{
  "id": "0711.2809",
  "version": "v2",
  "published": "2007-11-18T18:50:22.000Z",
  "updated": "2008-06-13T18:47:09.000Z",
  "title": "Root Systems for Levi Factors and Borel-de Siebenthal Theory",
  "authors": [
    "Bertram Kostant"
  ],
  "comment": "28 pages, plain tex",
  "categories": [
    "math.RT",
    "math.GR"
  ],
  "abstract": "Let $\\frak{m}$ be a Levi factor of a proper parabolic subalgebra $\\frak{q}$ of a complex semisimple Lie algebra $\\frak{g}$. Let $\\frak{t} = cent \\frak{m}$. A nonzero element $\\nu \\in \\frak{t}^*$ is called a $\\frak {t}$-root if the corresponding adjoint weight space $\\frak{g}_{nu}$ is not zero. If $\\nu$ is a $\\frak{t}$-root, some time ago we proved that $\\frak{g}_{\\nu}$ is $ad \\frak{m}$ irreducible. Based on this result we develop in the present paper a theory of $\\frak{t}$-roots which replicates much of the structure of classical root theory (case where $\\frak{t}$ is a Cartan subalgebra). The results are applied to obtain new reults about the structure of the nilradical $\\frak{n}$ of $\\frak{q}$. Also applications in the case where $dim \\frak{t}=1$ are used in Borel-de Siebenthal theory to determine irreducibility theorems for certain equal rank subalgebras of $\\frak{g}$. In fact the irreducibility results readily yield a proof of the main assertions of the Borel-de Siebenthal theory.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-06-13T18:47:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20Gxx",
      "22E10",
      "22E25",
      "22E46"
    ],
    "keywords": [
      "borel-de siebenthal theory",
      "levi factor",
      "root systems",
      "complex semisimple lie algebra",
      "irreducibility results readily yield"
    ],
    "note": {
      "typesetting": "Plain TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.2809K"
    }
  }
}