{
  "id": "1205.5983",
  "version": "v2",
  "published": "2012-05-27T16:12:16.000Z",
  "updated": "2013-05-05T10:40:32.000Z",
  "title": "Abelian ideals of a Borel subalgebra and root systems",
  "authors": [
    "Dmitri I. Panyushev"
  ],
  "comment": "17 pages, final version; to appear in J. Europ. Math. Soc",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "Let $g$ be a simple Lie algebra and $Ab$ the poset of non-trivial abelian ideals of a fixed Borel subalgebra of $g$. In 2003 (IMRN, no.35, 1889--1913), we constructed a partition of $Ab$ into the subposets $Ab_\\mu$, parameterised by the long positive roots of $g$, and established some properties of these subposets. In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation of abelian ideals and to the centralisers of abelian ideals. We also prove that the poset of positive roots of $g$ is a join-semilattice.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-05-05T10:40:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B20",
      "17B22",
      "20F55"
    ],
    "keywords": [
      "root systems",
      "non-trivial abelian ideals",
      "simple lie algebra",
      "fixed borel subalgebra",
      "long positive roots"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.5983P"
    }
  }
}