{
  "id": "2203.15861",
  "version": "v1",
  "published": "2022-03-29T19:03:20.000Z",
  "updated": "2022-03-29T19:03:20.000Z",
  "title": "Non-Lie subgroups in Lie groups over local fields of positive characteristic",
  "authors": [
    "Helge Glockner"
  ],
  "comment": "11 pages, LaTeX",
  "categories": [
    "math.GR"
  ],
  "abstract": "By Cartan's Theorem, every closed subgroup $H$ of a real (or $p$-adic) Lie group $G$ is a Lie subgroup. For Lie groups over a local field ${\\mathbb K}$ of positive characteristic, the analogous conclusion is known to be wrong. We show more: There exists a ${\\mathbb K}$-analytic Lie group $G$ and a non-discrete, compact subgroup $H$ such that, for every ${\\mathbb K}$-analytic manifold $M$, every ${\\mathbb K}$-analytic map $f\\colon M\\to G$ with $f(M)\\subseteq H$ is locally constant. In particular, the set $H$ does not admit a non-discrete ${\\mathbb K}$-analytic manifold structure which makes the inclusion of $H$ into $G$ a ${\\mathbb K}$-analytic map. We can achieve that, moreover, $H$ does not admit a ${\\mathbb K}$-analytic Lie group structure compatible with the topological group structure induced by $G$ on $H$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-29T19:03:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E20",
      "22E35",
      "22E50",
      "32P05"
    ],
    "keywords": [
      "local field",
      "positive characteristic",
      "non-lie subgroups",
      "analytic map",
      "analytic lie group structure compatible"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}