{
  "id": "0903.0398",
  "version": "v1",
  "published": "2009-03-02T22:19:26.000Z",
  "updated": "2009-03-02T22:19:26.000Z",
  "title": "On the Dynkin index of a principal $\\mathfrak{sl}_2$-subalgebra",
  "authors": [
    "Dmitri I. Panyushev"
  ],
  "comment": "6 pages, to appear in \"Advances in Math\"",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $g$ be a simple Lie algebra over an algebraically closed field of characteristic zero. The goal of this note is to prove a closed formula for the Dynkin index of a principal $sl_2$-subalgebra of $g$. The key step in the proof uses the \"strange formula\" of Freudenthal--de Vries. As an application, we (1) compute the Dynkin index any simple $g$-module regarded as $sl_2$-module and (2) obtain an identity connecting the exponents of $g$ and the dual Coxeter numbers of both $g$ and the Langlands dual $g^\\vee$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-02T22:19:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B20"
    ],
    "keywords": [
      "dynkin index",
      "subalgebra",
      "simple lie algebra",
      "dual coxeter numbers",
      "langlands dual"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.0398P"
    }
  }
}