{
  "id": "math-ph/0107023",
  "version": "v2",
  "published": "2001-07-23T10:33:11.000Z",
  "updated": "2002-02-13T17:33:36.000Z",
  "title": "Quantization as a functor",
  "authors": [
    "N. P. Landsman"
  ],
  "comment": "20 pages LaTeX. Major revision",
  "categories": [
    "math-ph",
    "math.MP",
    "math.OA",
    "math.SG"
  ],
  "abstract": "Notwithstanding known obstructions to this idea, we formulate an attempt to turn quantization into a functorial procedure. We define a category PO of Poisson manifolds, whose objects are integrable Poisson manifolds and whose arrows are isomorphism classes of regular Weinstein dual pairs; it follows that identity arrows are symplectic groupoids, and that two objects are isomorphic in PO iff they are Morita equivalent in the sense of P. Xu. It has a subcategory LPO that has duals of integrable Lie algebroids as objects and cotangent bundles as arrows. We argue that naive C*-algebraic quantization should be functorial from LPO to the well-known category KK, whose objects are separable C*-algebras and whose arrows are Kasparov's KK-groups. This limited functoriality of quantization would already imply the Atiyah-Singer index theorem, as well as its far-reaching generalizations developed by Connes and others. In the category KK, isomorphism of objects implies isomorphism of K-theory groups, so that the functoriality of quantization on all of PO would imply that Morita equivalent Poisson algebras are quantized by C*-algebras with isomorphic K-theories. Finally, we argue that the correct codomain for the possible functoriality of quantization is the category RKK(I), which takes the deformation aspect of quantization into account.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2002-02-13T17:33:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81S10",
      "46L65",
      "53D20",
      "53D55"
    ],
    "keywords": [
      "quantization",
      "poisson manifolds",
      "morita equivalent poisson algebras",
      "regular weinstein dual pairs",
      "objects implies isomorphism"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.ph...7023L"
    }
  }
}