{
  "id": "2404.07372",
  "version": "v1",
  "published": "2024-04-10T22:12:19.000Z",
  "updated": "2024-04-10T22:12:19.000Z",
  "title": "Cyclic wide subalgebras of semisimple Lie algebras",
  "authors": [
    "Andrew Douglas",
    "Joe Repka"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathfrak{s}$ $\\ltimes$ $\\mathfrak{r}$ be a Levi decomposable Lie algebra, with Levi factor $\\mathfrak{s}$, and radical $\\mathfrak{r}$. A module $V$ of $\\mathfrak{s}$ $\\ltimes$ $\\mathfrak{r}$ is cyclic indecomposable if it is indecomposable and the quotient module $V /\\mathfrak{r}\\cdot V$ is a simple $\\mathfrak{s}$-module. A Levi decomposable subalgebra of a semisimple Lie algebra is cyclic wide if the restriction of every simple module of the semisimple Lie algebra to the subalgebra is cyclic indecomposable. We establish a condition for a regular Levi decomposable subalgebra of a semisimple Lie algebra to be cyclic wide. Then, in the case of a regular Levi decomposable subalgebra whose radical is an ad-nilpotent subalgebra, we show that the condition is necessary and sufficient for the subalgebra to be cyclic wide. All Lie algebras, and modules in this article are finite-dimensional, and over the complex numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-10T22:12:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B05",
      "17B10",
      "17B20",
      "17B22",
      "17B30"
    ],
    "keywords": [
      "semisimple lie algebra",
      "cyclic wide subalgebras",
      "regular levi decomposable subalgebra",
      "levi decomposable lie algebra",
      "complex numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}