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

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 ${\mathrm{C}}^{\infty }(M)-$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 ${\mathrm{C}}^{\infty }(M)-$module.

16 pages. arXiv admin note: substantial text overlap with arXiv:2007.14649