{
  "id": "1403.2481",
  "version": "v1",
  "published": "2014-03-11T06:47:50.000Z",
  "updated": "2014-03-11T06:47:50.000Z",
  "title": "Three results on representations of Mackey Lie algebras",
  "authors": [
    "Alexandru Chirvasitu"
  ],
  "comment": "9 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "I. Penkov and V. Serganova have recently introduced, for any non-degenerate pairing $W\\otimes V\\to\\mathbb C$ of vector spaces, the Lie algebra $\\mathfrak{gl}^M=\\mathfrak{gl}^M(V,W)$ consisting of endomorphisms of $V$ whose duals preserve $W\\subseteq V^*$. In their work, the category $\\mathbb{T}_{\\mathfrak{gl}^M}$ of $\\mathfrak{gl}^M$-modules which are finite length subquotients of the tensor algebra $T(W\\otimes V)$ is singled out and studied. In this note we solve three problems posed by these authors concerning the categories $\\mathbb{T}_{\\mathfrak{gl}^M}$. Denoting by $\\mathbb{T}_{V\\otimes W}$ the category with the same objects as $\\mathbb{T}_{\\mathfrak{gl}^M}$ but regarded as $V\\otimes W$-modules, we first show that when $W$ and $V$ are paired by dual bases, the functor $\\mathbb{T}_{\\mathfrak{gl}^M}\\to \\mathbb{T}_{V\\otimes W}$ taking a module to its largest weight submodule with respect to a sufficiently nice Cartan subalgebra of $V\\otimes W$ is a tensor equivalence. Secondly, we prove that when $W$ and $V$ are countable-dimensional, the objects of $\\mathbb{T}_{\\mathrm{End}(V)}$ have finite length as $\\mathfrak{gl}^M$-modules. Finally, under the same hypotheses, we compute the socle filtration of a simple object in $\\mathbb{T}_{\\mathrm{End}(V)}$ as a $\\mathfrak{gl}^M$-module.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-11T06:47:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B20",
      "17B65"
    ],
    "keywords": [
      "mackey lie algebras",
      "representations",
      "finite length subquotients",
      "sufficiently nice cartan subalgebra",
      "largest weight submodule"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1285895,
      "adsabs": "2014arXiv1403.2481C"
    }
  }
}