Generalized Weyl algebras: category O and graded Morita equivalence

TL;DR

This paper investigates the structure and representation theory of graded modules over generalized Weyl algebras, establishing decomposition results, conditions for category O to have projective generators, and classifying Morita equivalences.

Contribution

It introduces a new category of graded modules similar to category O, provides conditions for Morita equivalences between GWAs, and fully characterizes Morita equivalences for classical GWAs.

Findings

Decomposition of graded Artinian modules over GWAs.

Existence of projective generators in category O under certain conditions.

Complete classification of strongly graded Morita equivalences for classical GWAs.

Abstract

We study the structural and homological properties of graded Artinian modules over generalized Weyl algebras (GWAs), and this leads to a decomposition result for the category of graded Artinian modules. Then we define and examine a category of graded modules analogous to the BGG category O. We discover a condition on the data defining the GWA that ensures O has a system of projective generators. Under this condition, O has nice representation-theoretic properties. There is also a decomposition result for O. Next, we give a necessary condition for there to be a strongly graded Morita equivalence between two GWAs. We define a new algebra related to GWAs, and use it to produce some strongly graded Morita equivalences. Finally, we give a complete answer to the strongly graded Morita problem for classical GWAs.