{
  "id": "2301.11383",
  "version": "v1",
  "published": "2023-01-26T19:53:39.000Z",
  "updated": "2023-01-26T19:53:39.000Z",
  "title": "Tensor products and intertwining operators between two uniserial representations of the Galilean Lie algebra $\\mathfrak{sl}(2)\\ltimes \\mathfrak{h}_n$",
  "authors": [
    "Leandro Cagliero",
    "Iván Gómez Rivera"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2201.10605",
  "categories": [
    "math.RT",
    "math.KT",
    "math.RA"
  ],
  "abstract": "Let $\\mathfrak{sl}(2)\\ltimes \\mathfrak{h}_n$, $n\\ge 1$, be the Galilean Lie algebra over a field of characteristic zero, where $\\mathfrak{h}_{n}$ is the Heisenberg Lie algebra of dimension $2n+1$, and $\\mathfrak{sl}(2)$ acts on $\\mathfrak{h}_{n}$ so that $\\mathfrak{h}_n\\simeq V(2n-1)\\oplus V(0)$ as $\\mathfrak{sl}(2)$-modules (here $V(k)$ denotes the irreducible $\\mathfrak{sl}(2)$-module of highest weight $k$). The isomorphism classes of uniserial $\\big(\\mathfrak{sl}(2)\\ltimes \\mathfrak{h}_n\\big)$-modules are known. In this paper we study the tensor product of two uniserial representations of $\\mathfrak{sl}(2)\\ltimes \\mathfrak{h}_n$. Among other things, we obtain the $\\mathfrak{sl}(2)$-module structure of the socle of $V\\otimes W$ and we describe the space of intertwining operators $\\text{Hom}_{\\mathfrak{sl}(2)\\ltimes \\mathfrak{h}_n}(V,W)$, where $V$ and $W$ are uniserial representations of $\\mathfrak{sl}(2)\\ltimes \\mathfrak{h}_n$. This article extends a previous work in which we obtained analogous results for the Lie algebra $\\mathfrak{sl}(2)\\ltimes \\mathfrak{a}_m$ where $\\mathfrak{a}_m$ is the abelian Lie algebra and $\\mathfrak{sl}(2)$ acts so that $\\mathfrak{a}_m\\simeq V(m-1)$ as $\\mathfrak{sl}(2)$-modules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-26T19:53:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "18M20",
      "22E27"
    ],
    "keywords": [
      "galilean lie algebra",
      "uniserial representations",
      "tensor product",
      "intertwining operators",
      "abelian lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}