{
  "id": "1305.4059",
  "version": "v2",
  "published": "2013-05-17T12:25:54.000Z",
  "updated": "2015-12-22T16:24:28.000Z",
  "title": "Irreducible representations of untwisted affine Kac-Moody algebras",
  "authors": [
    "Xiangqian Guo",
    "Kaiming Zhao"
  ],
  "comment": "36 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "In this paper we construct a class of new irreducible modules over untwisted affine Kac-Moody algebras $\\widetilde{\\mathfrak{g}}$, generalizing and including both highest weight modules and Whittaker modules. These modules allow us to obtain a complete classification of irreducible $\\widetilde{\\mathfrak{g}}$-modules on which the action of each root vector in $\\widetilde{\\mathfrak{n}}_+$ is locally finite, where $\\widetilde{\\mathfrak{n}}_+$ is the locally nilpotent subalgebra (or positive part) of $\\widetilde{\\mathfrak{g}}$. The necessary and sufficient conditions for two such irreducible $\\widetilde{\\mathfrak{g}}$-modules to be isomorphic are also determined. In the second part of the paper, we use the \"shifting technique\" to obtain a necessary and sufficient condition for the tensor product of irreducible integrable loop $\\widetilde{\\mathfrak{g}}$-modules and irreducible integrable highest weight $\\widetilde{\\mathfrak{g}}$-modules to be simple. This tensor product problem was originally studied by Chari and Pressley 28 years ago.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-17T12:25:54.000Z",
      "abstract": "In this paper we construct a class of new irreducible modules over the untwisted affine Kac-Moody algebras $\\tg$, generalizing and including both highest weight modules and Whittaker modules. The necessary and sufficient conditions for two such irreducible $\\tg$-modules to be isomorphic are also determined. These modules allow us to obtain a complete classification of irreducible $\\tg$-modules on which the action of each weight vector in $\\tn_+$ is locally finite, where $\\tn_+$ is the locally nilpotent subalgebra(or positive part) of $\\tg$.",
      "comment": "23 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-12-22T16:24:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B05",
      "17B30",
      "17B65",
      "17B67"
    ],
    "keywords": [
      "untwisted affine kac-moody algebras",
      "irreducible representations",
      "highest weight modules",
      "weight vector",
      "complete classification"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.4059G"
    }
  }
}