Classical Poisson algebra of a vector bundle : Lie-algebraic characterization

TL;DR

This paper characterizes vector bundles via the Lie algebra of symbols of linear operators acting on their sections, providing a Lie algebraic perspective that distinguishes the bundle structure.

Contribution

It offers a novel Lie algebraic characterization of vector bundles using symbols of linear operators, extending previous results to first-order operators without certain module assumptions.

Findings

Characterization of vector bundles via Lie algebra of symbols

Extension to first-order linear operators without module assumptions

Enhanced understanding of the algebraic structure of vector bundles

Abstract

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 and that the vector bundle is of rank $n>1.$ We improve this result for the Lie algebra $\mathcal{S}^1(\mathcal{P}(E,M))$ of symbols of first-order linear operators. We obtain a Lie algebraic characterization of vector bundles with $\mathcal{S}^1(\mathcal{P}(E,M))$ without the hypothesis of being seen as a ${\rm C}^\infty(M)-$module.