{
  "id": "1712.07035",
  "version": "v1",
  "published": "2017-12-19T16:42:46.000Z",
  "updated": "2017-12-19T16:42:46.000Z",
  "title": "Lie 2-algebroids and matched pairs of 2-representations - a geometric approach",
  "authors": [
    "Madeleine Jotz Lean"
  ],
  "comment": "This is the improved and completed second part of the work arXiv:1504.00880 (N-manifolds of degree 2 and metric double vector bundles)",
  "categories": [
    "math.DG"
  ],
  "abstract": "Li-Bland's correspondence between linear Courant algebroids and Lie $2$-algebroids is explained and shown to be an equivalence of categories. Decomposed VB-Courant algebroids are shown to be equivalent to split Lie 2-algebroids in the same manner as decomposed VB-algebroids are equivalent to 2-term representations up to homotopy (Gracia-Saz and Mehta). Several classes of examples are discussed, yielding new examples of split Lie 2-algebroids. We prove that the bicrossproduct of a matched pair of $2$-representations is a split Lie $2$-algebroid and we explain this result geometrically, as a consequence of the equivalence of VB-Courant algebroids and Lie $2$-algebroids. This explains in particular how the two notions of double\" of a matched pair of representations are geometrically related. In the same manner, we explain the geometric link between the two notions of double of a Lie bialgebroid.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-19T16:42:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53B05",
      "53D17"
    ],
    "keywords": [
      "matched pair",
      "geometric approach",
      "split lie",
      "linear courant algebroids",
      "representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}