{
  "schema_version": 1,
  "problem_number": "OWR-14299906-003",
  "title": "On a Question of Freslon, Gerontogiannis and Skalski about Quantum Isometries of Cuntz–Krieger Algebras",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a primitive 0–1 matrix A, Freslon, Gerontogiannis and Skalski computed the quantum isometry group G_A^∞ of the log-Laplacian spectral triple on the Cuntz–Krieger algebra O_A. They showed that the quantum automorphism group QAut(A) is a quantum subgroup of a larger quantum group G_A^1 and asked if it is one of G_A^∞ when QAut(A) ≠ Aut(A). We show that the answer is negative in general. For the Cuntz algebras O_N and for A = J_N − I_N, with N ≥ 4, the quantum permutation group S_N^+ = QAut(A) is not a quantum subgroup of G_A^∞. For every primitive A with QAut(A) ≠ Aut(A), the canonical inclusion of QAut(A) in G_A^1 does not extend to G_A^∞, and the natural action of QAut(A) on O_A is not isometric for this spectral triple. Quantum groups such as S_M^+, M ≥ 4, cannot sit in G_A^∞ through an embedding that fixes a vertex. On the other hand, for two explicit primitive 4 × 4 matrices, the non-classical QAut(A) (the dual of the infinite dihedral group, and the hyperoctahedral quantum group H_2^+) is a quantum subgroup of G_A^∞. For general A it remains open whether S_M^+, M ≥ 4, can sit in G_A^∞ through an embedding that fixes no vertex. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.OA",
    "math.QA"
  ],
  "keywords": [
    "quantum isometry group",
    "Cuntz–Krieger algebra",
    "Cuntz algebra",
    "quantum automorphism group of a graph",
    "quantum permutation group",
    "compact quantum group",
    "spectral triple",
    "Oberwolfach Reports",
    "OWR-14299906-003",
    "math.OA",
    "math.QA",
    "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-14299906-003/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299906-003/paper.pdf?v=0d77afa40dda",
  "doi": "10.5281/zenodo.23071235",
  "zenodo_record_url": "https://zenodo.org/records/23071235",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The note answers the question negatively in general: for the Cuntz algebras and for J_N − I_N (N ≥ 4) the quantum permutation group is not a quantum subgroup of the quantum isometry group, and the canonical inclusion never extends. Positive examples exist for two explicit 4 × 4 matrices. A characterization of the positive cases, and embeddings that fix no vertex, remain open.",
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    "source.zip": {
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    "verification_report.md": {
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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."
}