{
  "id": "2001.01355",
  "version": "v1",
  "published": "2020-01-06T01:45:00.000Z",
  "updated": "2020-01-06T01:45:00.000Z",
  "title": "Homotopy Poisson algebras, Maurer-Cartan elements and Dirac structures of CLWX 2-algebroids",
  "authors": [
    "Jiefeng Liu",
    "Yunhe Sheng"
  ],
  "comment": "30 pages, to appear in J. Noncommutative Geom",
  "categories": [
    "math.DG",
    "math-ph",
    "math.MP",
    "math.SG"
  ],
  "abstract": "In this paper, we construct a homotopy Poisson algebra of degree 3 associated to a split Lie 2-algebroid, by which we give a new approach to characterize a split Lie 2-bialgebroid. We develop the differential calculus associated to a split Lie 2-algebroid and establish the Manin triple theory for split Lie 2-algebroids. More precisely, we give the notion of a strict Dirac structure and define a Manin triple for split Lie 2-algebroids to be a CLWX 2-algebroid with two transversal strict Dirac structures. We show that there is a one-to-one correspondence between Manin triples for split Lie 2-algebroids and split Lie 2-bialgebroids. We further introduce the notion of a weak Dirac structure of a CLWX 2-algebroid and show that the graph of a Maurer-Cartan element of the homotopy Poisson algebra of degree 3 associated to a split Lie 2-bialgebroid is a weak Dirac structure. Various examples including the string Lie 2-algebra, split Lie 2-algebroids constructed from integrable distributions and left-symmetric algebroids are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-06T01:45:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "split lie",
      "homotopy poisson algebra",
      "maurer-cartan element",
      "weak dirac structure",
      "transversal strict dirac structures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}