{
  "id": "1512.08362",
  "version": "v1",
  "published": "2015-12-28T10:05:04.000Z",
  "updated": "2015-12-28T10:05:04.000Z",
  "title": "Path algebras of quivers and representations of locally finite Lie algebras",
  "authors": [
    "J. Hennig",
    "S. J. Sierra"
  ],
  "comment": "30 pages",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "We explore the (noncommutative) geometry of locally simple representations of the diagonal locally finite Lie algebras $\\mathfrak{sl}(n^\\infty)$, $\\mathfrak o(n^\\infty)$, and $\\mathfrak{sp}(n^\\infty)$. Let $\\mathfrak g_\\infty$ be one of these Lie algebras, and let $I \\subseteq U(\\mathfrak g_\\infty)$ be the annihilator of a locally simple $\\mathfrak g_\\infty$-module. We show that for each such $I$, there is a quiver $Q$ so that locally simple $\\mathfrak g_\\infty$-modules with annihilator $I$ are parameterised by \"points\" in the \"noncommutative space\" corresponding to the path algebra of $Q$. Methods of noncommutative algebraic geometry are key to this correspondence. We classify the quivers that arise and relate them to characters of symmetric groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-28T10:05:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B65",
      "16W50",
      "16E50",
      "16G20",
      "16D90"
    ],
    "keywords": [
      "path algebra",
      "diagonal locally finite lie algebras",
      "annihilator",
      "locally simple representations",
      "symmetric groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}