{
  "id": "0712.1398",
  "version": "v2",
  "published": "2007-12-10T06:12:26.000Z",
  "updated": "2008-06-05T15:10:11.000Z",
  "title": "Prolongations of Lie algebras and applications",
  "authors": [
    "Paul-Andi Nagy"
  ],
  "comment": "New version. Proofs shortened, one section added on flat connections with 3-form torsion",
  "journal": "Journal of Lie Theory 23 (2013), No. 1, 1-33",
  "categories": [
    "math.DG"
  ],
  "abstract": "We study the skew-symmetric prolongation of a Lie subalgebra $\\g \\subseteq \\mathfrak{so}(n)$, in other words the intersection $\\Lambda^3 \\cap (\\Lambda^1 \\otimes \\g)$.We compute this space in full generality. Applications include uniqueness results for connections with skew-symmetric torsion and also the proof of the Euclidean version of a conjecture posed in \\cite{ofarill} concerning a class of Pl\\\"ucker-type embeddings. We also derive a classification of the metric k-Lie algebras (or Filipov algebras), in positive signature and finite dimension. Prolongations of Lie algebras can also be used to finish the classification, started in \\cite{datri}, of manifolds admitting Killing frames, or equivalently flat connections with 3-form torsion. Next we study specific properties of invariant 4-forms of a given metric representation and apply these considerations to classify the holonomy representation of metric connections with vectorial torsion, that is with torsion contained in $\\Lambda^1 \\subseteq \\Lambda^1 \\otimes \\Lambda^2$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-06-05T15:10:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C12",
      "53C24",
      "53C55"
    ],
    "keywords": [
      "applications",
      "study specific properties",
      "metric k-lie algebras",
      "filipov algebras",
      "vectorial torsion"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.1398N",
      "inspire": 1419902
    }
  }
}