{
  "id": "2402.16676",
  "version": "v2",
  "published": "2024-02-26T15:53:16.000Z",
  "updated": "2024-09-04T00:16:01.000Z",
  "title": "Tensor K-matrices for quantum symmetric pairs",
  "authors": [
    "Andrea Appel",
    "Bart Vlaar"
  ],
  "comment": "52 pages. Minor revision: Appendix A added; bibliography updated and expanded; minor changes to the introduction",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, $U_q(\\mathfrak{g})$ its quantum group, and $U_q(\\mathfrak{k}) \\subset U_q(\\mathfrak{g})$ a quantum symmetric pair subalgebra determined by a Lie algebra automorphism $\\theta$. We introduce a category $W_\\theta$ of weight $U_q(\\mathfrak{k})$-modules, which is acted upon by the category of weight $U_q(\\mathfrak{g})$-modules via tensor products. We construct a universal tensor K-matrix $\\mathbb{K}$ (that is, a solution of a reflection equation) in a completion of $U_q(\\mathfrak{k}) \\otimes U_q(\\mathfrak{g})$. This yields a natural operator on any tensor product $M \\otimes V$, where $M\\in W_\\theta$ and $V\\in \\mathcal{O}_\\theta$, that is $V$ is a $U_q(\\mathfrak{g})$-module in category $\\mathcal{O}$ satisfying an integrability property determined by $\\theta$. $W_\\theta$ is equipped with a structure of a bimodule category over $\\mathcal{O}_\\theta$ and the action of $\\mathbb{K}$ is encoded by a new categorical structure, which we call a boundary structure on $W_\\theta$. This generalizes a result of Kolb which describes a braided module structure on finite-dimensional $U_q(\\mathfrak{k})$-modules when $\\mathfrak{g}$ is finite-dimensional. We apply our construction to the case of an affine Lie algebra, where it yields a formal tensor K-matrix valued in the endomorphisms of the tensor product of any module in $W_\\theta$ and any finite-dimensional module over the corresponding quantum affine algebra. This formal series is normalized to a trigonometric K-matrix if the factors in the tensor product are both finite-dimensional irreducible modules over the quantum affine algebra.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-09-04T00:16:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81R50",
      "16T25",
      "17B37",
      "81R12"
    ],
    "keywords": [
      "tensor product",
      "quantum affine algebra",
      "quantum symmetric pair subalgebra",
      "finite-dimensional",
      "universal tensor k-matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}