{
  "id": "2207.13946",
  "version": "v1",
  "published": "2022-07-28T08:27:45.000Z",
  "updated": "2022-07-28T08:27:45.000Z",
  "title": "Commutation relations of $\\mathfrak g\\_2$ and the incidence geometry of the Fano plane",
  "authors": [
    "Michel Rausch de Traubenberg",
    "M J Slupinski"
  ],
  "categories": [
    "math.CO",
    "math-ph",
    "math.MP",
    "math.RT"
  ],
  "abstract": "We continue our study and classification of structures on the Fano plane ${\\cal F}$ and its dual ${\\cal F}^\\ast$ involved in the construction of octonions and the Lie algebra $\\mathfrak g_2 (\\mathbb F)$ over a field $\\mathbb F$. These are a \"composition factor\": ${\\cal F}\\times {\\cal F} \\to\\{-1, 1\\}$, inducing an octonion multiplication, and a function $\\delta^\\ast : Aut({\\cal F}) \\times {\\cal F}^\\ast \\to \\{-1, 1\\}$ such that $g \\in Aut({\\cal F})$ can be lifted to an automorphism of the octonions iff $\\delta^\\ast(g, \\cdot)$ is the Radon transform of a function on ${\\cal F}$. We lift the action of $Aut({\\cal F})$ on ${\\cal F}$ to the action of a non-trivial eight-fold covering $Aut({\\cal F})$ on a twofold covering $\\hat {\\cal F}$ of ${\\cal F}$ contained in the octonions. This extends tautologically to an action on the octonions by automorphism. Finally, we associate to incident point-line pairs a generating set of $\\mathfrak g_2 (\\mathbb F)$ and express brackets in terms of the incidence geometry of ${\\cal F}$ and $\\epsilon$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-28T08:27:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fano plane",
      "incidence geometry",
      "commutation relations",
      "incident point-line pairs",
      "non-trivial eight-fold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}