{
  "id": "2010.04358",
  "version": "v1",
  "published": "2020-10-09T03:56:48.000Z",
  "updated": "2020-10-09T03:56:48.000Z",
  "title": "Abelian Ideals and the Variety of Lagrangian Subalgebras",
  "authors": [
    "Sam Evens",
    "Yu Li"
  ],
  "categories": [
    "math.RT",
    "math.AG",
    "math.SG"
  ],
  "abstract": "For a semisimple algebraic group $G$ of adjoint type with Lie algebra $\\mathfrak g$ over the complex numbers, we establish a bijection between the set of closed orbits of the group $G \\ltimes \\mathfrak g^{\\ast}$ acting on the variety of Lagrangian subalgebras of $\\mathfrak g \\ltimes \\mathfrak g^{\\ast}$ and the set of abelian ideals of a fixed Borel subalgebra of $\\mathfrak g$. In particular, the number of such orbits equals $2^{\\text{rk} \\mathfrak g}$ by Peterson's theorem on abelian ideals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-09T03:56:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "abelian ideals",
      "lagrangian subalgebras",
      "semisimple algebraic group",
      "adjoint type",
      "petersons theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}