On a Poisson-algebraic characterization of vector bundles

TL;DR

This paper demonstrates that the Poisson algebra structure of symbol algebras of differential operators uniquely characterizes vector bundles, unlike their algebraic structure alone, which is insufficient.

Contribution

It shows that the Poisson algebra structure of symbol algebras fully characterizes vector bundles without relying on the module structure.

Findings

The algebra of symbols decomposes into an ideal and polynomial functions on the cotangent bundle.

The algebraic structure alone cannot distinguish vector bundles.

The Poisson algebra structure uniquely characterizes vector bundles.

Abstract

We prove that the $\mathbb{R}-$algebra $\mathcal{S}(\mathcal{P}(E,M)) $ of symbols of differential operators acting on the sections of the vector bundle $E\to M$ decompose into the sum \[ \mathcal{S}(\mathcal{P}(E,M))=\mathcal{J}(E)\oplus {\rm Pol}(T^*M) \] where $\mathcal{J}(E)$ is an ideal of $\mathcal{S}(\mathcal{P}(E,M))$ in which product of two elements is always zero. This induces that $\mathcal{S}(\mathcal{P}(E,M))$ cannot characterize $E \to M$ with its only structure of $\mathbb{R}-$ algebra. We prove that with its Poisson algebra structure, $\mathcal{S}(\mathcal{P}(E,M))$ characterizes the vector bundle $E\to M$ without the requirement to be considered as a ${\rm C}^\infty(M)-$module.