{
  "id": "1412.8071",
  "version": "v1",
  "published": "2014-12-27T18:05:05.000Z",
  "updated": "2014-12-27T18:05:05.000Z",
  "title": "Quantization and injective submodules of differential operator modules",
  "authors": [
    "Charles H. Conley",
    "Dimitar Grantcharov"
  ],
  "comment": "26 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "The Lie algebra of vector fields on $R^m$ acts naturally on the spaces of differential operators between tensor field modules. Its projective subalgebra is isomorphic to $sl_{m+1}$, and its affine subalgebra is a maximal parabolic subalgebra of the projective subalgebra with Levi factor $gl_m$. We prove two results. First, we realize all injective objects of the parabolic category O$^{gl_m}(sl_{m+1})$ of $gl_m$-finite $sl_{m+1}$-modules as submodules of differential operator modules. Second, we study projective quantizations of differential operator modules, i.e., $sl_{m+1}$-invariant splittings of their order filtrations. In the case of modules of differential operators from a tensor density module to an arbitrary tensor field module, we determine when there exists a unique projective quantization, when there exists no projective quantization, and when there exist multiple projective quantizations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-27T18:05:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B66"
    ],
    "keywords": [
      "differential operator modules",
      "projective quantization",
      "injective submodules",
      "arbitrary tensor field module",
      "projective subalgebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}