{
  "id": "1609.01549",
  "version": "v1",
  "published": "2016-09-06T13:45:08.000Z",
  "updated": "2016-09-06T13:45:08.000Z",
  "title": "A new approach to representations of $3$-Lie algebras and abelian extensions",
  "authors": [
    "Jiefeng Liu",
    "Abdenacer Makhlouf",
    "Yunhe Sheng"
  ],
  "comment": "17 pages",
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP",
    "math.RA"
  ],
  "abstract": "In this paper, we introduce the notion of generalized representation of a $3$-Lie algebra, by which we obtain a generalized semidirect product $3$-Lie algebra. Moreover, we develop the corresponding cohomology theory. Various examples of generalized representations of 3-Lie algebras and computation of 2-cocycles of the new cohomology are provided. Also, we show that a split abelian extension of a 3-Lie algebra is isomorphic to a generalized semidirect product $3$-Lie algebra. Furthermore, we describe general abelian extensions of 3-Lie algebras using Maurer-Cartan elements.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-06T13:45:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lie algebra",
      "generalized semidirect product",
      "split abelian extension",
      "general abelian extensions",
      "generalized representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}