{
  "id": "1802.07547",
  "version": "v1",
  "published": "2018-02-21T12:55:35.000Z",
  "updated": "2018-02-21T12:55:35.000Z",
  "title": "On some realizations of globally exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3 $-symmetric spaces $G/K$, $G=G_2, F_4, E_6$, Part I",
  "authors": [
    "Toshikazu Miyashita"
  ],
  "comment": "44 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "R. Lutz introduced the notion of $\\varGamma$-symmetric space as a generalization of the classical notion of symmetric space in 1981, where $\\varGamma$ is a finite abelian group. In the present article, as $\\varGamma=\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$, we give the automorphisms $\\tilde{\\sigma}_3, \\tilde{\\tau}_3$ of order $3$ on the connected compact exceptional Lie groups $G=G_2, F_4,E_6$ %and construct $\\varGamma=\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$ as the elements of order $3$ in $\\Aut(G)$, explicitly and determine the structure of the group $G^{\\sigma_3} \\cap G^{\\tau_3}$ using homomorphism theorem elementary. These amount to some global realizations of exceptional $\\varmathbb{Z}_3 \\times \\varmathbb{Z}_3$-symmetric spaces $G/K$, where $(G^{\\sigma_3} \\cap G^{\\tau_3})_0 \\subseteq K \\subseteq G^{\\sigma_3} \\cap G^{\\tau_3}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-21T12:55:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C30",
      "53C35",
      "17B40"
    ],
    "keywords": [
      "symmetric space",
      "globally exceptional",
      "connected compact exceptional lie groups",
      "finite abelian group",
      "homomorphism theorem elementary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}