Module structure of Weyl algebras

TL;DR

This paper reviews the historical development and recent advances in understanding the module structure of Weyl algebras, including constructions of simple modules and their geometric and algebraic implications.

Contribution

It traces the origins of Stafford's work, explains the construction of non-holonomic simple modules, and discusses recent developments related to Weyl algebras and their modules.

Findings

Construction of non-holonomic simple modules over Weyl algebras

Parametrization of right ideals in the first Weyl algebra

Connections to Calogero-Moser spaces and quantized symplectic singularities

Abstract

The seminal paper "J.T. Stafford, Module structure of Weyl algebras, J. London Math. Soc. (2) 18 (1978), no. 3, 429--442" was a major step forward in our understanding of Weyl algebras. Beginning with Serre's Theorem on free summands of projective modules and Bass' Stable Range Theorem in commutative algebra, we attempt to trace the origins of this work and explain how it led to Stafford's construction of non-holonomic simple modules over Weyl algebras. We also describe Bernstein-Lunt's geometric construction of infinite families of non-holonomic simple modules. We recall more recent developments related to Weyl algebras, especially that of parametrizing right ideals in the first Weyl algebra and its relation to Calogero-Moser spaces. Finally, we revisit Stafford's results in the context of quantized symplectic singularities, where they lead naturally to open problems on the behaviour of simple modules.