On a Lie algebraic characterization of vector bundles

We prove that a vector bundle $\pi\colon E\to M$ is characterized by the Lie algebra generated by all differential operators on $E$ which are eigenvectors of the Lie derivative in the direction of the Euler vector field. Our result is of Pursell–Shanks type but it is remarkable in the sense that it is the whole fibration that is characterized here. The proof relies on a theorem of [Lecomte P., J. Math. Pures Appl. (9) 60 (1981), 229–239] and inherits the same hypotheses. In particular, our characterization holds only for vector bundles of rank greater than 1.