On Gel'fand-Kolmogoroff type results

TL;DR

This paper establishes that a vector bundle can be uniquely characterized by the algebraic structure of its differential operators and symbols, extending Gel'fand-Kolmogoroff type theorems to these algebraic structures.

Contribution

It proves that the associative algebra of symbols of differential operators determines the vector bundle, generalizing classical results to new algebraic contexts.

Findings

Vector bundle characterized by algebra of differential operator symbols

Extension of Gel'fand-Kolmogoroff theorem to polynomial functions on cotangent bundle

Algebraic structures encode geometric information about vector bundles

Abstract

We prove that a vector bundle $ E \to M$ is characterized by the associative structure of the space of symbols of 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. We also obtain similar result with the $\mathbb{R}-$ algebra of smooth functions which are polynomial along the fibers of $E.$ This allows us to deduce a Gel'fand-Kolmogoroff type result for the $\mathbb{R}-$algebra ${\rm Pol}(T^*(M))$ of symbols of the differential operators of $M.$