{
  "id": "0909.3746",
  "version": "v3",
  "published": "2009-09-21T12:04:55.000Z",
  "updated": "2014-05-05T12:52:14.000Z",
  "title": "Quiver grassmannians, quiver varieties and the preprojective algebra",
  "authors": [
    "Alistair Savage",
    "Peter Tingley"
  ],
  "comment": "30 pages. v2: minor corrections and notation changes, some proofs simplified. v3: Some statements and their proofs corrected. This version incorporates an erratum to the published version. See Appendix B for details",
  "journal": "Pacific J. Math. 251-2 (2011), 393--429",
  "doi": "10.2140/pjm.2011.251.393",
  "categories": [
    "math.RT",
    "math.AG",
    "math.RA"
  ],
  "abstract": "Quivers play an important role in the representation theory of algebras, with a key ingredient being the path algebra and the preprojective algebra. Quiver grassmannians are varieties of submodules of a fixed module of the path or preprojective algebra. In the current paper, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective modules are homeomorphic to the lagrangian quiver varieties of Nakajima which have been well studied in the context of geometric representation theory. We then refine this result by finding quiver grassmannians which are homeomorphic to the Demazure quiver varieties introduced by the first author, and others which are homeomorphic to the graded/cyclic quiver varieties defined by Nakajima. The Demazure quiver grassmannians allow us to describe injective objects in the category of locally nilpotent modules of the preprojective algebra. We conclude by relating our construction to a similar one of Lusztig using projectives in place of injectives.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-05-05T12:52:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "17B10"
    ],
    "keywords": [
      "preprojective algebra",
      "demazure quiver varieties",
      "lagrangian quiver varieties",
      "geometric representation theory",
      "demazure quiver grassmannians"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.3746S"
    }
  }
}