{
  "id": "1501.06362",
  "version": "v1",
  "published": "2015-01-26T12:24:50.000Z",
  "updated": "2015-01-26T12:24:50.000Z",
  "title": "Automorphism Groups of Affine Varieties and a Characterization of Affine n-Space",
  "authors": [
    "Hanspeter Kraft"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AG",
    "math.GR"
  ],
  "abstract": "We show that the automorphism group of affine n-space $A^n$ determines $A^n$ up to isomorphism: If $X$ is a connected affine variety such that $Aut(X)$ is isomorphic to $Aut(A^n)$ as ind-groups, then $X$ is isomorphic to $A^n$ as a variety. We also show that every finite group and every torus appears as $Aut(X)$ for a suitable affine variety $X$, but that $Aut(X)$ cannot be isomorphic to a semisimple group. In fact, if $Aut(X)$ is finite dimensional and $X$ not isomorphic to the affine line $A^1$, then the connected component $Aut(X)^0$ is a torus. Concerning the structure of $Aut(A^n)$ we prove that any homomorphism $Aut(A^n) \\to G$ of ind-groups either factors through the Jacobian determinant $jac\\colon Aut(A^n) \\to k^*$, or it is a closed immersion. For $SAut(A^n):=\\ker(jac)$ we show that every nontrivial homomorphism $SAut(A^n) \\to G$ is a closed immersion. Finally, we prove that every non-trivial homomorphism $SAut(A^n) \\to SAut(A^n)$ is an automorphism, and is given by conjugation with an element from $Aut(A^n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-26T12:24:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L30",
      "20G99",
      "22F50",
      "54H15"
    ],
    "keywords": [
      "automorphism group",
      "affine n-space",
      "characterization",
      "isomorphic",
      "closed immersion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150106362K"
    }
  }
}