On Pursell-Shanks type results

TL;DR

This paper characterizes vector bundles using Lie algebraic methods applied to differential operators and their symbols, extending classical results without relying on the module structure over smooth functions.

Contribution

It provides new Lie algebraic characterizations of vector bundles via differential operators and their symbols, removing the need for the module assumption.

Findings

Lie algebra $ ext{Der}( ext{Diff}(E,M))$ characterizes the vector bundle

Symbol Lie algebras $ ext{S}( ext{Diff}(E,M))$ characterize the bundle

Results hold without assuming the module structure over ${ m C}^inity(M)$

Abstract

We prove a Lie-algebraic characterization of vector bundle for the Lie algebra $\mathcal{D}(E,M),$ seen as ${\rm C}^\infty(M)-$module, of all linear operators acting on sections of a vector bundle $E\to M$. We obtain similar result for its Lie subalgebra $\mathcal{D}^1(E,M)$ of all linear first-order differential operators. Thanks to a well-chosen filtration, $\mathcal{D}(E,M)$ becomes $\mathcal{P}(E,M)$ and we prove that $\mathcal{P}^1(E,M)$ characterizes the vector bundle without the hypothesis of being seen as ${\rm C}^\infty(M)-$module. We prove that the Lie algebra $\mathcal{S}(\mathcal{P}(E,M))$ of symbols of linear operators acting on smooth sections of a vector bundle $E\to M,$ characterizes it. To obtain this, we assume that $\mathcal{S}(\mathcal{P}(E,M))$ is seen as ${\rm C}^\infty(M)-$module. We obtain a similar result with the Lie algebra $\mathcal{S}^1(\mathcal{P}(E,M))$ of symbols of first-order linear operators without the hypothesis of being seen as a ${\rm C}^\infty(M)-$module.