{
  "id": "2010.06996",
  "version": "v1",
  "published": "2020-10-14T12:17:58.000Z",
  "updated": "2020-10-14T12:17:58.000Z",
  "title": "Representations of shifted quantum affine algebras",
  "authors": [
    "David Hernandez"
  ],
  "comment": "63 pages",
  "categories": [
    "math.RT",
    "hep-th",
    "math.QA"
  ],
  "abstract": "We develop the representation theory of shifted quantum affine algebras $\\mathcal{U}_q^\\mu(\\hat{\\mathfrak{g}})$ and of their truncations which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge theories. Our direct approach is based on relations that we establish with the category $\\mathcal{O}$ of representations of the quantum affine Borel algebra $\\mathcal{U}_q(\\hat{\\mathfrak{b}})$ and on associated quantum integrable models we have previously studied. We introduce the category $\\mathcal{O}^\\mu$ of representations of $\\mathcal{U}_q^\\mu(\\hat{\\mathfrak{g}})$ and we classify its simple objects. For $\\mathfrak{g} = sl_2$ we prove the existence of evaluation morphisms to $q$-oscillator algebras. We establish the existence of a fusion product and we get a ring structure on the sum of the Grothendieck groups $K_0(\\mathcal{O}^\\mu)$. We introduce induction and restriction functors to the category $\\mathcal{O}$ of $\\mathcal{U}_q(\\mathfrak{b})$. As a by product we classify simple finite-dimensional representations of $\\mathcal{U}_q^\\mu(\\hat{\\mathfrak{g}})$ and we obtain a cluster algebra structure on the Grothendieck ring of finite-dimensional representations. We establish a necessary condition for a simple representation to descend to a truncation, which is also sufficient for $\\mathfrak{g} = sl_2$. We introduce a related partial ordering on simple modules and we prove a truncation has only a finite number of simple representations. We state a conjecture on the parametrization of simple modules of a non simply-laced truncation in terms of the Langlands dual Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations proved by using the Baxter polynomiality of quantum integrable models.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-14T12:17:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "shifted quantum affine algebras",
      "simple finite-dimensional representations",
      "quantum integrable models",
      "susy quiver gauge theories",
      "langlands dual lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 63,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}