{
  "id": "math/0612853",
  "version": "v4",
  "published": "2006-12-29T15:14:54.000Z",
  "updated": "2007-12-21T17:38:06.000Z",
  "title": "Deformations of type D Kleinian singularities",
  "authors": [
    "Paul Boddington"
  ],
  "comment": "25 pages. Section 2 rewritten, proof of main theorem shortened slightly",
  "categories": [
    "math.RA"
  ],
  "abstract": "For $n\\geq 4$ we shall construct a family $D(q)$ of non-commutative deformations of the coordinate algebra of a Kleinian singularity of type $D_n$ depending on a polynomial $q$ of degree $n$. We shall prove that every deformation of a type $D$ Kleinian singularity which is not commutative is isomorphic to some $D(q)$. We shall then consider in type $D$ the family of deformations $\\mathcal{O}^{\\boldsymbol{\\lambda}}$ constructed by Crawley-Boevey and Holland. For each $\\mathcal{O}^{\\boldsymbol{\\lambda}}$ which is not commutative we shall exhibit an explicit isomorphism $D(q)\\cong \\mathcal{O}^{\\boldsymbol{\\lambda}}$ for a suitable choice of $q$. This will enable us to prove that every deformation of a Kleinian singularity of type $D_n$ is isomorphic to some $\\mathcal{O}^{\\boldsymbol{\\lambda}}$ and determine when two $\\mathcal{O}^{\\boldsymbol{\\lambda}}$ are isomorphic.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2007-12-21T17:38:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16S30",
      "32S30"
    ],
    "keywords": [
      "kleinian singularity",
      "isomorphic",
      "coordinate algebra",
      "explicit isomorphism",
      "polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....12853B"
    }
  }
}