Classification of quadratic and cubic PBW algebras on three generators
Natalia Iyudu, Stanislav Shkarin

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 , 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 . 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.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Nonlinear Waves and Solitons
