On a Lie Algebraic Characterization of Vector Bundles

TL;DR

This paper characterizes vector bundles using the Lie algebra generated by specific differential operators, extending classical results to entire fibrations for bundles of rank greater than one.

Contribution

It provides a Lie algebraic characterization of vector bundles that encompasses the entire fibration, building on and extending Pursell-Shanks type results.

Findings

Characterization holds for vector bundles of rank > 1

Uses Lie algebra generated by eigenvector differential operators

Extends classical Lie algebraic characterizations to entire fibrations

Abstract

We prove that a vector bundle $\pi : E \to M$ 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.