On a Poisson-algebraic characterization of 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.