{
  "id": "0903.3049",
  "version": "v3",
  "published": "2009-03-17T21:03:16.000Z",
  "updated": "2009-06-15T20:42:30.000Z",
  "title": "The group of automorphisms of the algebra of one-sided inverses of a polynomial algebra",
  "authors": [
    "V. V. Bavula"
  ],
  "comment": "41 pages",
  "categories": [
    "math.AG",
    "math.RA"
  ],
  "abstract": "The algebra $\\mS_n$ in the title is obtained from a polynomial algebra $P_n$ in $n$ variables by adding commuting, {\\em left} (but not two-sided) inverses of the canonical generators of $P_n$. Ignoring non-Noetherian property, the algebra $\\mS_n$ belongs to a family of algebras like the Weyl algebra $A_n$ and the polynomial algebra $P_{2n}$. The group of automorphisms $G_n$ of the algebra $\\mS_n$ is found: $$ G_n=S_n\\ltimes \\mT^n\\ltimes \\Inn (\\mS_n) \\supseteq S_n\\ltimes \\mT^n\\ltimes \\underbrace{\\GL_\\infty (K)\\ltimes... \\ltimes \\GL_\\infty (K)}_{2^n-1 {\\rm times}}=:G_n' $$ where $S_n$ is the symmetric group, $\\mT^n$ is the $n$-dimensional torus, $\\Inn (\\mS_n)$ is the group of inner automorphisms of $\\mS_n$ (which is huge), and $\\GL_\\infty (K)$ is the group of invertible infinite dimensional matrices. This result may help in understanding of the structure of the groups of automorphisms of the Weyl algebra $A_n$ and the polynomial algebra $P_{2n}$. An analog of the {\\em Jacobian homomorphism}: $\\Aut_{K-{\\rm alg}}(P_{2n})\\ra K^*$, so-called, the {\\em global determinant} is introduced for the group $G_n'$ (notice that the algebra $\\mS_n$ is {\\em noncommutative} and neither left nor right Noetherian).",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-06-15T20:42:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16E10",
      "16G99",
      "16D25",
      "16D60"
    ],
    "keywords": [
      "polynomial algebra",
      "one-sided inverses",
      "weyl algebra",
      "invertible infinite dimensional matrices",
      "right noetherian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.3049B"
    }
  }
}