{
  "id": "2201.10605",
  "version": "v1",
  "published": "2022-01-25T20:02:00.000Z",
  "updated": "2022-01-25T20:02:00.000Z",
  "title": "Tensor products and intertwining operators for uniserial representations of the Lie algebra $\\mathfrak{sl}(2)\\ltimes V(m)$",
  "authors": [
    "Leandro Cagliero",
    "Iván Gómez Rivera"
  ],
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Let $\\mathfrak{g}_m=\\mathfrak{sl}(2)\\ltimes V(m)$, $m\\ge 1$, where $V(m)$ is the irreducible $\\mathfrak{sl}(2)$-module of dimension $m+1$ viewed as an abelian Lie algebra. It is known that the isomorphism classes of uniserial $\\mathfrak{g}_m$-modules consist of a family, say of type $Z$, containing modules of arbitrary composition length, and some exceptional modules with composition length $\\le 4$. Let $V$ and $W$ be two uniserial $\\mathfrak{g}_m$-modules of type $Z$. In this paper we obtain the $\\mathfrak{sl}(2)$-module decomposition of $\\text{soc}(V\\otimes W)$ by giving explicitly the highest weight vectors. It turns out that $\\text{soc}(V\\otimes W)$ is multiplicity free. Roughly speaking, $\\text{soc}(V\\otimes W)=\\text{soc}(V)\\otimes \\text{soc}(W)$ in half of the cases, and in these cases we obtain the full socle series of $V\\otimes W$ by proving that $ \\text{soc}^{t+1}(V\\otimes W)=\\sum_{i=0}^{t} \\text{soc}^{i+1}(V)\\otimes \\text{soc}^{t+1-i}(W)$ for all $t\\ge0$. As applications of these results, we obtain for which $V$ and $W$, the space of $\\mathfrak{g}_m$-module homomorphisms $\\text{Hom}_{\\mathfrak{g}_m}(V,W)$ is not zero, in which case is 1-dimensional. Finally we prove, for $m\\ne 2$, that if $U$ is the tensor product of two uniserial $\\mathfrak{g}_m$-modules of type $Z$, then the factors are determined by $U$. We provide a procedure to identify the factors from $U$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-25T20:02:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B30",
      "22E27",
      "16G10"
    ],
    "keywords": [
      "tensor product",
      "uniserial representations",
      "intertwining operators",
      "arbitrary composition length",
      "highest weight vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}