{
  "id": "2203.16503",
  "version": "v1",
  "published": "2022-03-30T17:40:14.000Z",
  "updated": "2022-03-30T17:40:14.000Z",
  "title": "Rational K-matrices for finite-dimensional representations of quantum affine algebras",
  "authors": [
    "Andrea Appel",
    "Bart Vlaar"
  ],
  "comment": "37 pages",
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP",
    "math.QA"
  ],
  "abstract": "Let $\\mathfrak{g}$ be a complex simple Lie algebra. We prove that every finite-dimensional representation of the (untwisted) quantum affine algebra $U_qL\\mathfrak{g}$ gives rise to a family of spectral K-matrices, namely solutions of Cherednik's generalized reflection equation, which depends upon the choice of a quantum affine symmetric pair $U_q\\mathfrak{k}\\subset U_qL\\mathfrak{g}$. Moreover, we prove that every irreducible representation over $U_qL\\mathfrak{g}$ remains generically irreducible under restriction to $U_q\\mathfrak{k}$. From the latter result, we deduce that every obtained K-matrix can be normalized to a matrix-valued rational function in a multiplicative parameter, known in the study of quantum integrability as a trigonometric K-matrix. Finally, we show that our construction recovers many of the known solutions of the standard reflection equation and gives rise to a large class of new solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-30T17:40:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81R50",
      "16T25",
      "17B37",
      "81R12"
    ],
    "keywords": [
      "quantum affine algebra",
      "finite-dimensional representation",
      "rational k-matrices",
      "complex simple lie algebra",
      "quantum affine symmetric pair"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}