koszul and Artin-Schelter regular algebras, up to isomorphism.

Findings

Classified quadratic algebras with Hilbert series (1-t)^{-3}

Classified cubic algebras with Hilbert series (1+t)^{-1}(1-t)^{-3}

Extended known results on Artin-Schelter regular algebras

Abstract

We give a complete classification of quadratic algebras A, with Hilbert series $H_A=(1-t)^{-3}$, which is the Hilbert series of commutative polynomials on 3 variables. Koszul algebras as well as algebras with quadratic Gr\"obner basis among them are identified. We also give a complete classification of cubic algebras A with Hilbert series $H_A=(1+t)^{-1}(1-t)^{-3}$. These two classes of algebras contain all Artin-Schelter regular algebras of global dimension 3. As far as the latter are concerned, our results extend well-known results of Artin and Schelter by providing a classification up to an algebra isomorphism.