{
  "id": "0709.4232",
  "version": "v1",
  "published": "2007-09-26T18:30:49.000Z",
  "updated": "2007-09-26T18:30:49.000Z",
  "title": "Q-manifolds and Mackenzie theory: an overview",
  "authors": [
    "Theodore Voronov"
  ],
  "comment": "LaTeX, 17 pages; based on a talk at ESI, August/September 2007",
  "categories": [
    "math.DG",
    "math-ph",
    "math.MP",
    "math.SG"
  ],
  "abstract": "This text is meant to be a brief overview of the topics announced in the title and is based on my talk in Vienna (August/September 2007). It does not contain new results (except probably for a remark concerning Q-manifold homology, which I wish to elaborate elsewhere). \"Mackenzie theory\" stands for the rich circle of notions that have been put forward by Kirill Mackenzie (solo or in collaboration): double structures such as double Lie groupoids and double Lie algebroids, Lie bialgebroids and their doubles, nontrivial dualities for double and multiple vector bundles, etc. \"Q-manifolds\" are (super)manifolds with a homological vector field, i.e., a self-commuting odd vector field. They may have an extra Z-grading (called weight) not necessarily linked with the Z_2-grading (parity). I discuss double Lie algebroids (discovered by Mackenzie) and explain how this quite complicated fundamental notion is equivalent to a very simple one if the language of Q-manifolds is used. In particular, it shows how the two seemingly different notions of a \"Drinfeld double\" of a Lie bialgebroid due to Mackenzie and Roytenberg respectively, turn out to be the same thing if properly understood.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-09-26T18:30:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53D17",
      "17B62",
      "17B66",
      "18D05",
      "58A50",
      "58C50",
      "58H05"
    ],
    "keywords": [
      "mackenzie theory",
      "double lie algebroids",
      "lie bialgebroid",
      "remark concerning q-manifold homology",
      "multiple vector bundles"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.4232V"
    }
  }
}