{
  "id": "math/0611799",
  "version": "v2",
  "published": "2006-11-26T21:25:22.000Z",
  "updated": "2006-12-22T22:33:40.000Z",
  "title": "Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids",
  "authors": [
    "K. C. H. Mackenzie"
  ],
  "comment": "Reference added. Some rewording in introduction. 34 pages. Uses Xy-pic",
  "categories": [
    "math.DG",
    "math.CT",
    "math.SG"
  ],
  "abstract": "The word `double' was used by Ehresmann to mean `an object X in the category of all X'. Double categories, double groupoids and double vector bundles are instances, but the notion of Lie algebroid cannot readily be doubled in the Ehresmann sense, since a Lie algebroid bracket cannot be defined diagrammatically. In this paper we use the duality of double vector bundles to define a notion of double Lie algebroid, and we show that this abstracts the infinitesimal structure (at second order) of a double Lie groupoid. We further show that the cotangent of either Lie algebroid in a Lie bialgebroid has a double Lie algebroid structure, and that a pair of Lie algebroid structures on dual vector bundles forms a Lie bialgebroid if and only if the structures which they canonically induce on their cotangents form a double Lie algebroid. In particular, the Drinfel'd double of a Lie bialgebra has a double Lie algebroid structure. We also show that matched pairs of Lie algebroids, as used by J.-H. Lu in the classification of Poisson group actions, are in bijective correspondence with vacant double Lie algebroids.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-12-22T22:33:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53D17",
      "17B62",
      "17B66",
      "18D05",
      "22A22",
      "58H05"
    ],
    "keywords": [
      "lie bialgebroid",
      "ehresmann doubles",
      "drinfeld double",
      "double lie algebroid structure",
      "double vector bundles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....11799M"
    }
  }
}