{
  "id": "1108.4294",
  "version": "v2",
  "published": "2011-08-22T12:45:13.000Z",
  "updated": "2012-01-07T17:58:04.000Z",
  "title": "A Geometric Approach to Noncommutative Principal Torus Bundles",
  "authors": [
    "Stefan Wagner"
  ],
  "comment": "This paper is an extended version of \"Smooth Localization in Noncommutative Geometry\", arxiv:1108.4294v1 [math.DG], 22 Aug 2011, with an application to the noncommutative geometry of principal torus bundles. All comments are welcome. 43 pages",
  "journal": "Proc. London Math. Soc. (2013)",
  "doi": "10.1112/plms/pds073",
  "categories": [
    "math.DG",
    "math-ph",
    "math.MP"
  ],
  "abstract": "A (smooth) dynamical system with transformation group $\\mathbb{T}^n$ is a triple $(A,\\mathbb{T}^n,\\alpha)$, consisting of a unital locally convex algebra $A$, the $n$-torus $\\mathbb{T}^n$ and a group homomorphism $\\alpha:\\mathbb{T}^n\\rightarrow\\Aut(A)$, which induces a (smooth) continuous action of $\\mathbb{T}^n$ on $A$. In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal torus bundles based on such dynamical systems. Our approach is inspired by the classical setting: In fact, after recalling the definition of a trivial noncommutative principal torus bundle, we introduce a convenient (smooth) localization method for noncommutative algebras and say that a dynamical system $(A,\\mathbb{T}^n,\\alpha)$ is called a noncommutative principal $\\mathbb{T}^n$-bundle, if localization leads to a trivial noncommutative principal $\\mathbb{T}^n$-bundle. We prove that this approach extends the classical theory of principal torus bundles and present a bunch of (non-trivial) noncommutative examples.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-01-07T17:58:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L87",
      "55R10",
      "37B05",
      "17A60"
    ],
    "keywords": [
      "geometric approach",
      "dynamical system",
      "trivial noncommutative principal torus bundle",
      "unital locally convex algebra",
      "approach extends"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.4294W"
    }
  }
}