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On a Poisson-algebraic characterization of vector bundles

Elie Zihindula Mushengezi

Abstract

We prove that the $\mathbb{R}-$algebra $\mathcal{S}(\mathcal{P}(E,M)) $ of symbols of differential operators acting on the sections of the vector bundle $E\to M$ decompose into the sum \[ \mathcal{S}(\mathcal{P}(E,M))=\mathcal{J}(E)\oplus {\rm Pol}(T^*M) \] where $\mathcal{J}(E)$ is an ideal of $\mathcal{S}(\mathcal{P}(E,M))$ in which product of two elements is always zero. This induces that $\mathcal{S}(\mathcal{P}(E,M))$ cannot characterize $E \to M$ with its only structure of $\mathbb{R}-$ algebra. We prove that with its Poisson algebra structure, $\mathcal{S}(\mathcal{P}(E,M))$ characterizes the vector bundle $E\to M$ without the requirement to be considered as a ${\rm C}^\infty(M)-$module.

On a Poisson-algebraic characterization of vector bundles

Abstract

We prove that the algebra of symbols of differential operators acting on the sections of the vector bundle decompose into the sum where is an ideal of in which product of two elements is always zero. This induces that cannot characterize with its only structure of algebra. We prove that with its Poisson algebra structure, characterizes the vector bundle without the requirement to be considered as a module.

Paper Structure

This paper contains 7 sections, 9 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. The Lie algebra of linear operators of a vector bundle
  3. The particular case of the $\mathbb{R}-$algebra $\sigma_{pson}(gl(E))$
  4. The $\mathbb{R}-$algebra $\mathcal{S}(\mathcal{P}(E,M))$
  5. From local expression
  6. Global point of view
  7. Poisson-algebraic characterization

Key Result

Proposition 3.1

When the $\mathbb{R}-$vector space $\sigma_{pson}(gl(E))=sl(E)\oplus {\rm C}^\infty(M)\,id$ is provided with the multiplication defined in (multi sgle) above, we have

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Proposition 4.6
  • ...and 3 more