{
  "abstract": "We construct spectral-parameter solutions of the quantum Yang--Baxter equation from the associativity condition of the universal symmetric $2$-algebraic two-valued group, whose multiplication law depends on four coefficients $k_2,k_4,k_6,k_8$ subject to the single relation $4k_8-k_4^2+k_2k_6=0$ -- which coincides with the three-dimensional WDVV equation for Dubrovin's normalised potential. Every three-dimensional Frobenius manifold with a cyclic vector field carries such a spectral two-valued group, whose finite branch points are the eigenvalues of the multiplication operator. For an arbitrary unital algebra we show that a two-parameter spectral $R$-operator built from multiplication and unit has Yang--Baxter defect proportional to an explicit tensor built from the associator; for tangent Frobenius algebras this yields a regular, unitary $9\\times9$ $R$-matrix satisfying the quantum Yang--Baxter equation exactly at WDVV points, with spectral parameter the Abel coordinate of a bielliptic curve whose elliptic quotient is the spectral curve, together with an associated integrable chain. The $A_3$ Frobenius manifold is worked out explicitly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02354",
  "author_total": 1,
  "authors": [
    {
      "name": "Victor M. Buchstaber"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d2b5nnx7vv29x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02354",
  "fulltext_url": null,
  "id": "d2b5nnx7vv29x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02354",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "nlin.SI"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.02354",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d2b5nnx7vv29x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:45Z",
      "native_id": "2610.02354",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02354"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "A solution to the quantum Yang--Baxter equation associated with a universal two-valued algebraic group",
  "title_text": "A solution to the quantum Yang--Baxter equation associated with a universal two-valued algebraic group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/d2b5nnx7vv29x",
  "venue": "arXiv",
  "versions": []
}