{
  "id": "1507.01349",
  "version": "v1",
  "published": "2015-07-06T08:10:10.000Z",
  "updated": "2015-07-06T08:10:10.000Z",
  "title": "Leibniz algebras associated with representations of the Diamond Lie algebra",
  "authors": [
    "S. Uguz",
    "I. A. Karimjanov",
    "B. A. Omirov"
  ],
  "comment": "18 pages",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "In this paper we describe some Leibniz algebras whose corresponding Lie algebra is four-dimensional Diamond Lie algebra $\\mathfrak{D}$ and the ideal generated by the squares of elements (further denoted by $I$) is a right $\\mathfrak{D}$-module. Using description \\cite{Cas} of representations of algebra $\\mathfrak{D}$ in $\\mathfrak{sl}(3,{\\mathbb{C}})$ and $\\mathfrak{sp}(4,{\\mathbb{F}})$ where ${\\mathbb{F}}={\\mathbb{R}}$ or ${\\mathbb{C}}$ we obtain the classification of above mentioned Leibniz algebras. Moreover, Fock representation of Heisenberg Lie algebra was extended to the case of the algebra $\\mathfrak{D}.$ Classification of Leibniz algebras with corresponding Lie algebra $\\mathfrak{D}$ and with the ideal $I$ as a Fock right $\\mathfrak{D}$-module is presented. The linear integrable deformations in terms of the second cohomology groups of obtained finite-dimensional Leibniz algebras are described. Two computer programs in Mathematica 10 which help to calculate for a given Leibniz algebra the general form of elements of spaces $BL^2$ and $ZL^2$ are constructed, as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-06T08:10:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17A32",
      "17B30",
      "17B10"
    ],
    "keywords": [
      "corresponding lie algebra",
      "four-dimensional diamond lie algebra",
      "finite-dimensional leibniz algebras",
      "heisenberg lie algebra",
      "second cohomology groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150701349U"
    }
  }
}