{
  "id": "2203.03848",
  "version": "v1",
  "published": "2022-03-08T04:42:10.000Z",
  "updated": "2022-03-08T04:42:10.000Z",
  "title": "Reduction of Structure to Parabolic Subgroups",
  "authors": [
    "Danny Ofek"
  ],
  "comment": "22 pages",
  "categories": [
    "math.GR",
    "math.AG",
    "math.RA"
  ],
  "abstract": "Let $G$ be an affine group over a field of characteristic not two. A $G$-torsor is called isotropic if it admits reduction of structure to a proper parabolic subgroup of $G$. This definition generalizes isotropy of affine groups and involutions of central simple algebras. When does $G$ admit anisotropic torsors? Building on work of J. Tits, we answer this question for simple groups. We also give an answer for connected and semisimple $G$ under certain restrictions on its root system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-08T04:42:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E72",
      "20G15",
      "20G07",
      "11E39",
      "16W10"
    ],
    "keywords": [
      "affine group",
      "definition generalizes isotropy",
      "proper parabolic subgroup",
      "central simple algebras",
      "admit anisotropic torsors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}