{
  "id": "2008.13587",
  "version": "v1",
  "published": "2020-08-31T13:17:25.000Z",
  "updated": "2020-08-31T13:17:25.000Z",
  "title": "On a Poisson-algebraic characterization of vector bundles",
  "authors": [
    "Zihindula Mushengezi Elie"
  ],
  "comment": "13 pages",
  "categories": [
    "math.DG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-31T13:17:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vector bundle",
      "poisson-algebraic characterization",
      "poisson algebra structure",
      "differential operators acting"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}