{
  "id": "2212.03775",
  "version": "v1",
  "published": "2022-12-07T16:47:11.000Z",
  "updated": "2022-12-07T16:47:11.000Z",
  "title": "Semisimple elements and the little Weyl group of real semisimple $Z_m$-graded Lie algebras",
  "authors": [
    "Willem de Graaf",
    "Hông Vân Lê"
  ],
  "comment": "Preliminary version, 18 p. Comments welcome! arXiv admin note: substantial text overlap with arXiv:2106.00246",
  "categories": [
    "math.RT",
    "math.DG",
    "math.GR",
    "math.RA"
  ],
  "abstract": "In this paper we introduce the notion of the algebraic $\\theta$-group of a semisimple $Z_m$-graded Lie algebra over a field $k$ of characteristic $0$ whose identity component is the $\\theta$-group. Using this, we extend a number of results concerning structure of semisimple elements in complex semisimple $Z_m$-graded Lie algebras and some results concerning real (and complex) semisimple Lie algebras to the case of real (and complex) semisimple $Z_m$-graded Lie algebras, using Galois cohomology theory and ad hoc techniques. In particular, we parameterize the conjugacy classes of Cartan subspaces in a real semisimple $Z_m$-graded Lie algebra $g$ in terms of Galois cohomology. We prove that if $m$ is a prime number or $g^c$ is a complex semisimple $Z_m$-graded Lie algebra of maximal rank then the centralizer $Z_{g^c}(p)$ of a homogeneous semisimple element $p$ and the stabilizer $W_p$ of $p$ under the Weyl group action define each other. As a result, we obtain a number of consequences on the conjugacy classes of homogeneous semisimple elements in $g^c$ and in its real forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-07T16:47:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E72",
      "20G05",
      "20G20"
    ],
    "keywords": [
      "graded lie algebra",
      "little weyl group",
      "real semisimple",
      "homogeneous semisimple element",
      "complex semisimple"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}