{
  "id": "2008.13495",
  "version": "v1",
  "published": "2020-08-31T11:23:27.000Z",
  "updated": "2020-08-31T11:23:27.000Z",
  "title": "Classical Poisson algebra of a vector bundle : Lie-algebraic characterization",
  "authors": [
    "Lecomte P. B. A",
    "Zihindula Mushengezi Elie"
  ],
  "comment": "16 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "We prove that the Lie algebra $\\mathcal{S}(\\mathcal{P}(E,M))$ of symbols of linear operators acting on smooth sections of a vector bundle $E\\to M,$ characterizes it. To obtain this, we assume that $\\mathcal{S}(\\mathcal{P}(E,M))$ is seen as ${\\rm C}^\\infty(M)-$module and that the vector bundle is of rank $n>1.$ We improve this result for the Lie algebra $\\mathcal{S}^1(\\mathcal{P}(E,M))$ of symbols of first-order linear operators. We obtain a Lie algebraic characterization of vector bundles with $\\mathcal{S}^1(\\mathcal{P}(E,M))$ without the hypothesis of being seen as a ${\\rm C}^\\infty(M)-$module.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-31T11:23:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vector bundle",
      "classical poisson algebra",
      "lie-algebraic characterization",
      "first-order linear operators",
      "lie algebraic characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}