{
  "id": "1007.3658",
  "version": "v5",
  "published": "2010-07-21T13:51:32.000Z",
  "updated": "2015-01-26T16:15:52.000Z",
  "title": "VB-groupoids and representation theory of Lie groupoids",
  "authors": [
    "Alfonso Gracia-Saz",
    "Rajan Amit Mehta"
  ],
  "comment": "v5: Introduction is completely rewritten, many other improvements in the exposition",
  "categories": [
    "math.DG",
    "math.SG"
  ],
  "abstract": "A VB-groupoid is a Lie groupoid equipped with a compatible linear structure. In this paper, we describe a correspondence, up to isomorphism, between VB-groupoids and 2-term representations up to homotopy of Lie groupoids. Under this correspondence, the tangent bundle of a Lie groupoid G corresponds to the \"adjoint representation\" of G. The value of this point of view is that the tangent bundle is canonical, whereas the adjoint representation is not. We define a cochain complex that is canonically associated to any VB-groupoid. The cohomology of this complex is isomorphic to the groupoid cohomology with values in the corresponding representations up to homotopy. When applied to the tangent bundle of a Lie groupoid, this construction produces a canonical complex that computes the cohomology with values in the adjoint representation. Finally, we give a classification of regular 2-term representations up to homotopy. By considering the adjoint representation, we find a new cohomological invariant associated to regular Lie groupoids.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-09-29T14:02:33.000Z",
      "abstract": "A VB-groupoid is a vector-bundle object in the category of Lie groupoids. In this paper, we explain how VB-groupoids are the intrinsic geometric objects that correspond to 2-term representations up to homotopy of Lie groupoids. In particular, the tangent bundle of a Lie groupoid is a VB-groupoid that corresponds to the adjoint representations up to homotopy. The value of this point of view is that the tangent bundle is canonical, whereas the adjoint representations up to homotopy depend on a choice of connection. In the process of describing the correspondence between VB-groupoids and representations up to homotopy, we define a cochain complex that is canonically associated to any VB-groupoid. The cohomology of this complex is isomorphic to the groupoid cohomology with values in the corresponding representations up to homotopy. The classification of regular VB-groupoids leads to a new cohomological invariant associated to regular Lie groupoids.",
      "comment": "34 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2015-01-26T16:15:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18D05",
      "22A22",
      "53D17"
    ],
    "keywords": [
      "vb-groupoid",
      "representation theory",
      "adjoint representations",
      "tangent bundle",
      "regular lie groupoids"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.3658G"
    }
  }
}