Quantizations of modules of differential operators

TL;DR

This paper surveys recent advances in the quantization of modules of differential operators on manifolds, focusing on projective and conformal cases, and explores their applications in cohomology, symmetries, and module structures.

Contribution

It provides a comprehensive overview of recent results on projective quantizations and their applications to various aspects of differential operator modules.

Findings

Recent results on projective quantizations are summarized.

Applications include cohomology, geometric equivalences, and symmetries.

Insights into indecomposable modules are discussed.

Abstract

Fix a manifold M, and let V be an infinite dimensional Lie algebra of vector fields on M. Assume that V contains a finite dimensional semisimple maximal subalgebra A, the projective or conformal subalgebra. A projective or conformal quantization of a V-module of differential operators on M is a decomposition into irreducible A-modules. We survey recent results on projective quantizations and their applications to cohomology, geometric equivalences and symmetries of differential operator modules, and indecomposable modules.