{
  "id": "2004.10065",
  "version": "v1",
  "published": "2020-04-21T14:57:01.000Z",
  "updated": "2020-04-21T14:57:01.000Z",
  "title": "Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation",
  "authors": [
    "Yuwang Hu",
    "Jiefeng Liu",
    "Yunhe Sheng"
  ],
  "comment": "16 pages",
  "journal": "JOURNAL OF MATHEMATICAL PHYSICS 59, 081702 (2018)",
  "categories": [
    "math-ph",
    "math.MP",
    "math.RA",
    "math.RT"
  ],
  "abstract": "In this paper, first we study infinitesimal deformations of a Lie algebra with a representation and introduce the notion of a Nijenhuis pair, which gives a trivial deformation of a Lie algebra with a representation. Then we introduce the notion of a Kupershmidt-(dual-)Nijenhuis structure on a Lie algebra with a representation, which is a generalization of the $r$-$n$ structure ($r$-matrix-Nijenhuis structure) introduced by Ravanpak, Rezaei-Aghdam and Haghighatdoost. We show that a Kupershmidt-(dual-)Nijenhuis structure gives rise to a hierarchy of Kupershmidt operators. Finally, we define a Rota-Baxter-Nijenhuis structure to be a Kupershmidt-Nijenhuis structure on a Lie algebra with respect to the adjoint representation, and study the relation between Rota-Baxter-Nijenhuis structures and $r$-matrix-Nijenhuis structures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-21T14:57:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lie algebra",
      "matrix-nijenhuis structure",
      "rota-baxter-nijenhuis structure",
      "study infinitesimal deformations",
      "trivial deformation"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AIP",
      "journal": "J. Math. Phys."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}