Lie algebra of homogeneous operators of a vector bundle

TL;DR

This paper establishes an isomorphism between two Lie algebras of differential operators on a vector bundle, enabling explicit computation of derivations and zero-weight derivations of fiberwise polynomial functions.

Contribution

It proves the equivalence of Lie algebras generated by differential operators related to the Euler vector field and Grothendieck construction, providing explicit descriptions.

Findings

Identifies the isomorphism between the two Lie algebras.

Computes all derivations of the fiberwise polynomial algebra.

Describes the Lie algebra of zero-weight derivations explicitly.

Abstract

We prove that for a vector bundle $ E \to M$, the Lie algebra $\mathcal{D}_{\mathcal{E}}(E)$ generated by all differential operators on $E$ which are eigenvectors of $L_{\mathcal{E}},$ the Lie derivative in the direction of the Euler vector field of $E,$ and the Lie algebra $\mathcal{D}_G(E)$ obtained by Grothendieck construction over the $\mathbb{R}-$algebra $\mathcal{A}(E):= {\rm Pol}(E)$ of fiberwise polynomial functions, coincide up an isomorphism. This allows us to compute all the derivations of the $\mathbb{R}-$algebra $\mathcal{A}(E)$ and to obtain an explicit description of the Lie algebra of zero-weight derivations of $\mathcal{A}(E).$