{
  "schema_version": 1,
  "problem_number": "OWR-17294-017",
  "title": "The Quantum Symmetriser of a Yang–Baxter Solution Need Not Be a Quasi-Isomorphism",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a set-theoretic solution (X, σ) of the Yang–Baxter equation, Farinati and García Galofre showed that a comparison map from braided (co)homology to the Hochschild (co)homology of the structure algebra A = kM(X, σ) can be chosen to be the quantum symmetriser of −σ, and that it factors through a complex A ⊗ B ⊗ A, where B is the Nichols algebra of −σ. They asked whether this complex is a resolution of A, and Lebed asked in a 2019 Oberwolfach report whether the quantum symmetriser is bijective for general solutions; the answer is known to be positive for involutive (characteristic zero) and idempotent solutions. We show that it is negative in general. For the permutation rack X = Z/4, x ◁ y = x + 1, a bijective non-degenerate solution, we give an explicit 2-cycle of weight 3 in A ⊗ B ⊗ A that is not a boundary; it comes from a cubic quantum Serre relation in B. With coefficients in the module on which X acts by 0, the image of the quantum symmetriser misses part of HH₃(A; k) for every quotient of the braided complex. We show that A ⊗ B ⊗ A is a resolution if and only if A is Koszul and B is the Koszul dual coalgebra of A, and that in characteristic zero every finite permutation rack whose permutation has a cycle of length divisible by 4 (resp. 3) fails in weight 3 (resp. 4). For finite non-degenerate solutions that are not involutive and have a finite-dimensional Nichols algebra, such as the dihedral quandle of order 3, a negative answer already follows from a remark of Farinati and García Galofre and a theorem of Jespers, Kubat and Van Antwerpen. By exact computation, 13 of the 29 isomorphism classes of bijective solutions on three points fail. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.QA",
    "math.RA",
    "math.KT"
  ],
  "keywords": [
    "Yang–Baxter equation",
    "set-theoretic solution",
    "quantum symmetriser",
    "Nichols algebra",
    "structure monoid",
    "Hochschild homology",
    "braided homology",
    "Koszul algebra",
    "permutation rack",
    "dihedral quandle",
    "counterexample",
    "Oberwolfach Reports",
    "OWR-17294-017",
    "math.QA",
    "math.RA",
    "math.KT",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-01",
  "publication_date": "2026-10-01",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-01",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-17294-017/",
  "pdf_url": "https://eulersolve.org/papers/owr-17294-017/paper.pdf?v=07c5f22ae676",
  "doi": "10.5281/zenodo.23072076",
  "zenodo_record_url": "https://zenodo.org/records/23072076",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Lebed's question (OWR 51/2019, p. 3237) and Farinati–García Galofre's Question 44 negatively, read as asking whether the quantum symmetriser becomes a quasi-isomorphism on a suitable quotient: explicit counterexample for the permutation rack Z/4 over any field of characteristic other than 2, a Koszul criterion, and a family of permutation racks in characteristic zero. The finite-dimensional case (e.g. the dihedral quandle of order 3) already follows from published results. Which solutions satisfy the property remains open.",
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    "source.zip": {
      "sha256": "c4c12e93e508b141ab763037bab6c6049b28f1058b21a460c5fac6bd9576c378"
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    "verification_report.md": {
      "sha256": "468a1b55b5639b41818f0cfa632a080a5b1bcc59bf9de6708ced063d74202d24"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}