{
  "id": "1404.7033",
  "version": "v2",
  "published": "2014-04-28T16:07:08.000Z",
  "updated": "2014-10-20T12:15:10.000Z",
  "title": "An exotic zoo of diffeomorphism groups on $\\mathbb R^n$",
  "authors": [
    "Andreas Kriegl",
    "Peter W. Michor",
    "Armin Rainer"
  ],
  "comment": "45 pages; some small corrections done",
  "categories": [
    "math.DG",
    "math.AP",
    "math.FA"
  ],
  "abstract": "Let $C^{[M]}$ be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight sequence $M=(M_k)$ is log-convex and has moderate growth. We prove that the groups ${\\operatorname{Diff}}\\mathcal{B}^{[M]}(\\mathbb{R}^n)$, ${\\operatorname{Diff}}W^{[M],p}(\\mathbb{R}^n)$, ${\\operatorname{Diff}}{\\mathcal{S}}{}_{[L]}^{[M]}(\\mathbb{R}^n)$, and ${\\operatorname{Diff}}\\mathcal{D}^{[M]}(\\mathbb{R}^n)$ of $C^{[M]}$-diffeomorphisms on $\\mathbb{R}^n$ which differ from the identity by a mapping in $\\mathcal{B}^{[M]}$ (global Denjoy--Carleman), $W^{[M],p}$ (Sobolev-Denjoy-Carleman), ${\\mathcal{S}}{}_{[L]}^{[M]}$ (Gelfand--Shilov), or $\\mathcal{D}^{[M]}$ (Denjoy-Carleman with compact support) are $C^{[M]}$-regular Lie groups. As an application we use the $R$-transform to show that the Hunter-Saxton PDE on the real line is well-posed in any of the classes $W^{[M],1}$, ${\\mathcal{S}}{}_{[L]}^{[M]}$, and $\\mathcal{D}^{[M]}$. Here we find some surprising groups with continuous left translations and $C^{[M]}$ right translations (called half-Lie groups), which, however, also admit $R$-transforms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-28T16:07:08.000Z",
      "comment": "45 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-20T12:15:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26E10",
      "46A17",
      "46E50",
      "46F05",
      "58B10",
      "58B25",
      "58C25",
      "58D05",
      "58D15",
      "35Q31"
    ],
    "keywords": [
      "diffeomorphism groups",
      "exotic zoo",
      "regular lie groups",
      "denjoy-carleman class",
      "right translations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.7033K"
    }
  }
}