On Pursell-Shanks type results

Abstract

We prove a Lie-algebraic characterization of vector bundle for the Lie algebra $\mathcal{D}(E,M),$ seen as ${\mathrm{C}}^{\infty }(M)-$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 ${\mathrm{C}}^{\infty }(M)-$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 ${\mathrm{C}}^{\infty }(M)-$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 ${\mathrm{C}}^{\infty }(M)-$module.

27 pages