{
  "id": "2410.22819",
  "version": "v1",
  "published": "2024-10-30T08:55:57.000Z",
  "updated": "2024-10-30T08:55:57.000Z",
  "title": "Whittaker modules of central extensions of Takiff superalgebras and finite supersymmetric $W$-algebras",
  "authors": [
    "Chih-Whi Chen",
    "Shun-Jen Cheng",
    "Uhi Rinn Suh"
  ],
  "comment": "27 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "For a basic classical Lie superalgebra $\\mathfrak s$, let $\\mathfrak g$ be the central extension of the Takiff superalgebra $\\mathfrak s\\otimes\\Lambda(\\theta)$, where $\\theta$ is an odd indeterminate. We study the category of $\\mathfrak g$-Whittaker modules associated with a nilcharacter $\\chi$ of $\\mathfrak g$ and show that it is equivalent to the category of $\\mathfrak s$-Whittaker modules associated with a nilcharacter of $\\mathfrak s$ determined by $\\chi$. In the case when $\\chi$ is regular, we obtain, as an application, an equivalence between the categories of modules over the supersymmetric finite $W$-algebras associated to the odd principal nilpotent element at non-critical levels and the category of the modules over the principal finite $W$-superalgebra associated to $\\mathfrak s$. Here, a supersymmetric finite $W$-algebra is conjecturally the Zhu algebra of a supersymmetric affine $W$-algebra. This allows us to classify and construct irreducible representations of a principal finite supersymmetric $W$-algebra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-30T08:55:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B20"
    ],
    "keywords": [
      "whittaker modules",
      "takiff superalgebra",
      "central extension",
      "odd principal nilpotent element",
      "supersymmetric finite"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}