{
  "id": "2404.11749",
  "version": "v1",
  "published": "2024-04-17T20:59:12.000Z",
  "updated": "2024-04-17T20:59:12.000Z",
  "title": "Weyl group twists and representations of quantum affine Borel algebras",
  "authors": [
    "Keyu Wang"
  ],
  "comment": "30 pages",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "We define categories $\\mathcal{O}^w$ of representations of Borel subalgebras $\\mathcal{U}_q\\mathfrak{b}$ of quantum affine algebras $\\mathcal{U}_q\\hat{\\mathfrak{g}}$, which come from the category $\\mathcal{O}$ twisted by Weyl group elements $w$. We construct inductive systems of finite-dimensional $\\mathcal{U}_q\\mathfrak{b}$-modules twisted by $w$, which provide representations in the category $\\mathcal{O}^w$. We also establish a classification of simple modules in these categories $\\mathcal{O}^w$. We explore convergent phenomenon of $q$-characters of representations of quantum affine algebras, which conjecturally give the $q$-characters of representations in $\\mathcal{O}^w$. Furthermore, we propose a conjecture concerning the relationship between the category $\\mathcal{O}$ and the twisted category $\\mathcal{O}^w$, and we propose a possible connection with shifted quantum affine algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-17T20:59:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum affine borel algebras",
      "weyl group twists",
      "representations",
      "shifted quantum affine algebras",
      "weyl group elements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}