{
  "id": "0906.0600",
  "version": "v3",
  "published": "2009-06-02T21:07:44.000Z",
  "updated": "2010-04-18T20:31:47.000Z",
  "title": "${\\rm K}_1(\\mS_1)$ and the group of automorphisms of the algebra $\\mS_2$ of one-sided inverses of a polynomial algebra in two variables",
  "authors": [
    "V. V. Bavula"
  ],
  "comment": "13 pages",
  "categories": [
    "math.AG",
    "math.RA"
  ],
  "abstract": "Explicit generators are found for the group $G_2$ of automorphisms of the algebra $\\mS_2$ of one-sided inverses of a polynomial algebra in two variables over a field of characteristic zero. Moreover, it is proved that $$ G_2\\simeq S_2\\ltimes \\mT^2\\ltimes \\Z\\ltimes ((K^*\\ltimes E_\\infty (\\mS_1))\\boxtimes_{\\GL_\\infty (K)}(K^*\\ltimes E_\\infty (\\mS_1)))$$ where $S_2$ is the symmetric group, $\\mT^2$ is the 2-dimensional torus, $E_\\infty (\\mS_1)$ is the subgroup of $\\GL_\\infty (\\mS_1)$ generated by the elementary matrices. In the proof, we use and prove several results on the index of operators, and the final argument in the proof is the fact that ${\\rm K}_1 (\\mS_1) \\simeq K^*$ proved in the paper. The algebras $\\mS_1$ and $\\mS_2$ are noncommutative, non-Noetherian, and not domains. The group of units of the algebra $\\mS_2$ is found (it is huge).",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-04-18T20:31:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14E07",
      "14H37",
      "14R10",
      "14R15"
    ],
    "keywords": [
      "polynomial algebra",
      "one-sided inverses",
      "automorphisms",
      "explicit generators",
      "characteristic zero"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.0600B"
    }
  }
}