Double Ore extensions versus graded skew PBW extensions

TL;DR

This paper establishes conditions under which graded double Ore extensions are equivalent to graded skew PBW extensions and demonstrates that such extensions preserve Artin-Schelter regularity and skew Calabi-Yau properties.

Contribution

It provides necessary and sufficient conditions for the equivalence of graded double Ore and skew PBW extensions, and proves regularity and Calabi-Yau properties are preserved.

Findings

Graded skew PBW extensions of Artin-Schelter regular algebras are Artin-Schelter regular.

Such extensions of connected skew Calabi-Yau algebras are skew Calabi-Yau of higher dimension.

Conditions for equivalence between double Ore and skew PBW extensions are characterized.

Abstract

In this paper we present necessary and sufficient conditions for a graded (trimmed) double Ore extension to be a graded (quasi-commutative) skew PBW extension. Using this fact, we prove that a graded skew PBW extension $A = \sigma(R)\langle x_1,x_2 \rangle$ of an Artin-Schelter regular algebra $R$ is Artin-Schelter regular. As a consequence, every graded skew PBW extension $A = \sigma(R)\langle x_1,x_2 \rangle$ of a connected skew Calabi-Yau algebra $R$ of dimension $d$ is skew Calabi-Yau of dimension $d+2$.