PBW deformations of Artin-Schelter regular algebras

TL;DR

This paper studies PBW deformations of Artin-Schelter regular algebras, linking their properties to geometric algebra structures and classifying low-dimensional cases with examples.

Contribution

It provides a classification of PBW deformations of 2-dimensional regular algebras and explores properties like skew Calabi-Yau and tensor product closure.

Findings

Homogenization of PBW deformations is regular and geometric.

Classified all PBW deformations of 2D regular algebras.

Presented examples of 3D deformations.

Abstract

We consider algebras that can be realized as PBW deformations of (Artin-Schelter) regular algebras. This is equivalent to the homogenization of the algebra being regular. It is shown that the homogenization, when it is a geometric algebra, contains a component whose points are in 1-1 correspondence with the simple modules of the deformation. We classify all PBW deformations of 2-dimensional regular algebras and give examples of 3-dimensional deformations. Other properties, such as the skew Calabi-Yau property and closure under tensor products, are considered.