{
  "id": "math/0610897",
  "version": "v1",
  "published": "2006-10-29T15:30:23.000Z",
  "updated": "2006-10-29T15:30:23.000Z",
  "title": "On Whittaker modules over a class of algebras similar to $U(sl_{2})$",
  "authors": [
    "Xin Tang"
  ],
  "comment": "11 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Motivated by the study of invariant rings of finite groups on the first Weyl algebras $A_{1}$ (\\cite{AHV}) and finding interesting families of new noetherian rings, a class of algebras similar to $U(sl_{2})$ were introduced and studied by Smith in \\cite{S}. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained in \\cite{S} and \\cite{Ku}. But it seems that not much has been done on the part of nonweight modules. In this note, we generalize Kostant's results in \\cite{K} on the Whittaker model for the universal enveloping algebras $U(\\frak g)$ of finite dimensional semisimple Lie algebras $\\frak g$ to Smith's algebras. As a result, a complete classification of irreducible Whittaker modules (which are definitely infinite dimensional) for Smith's algebras is obtained, and the submodule structure of any Whittaker module is also explicitly described.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-10-29T15:30:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B55",
      "17B50"
    ],
    "keywords": [
      "whittaker module",
      "algebras similar",
      "finite dimensional semisimple lie algebras",
      "smiths algebras",
      "first weyl algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10897T"
    }
  }
}