On Gel'fand-Kolmogoroff type results

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 $\mathrm{Pol}(T^{\ast }(M))$ of symbols of the differential operators of $M.$

17 pages

References in corpus (2)

• Affine bundles are affine spaces over modules
• Lie algebra of homogeneous operators of a vector bundle