{
  "id": "1201.1324",
  "version": "v1",
  "published": "2012-01-05T23:11:16.000Z",
  "updated": "2012-01-05T23:11:16.000Z",
  "title": "The geometry of blueprints. Part II: Tits-Weyl models of algebraic groups",
  "authors": [
    "Oliver Lorscheid"
  ],
  "comment": "87 pages",
  "categories": [
    "math.AG",
    "math.GR"
  ],
  "abstract": "This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called $\\mathbb{F}_1$, \\emph{the field with one element}. Based on Part I of The geometry of blueprints, we introduce the class of \\emph{Tits morphisms} between blue schemes. The resulting \\emph{Tits category} $\\textup{Sch}_\\mathcal{T}$ comes together with a base extension to (semiring) schemes and the so-called \\emph{Weyl extension} to sets. We prove for $\\mathcal{G}$ in a wide class of Chevalley groups---which includes the special and general linear groups, symplectic and special orthogonal groups, and all types of adjoint groups---that a linear representation of $\\mathcal{G}$ defines a model $G$ in $\\textup{Sch}_\\mathcal{T}$ whose Weyl extension is the Weyl group $W$ of $\\mathcal{G}$. We call such models \\emph{Tits-Weyl models}. The potential of Tits-Weyl models lies in \\textit{(a)} their intrinsic definition that is given by a linear representation; \\textit{(b)} the (yet to be formulated) unified approach towards thick and thin geometries; and \\textit{(c)} the extension of a Chevalley group to a functor on blueprints, which makes it, in particular, possible to consider Chevalley groups over semirings. This opens applications to idempotent analysis and tropical geometry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-05T23:11:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "08A99",
      "14A20",
      "14L99",
      "16Y60",
      "20M14"
    ],
    "keywords": [
      "algebraic groups",
      "blueprints",
      "chevalley group",
      "linear representation",
      "weyl group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 87,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.1324L"
    }
  }
}