{
  "id": "1708.06323",
  "version": "v1",
  "published": "2017-08-21T16:54:50.000Z",
  "updated": "2017-08-21T16:54:50.000Z",
  "title": "Quantum groups, Yang-Baxter maps and quasi-determinants",
  "authors": [
    "Zengo Tsuboi"
  ],
  "comment": "45 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "For any quasi-triangular Hopf algebra, there exists the universal R-matrix, which satisfies the Yang-Baxter equation. It is known that the adjoint action of the universal R-matrix on the elements of the tensor square of the algebra constitutes a quantum Yang-Baxter map, which satisfies the set-theoretic Yang-Baxter equation. The map has a zero curvature representation among L-operators defined as images of the universal R-matrix. We find that the zero curvature representation can be solved by the Gauss decomposition of a product of L-operators. Thereby obtained a quasi-determinant expression of the quantum Yang-Baxter map associated with the quantum algebra $U_{q}(gl(n))$. Moreover, the map is identified with products of quasi-Pl\\\"{u}cker coordinates over a matrix composed of the L-operators. We also consider the quasi-classical limit, where the underlying quantum algebra reduces to a Poisson algebra. The quasi-determinant expression of the quantum Yang-Baxter map reduces to ratios of determinants, which give a new expression of a classical Yang-Baxter map.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-21T16:54:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum groups",
      "universal r-matrix",
      "zero curvature representation",
      "quasi-determinant expression",
      "quantum yang-baxter map reduces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}