{
  "annotation_id": "897b9d64-cd9a-4eb3-8eab-568ba8e41a56",
  "date_created": "2026-02-17T05:53:08.863000Z",
  "date_modified": "2026-02-17T05:53:08.863000Z",
  "file_hash": "d6dc3f1ad0f23de34c5e8af31668a42d68443598731eface452482561a3c0902",
  "private": false,
  "record": {
    "abstract": "We present a description of the classical elliptic ${\\rm BC}_1$ Ruijsenaars-van Diejen model with 8 independent coupling constants through a pair of ${\\rm BC}_1$ type classical Sklyanin algebras generated by the (classical) quadratic reflection equation with non-dynamical XYZ $r$-matrix. For this purpose, we consider the classical version of the $L$-operator for the Ruijsenaars-van Diejen model proposed by O. Chalykh. In ${\\rm BC}_1$ case it is factorized to the product of two Lax matrices depending on 4 constants. Then we apply an IRF-Vertex type gauge transformation and obtain a product of the Lax matrices for the Zhukovsky-Volterra gyrostats. Each of them is described by the ${\\rm BC}_1$ version of the classical Sklyanin algebra. In particular case, when 4 pairs of constants coincide, the ${\\rm BC}_1$ Ruijsenaars-van Diejen model exactly coincides with the relativistic Zhukovsky-Volterra gyrostat. Explicit change of variables is obtained. We also consider another special case of the ${\\rm BC}_1$ Ruijsenaars-van Diejen model with 7 independent constants. We show that it can be reproduced by considering the transfer matrix of the classical 1-site XYZ chain with boundaries. In the end of the paper, using another gauge transformation we represent the Chalykh\u0027s Lax matrix in a form depending on the Sklyanin\u0027s generators.",
    "arxiv_id": "2601.06826",
    "authors": [
      "A. Mostovskii",
      "A. Zotov"
    ],
    "categories": [
      "math-ph",
      "hep-th",
      "math.MP",
      "nlin.SI"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "Classical elliptic ${\\rm BC}_1$ Ruijsenaars-van Diejen model: relation to Zhukovsky-Volterra gyrostat and 1-site classical XYZ model with boundaries",
    "url": "https://arxiv.org/abs/2601.06826",
    "version": "v2"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "5cf00df0-a3c0-48da-901c-cdc74e018360",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}