{
  "id": "1512.03469",
  "version": "v1",
  "published": "2015-12-10T22:31:43.000Z",
  "updated": "2015-12-10T22:31:43.000Z",
  "title": "Complete classification of $H$-type algebras: I",
  "authors": [
    "Kenro Furutani",
    "Irina Markina"
  ],
  "comment": "38 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathscr N$ be a 2-step nilpotent Lie algebra endowed with non-degenerate scalar product $\\langle.\\,,.\\rangle$ and let $\\mathscr N=V\\oplus_{\\perp}Z$, where $Z$ is the centre of the Lie algebra and $V$ its orthogonal complement with respect to the scalar product. We study the classification of the Lie algebras for which the space $V$ arises as a representation space of a Clifford algebra $\\Cl(\\mathbb R^{r,s})$ and the representation map $J\\colon \\Cl(\\mathbb R^{r,s})\\to(V)$ is related to the Lie algebra structure by $\\langle J_zv,w\\rangle=\\langle z,[v,w]\\rangle$ for all $z\\in \\mathbb R^{r,s}$ and $v,w\\in V$. The classification is based on the range of parameters $r$ and $s$ and is completed for the Clifford modules $V$, having minimal possible dimension, that are not necessary irreducible. We find the necessary condition for the existence of a Lie algebra isomorphism according to the range of integer parameters $0\\leq r,s<\\infty$. We present the constructive proof for the isomorphism map for isomorphic Lie algebras and defined the class of non-isomorphic Lie algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-10T22:31:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B60",
      "17B30",
      "17B70",
      "22E15"
    ],
    "keywords": [
      "type algebras",
      "complete classification",
      "lie algebra isomorphism",
      "lie algebra structure",
      "nilpotent lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151203469F"
    }
  }
}