Quantization and injective submodules of differential operator modules

TL;DR

This paper explores the structure of differential operator modules under the action of vector fields, realizing injective modules within a category and analyzing conditions for projective quantizations.

Contribution

It characterizes injective objects in a category of modules and determines conditions for the existence and uniqueness of projective quantizations.

Findings

All injective objects of the parabolic category are realized as submodules of differential operator modules.

Conditions for existence and uniqueness of projective quantizations are established.

The study provides a detailed classification of projective quantizations for differential operators between tensor modules.

Abstract

The Lie algebra of vector fields on $R^m$ acts naturally on the spaces of differential operators between tensor field modules. Its projective subalgebra is isomorphic to $sl_{m+1}$, and its affine subalgebra is a maximal parabolic subalgebra of the projective subalgebra with Levi factor $gl_m$. We prove two results. First, we realize all injective objects of the parabolic category O$^{gl_m}(sl_{m+1})$ of $gl_m$-finite $sl_{m+1}$-modules as submodules of differential operator modules. Second, we study projective quantizations of differential operator modules, i.e., $sl_{m+1}$-invariant splittings of their order filtrations. In the case of modules of differential operators from a tensor density module to an arbitrary tensor field module, we determine when there exists a unique projective quantization, when there exists no projective quantization, and when there exist multiple projective quantizations.