Rings graded equivalent to the Weyl algebra

TL;DR

This paper classifies all graded rings that are equivalent to the first Weyl algebra under Euler grading, revealing surprising examples like idealizers, using a Morita-type characterization.

Contribution

It provides a complete classification of graded rings equivalent to the Weyl algebra, including novel examples such as idealizers, via a Morita-type approach.

Findings

A complete classification of graded rings equivalent to the Weyl algebra.

Identification of idealizers in localizations as graded equivalents.

Application of Morita-type characterization to graded module categories.

Abstract

We consider the first Weyl algebra, A, in the Euler gradation, and completely classify graded rings B that are graded equivalent to A: that is, the categories gr-A and gr-B are equivalent. This includes some surprising examples: in particular, we show that A is graded equivalent to an idealizer in a localization of A. We obtain this classification as an application of a general Morita-type characterization of equivalences of graded module categories.