{
  "id": "2203.03261",
  "version": "v1",
  "published": "2022-03-07T10:28:04.000Z",
  "updated": "2022-03-07T10:28:04.000Z",
  "title": "Incidence geometry of the Fano plane and Freudenthal's ansatz for the construction of (split) octonions",
  "authors": [
    "Michel Rausch de Traubenberg",
    "Marcus J. Slupinski"
  ],
  "categories": [
    "math.CO",
    "math.RA"
  ],
  "abstract": "In this article we consider structures on a Fano plane ${\\cal F}$ which allow a generalisation of Freudenthal's construction of a norm and a bilinear multiplication law on an eight-dimensional vector space ${\\mathbb O\\_{\\cal F}}$ canonically associated to ${\\cal F}$. We first determine necessary and sufficient conditions in terms of the incidence geometry of ${\\cal F}$ for these structures to give rise to division composition algebras, and classify the corresponding structures using a logarithmic version of the multiplication. We then show how these results can be used to deduce analogous results in the split composition algebra case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-07T10:28:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fano plane",
      "incidence geometry",
      "freudenthals ansatz",
      "split composition algebra case",
      "bilinear multiplication law"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}