{
  "id": "1607.04949",
  "version": "v1",
  "published": "2016-07-18T05:03:57.000Z",
  "updated": "2016-07-18T05:03:57.000Z",
  "title": "Leibniz algebras associated with representations of Euclidean Lie algebra",
  "authors": [
    "J. Q. Adashev",
    "B. A. Omirov",
    "S. Uguz"
  ],
  "comment": "12 pages",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "In the present paper we describe Leibniz algebras with three-dimensional Euclidean Lie algebra $\\mathfrak{e}(2)$ as its liezation. Moreover, it is assumed that the ideal generated by the squares of elements of an algebra (denoted by $I$) as a right $\\mathfrak{e}(2)$-module is associated to representations of $\\mathfrak{e}(2)$ in $\\mathfrak{sl}_2({\\mathbb{C}})\\oplus \\mathfrak{sl}_2({\\mathbb{C}}), \\mathfrak{sl}_3({\\mathbb{C}})$ and $\\mathfrak{sp}_4(\\mathbb{C})$. Furthermore, we present the classification of Leibniz algebras with general Euclidean Lie algebra ${\\mathfrak{e(n)}}$ as its liezation $I$ being an $(n+1)$-dimensional right ${\\mathfrak{e(n)}}$-module defined by transformations of matrix realization of $\\mathfrak{e(n)}.$ Finally, we extend the notion of a Fock module over Heisenberg Lie algebra to the case of Diamond Lie algebra $\\mathfrak{D}_k$ and describe the structure of Leibniz algebras with corresponding Lie algebra $\\mathfrak{D}_k$ and with the ideal $I$ considered as a Fock $\\mathfrak{D}_k$-module.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-18T05:03:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17A32",
      "17B10",
      "17B30"
    ],
    "keywords": [
      "leibniz algebras",
      "representations",
      "general euclidean lie algebra",
      "three-dimensional euclidean lie algebra",
      "heisenberg lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}