Fiber-Wise Linear Differential Operators

TL;DR

This paper introduces fiber-wise linear differential operators on vector bundles, establishing their equivalence to polynomial derivations on line bundles over the dual bundle, with potential applications to Lie groupoids and algebroids.

Contribution

It defines a new class of differential operators on vector bundles and links them to derivations on line bundles, advancing the understanding of multiplicative differential operators.

Findings

Fiber-wise linear differential operators are equivalent to polynomial derivations.

The framework may facilitate defining multiplicative differential operators on Lie groupoids.

Discussion on linearization of differential operators around submanifolds.

Abstract

We define a new notion of fiber-wise linear differential operator on the total space of a vector bundle $E$. Our main result is that fiber-wise linear differential operators on $E$ are equivalent to (polynomial) derivations of an appropriate line bundle over $E^\ast$. We believe this might represent a first step towards a definition of multiplicative (resp. infinitesimally multiplicative) differential operators on a Lie groupoid (resp. a Lie algebroid). We also discuss the linearization of a differential operator around a submanifold.