Rings graded equivalent to the Weyl algebra

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. Given a Z-graded ring R, an autoequivalence F of gr-R, and a finitely generated graded projective right R-module P, we show how to construct a twisted endomorphism ring and prove:Theorem – The Z-graded rings R and S are graded equivalent if and only if there are an autoequivalence F of gr-R and a finitely generated graded projective right R-module P such that the modules {FnP} generate gr-R and .