{
  "id": "2312.06325",
  "version": "v1",
  "published": "2023-12-11T12:15:51.000Z",
  "updated": "2023-12-11T12:15:51.000Z",
  "title": "Representations of Toroidal and Full toroidal Lie algebras over polynomial algebras",
  "authors": [
    "Santanu Tantubay",
    "Priyanshu Chakraborty"
  ],
  "comment": "12 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Toroidal Lie algebras are $n$ variable generalizations of affine Kac-Moody Lie algebras. Full toroidal Lie algebra is the semidirect product of derived Lie algebra of toroidal Lie algebra and Witt algebra, also it can be thought of $n$-variable generalization of Affine-Virasoro algebras. Let $\\tilde{\\mathfrak{h}}$ be a Cartan subalgebra of a toroidal Lie algebra as well as full toroidal Lie algebra without containing the zero-degree central elements. In this paper, we classify the module structure on $U(\\tilde{\\mathfrak{h}})$ for all toroidal Lie algebras as well as full toroidal Lie algebras which are free $U(\\tilde{\\mathfrak{h}})$-modules of rank 1. These modules exist only for type $A_l (l\\geq 1)$, $C_l (l\\geq2)$ toroidal Lie algebras and the same is true for full toroidal Lie algebras. Also, we determined the irreducibility condition for these classes of modules for both the Lie algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-11T12:15:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B65",
      "17B66",
      "17B68"
    ],
    "keywords": [
      "full toroidal lie algebra",
      "polynomial algebras",
      "affine kac-moody lie algebras",
      "representations",
      "zero-degree central elements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}