{
  "id": "2011.14253",
  "version": "v1",
  "published": "2020-11-29T01:55:58.000Z",
  "updated": "2020-11-29T01:55:58.000Z",
  "title": "PBW theory for quantum affine algebras",
  "authors": [
    "Masaki Kashiwara",
    "Myungho Kim",
    "Se-jin Oh",
    "Euiyong Park"
  ],
  "comment": "63 pages. This is a full paper of the announcement: PBW theoretic approach to the module category of quantum affine algebras, arXiv:2005.04838v2",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "Let $U_q'(\\mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $\\mathcal{C}_{\\mathfrak{g}}$ be Hernandez-Leclerc's category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $\\mathcal{C}_{\\mathfrak{g}}$. We introduce the notion of a strong (complete) duality datum $D$ and prove that, when $D$ is strong, the induced duality functor $F_D$ sends simple modules to simple modules and preserves the invariants $\\Lambda$ and $\\Lambda^\\infty$ introduced by the authors. We next define the reflections $\\mathcal{S}_k$ and $\\mathcal{S}^{-1}_k$ acting on strong duality data $D$. We prove that if $D$ is a strong (resp.\\ complete) duality datum, then $\\mathcal{S}_k(D)$ and $\\mathcal{S}_k^{-1}(D)$ are also strong (resp.\\ complete ) duality data. We finally introduce the notion of affine cuspidal modules in $\\mathcal{C}_{\\mathfrak{g}}$ by using the duality functor $F_D$, and develop the cuspidal module theory for quantum affine algebras similarly to the quiver Hecke algebra case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-29T01:55:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B37",
      "81R50",
      "18D10"
    ],
    "keywords": [
      "quantum affine algebra",
      "duality datum",
      "pbw theory",
      "quantum affine schur-weyl duality functor",
      "cuspidal module"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 63,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}