{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1329",
  "content_hash": null,
  "slug": "kaplansky-sixth-semisimple-hopf-status-20260801",
  "type": "claim",
  "title": "Current checked status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-27. The MathOverflow answers prove special cases and cite the conjecture as open. The checked survey and geometric-invariant-theory work do not supply a proof for all finite-dimensional semisimple Hopf algebras or a counterexample.",
  "relevance": "Records the strongest checked neighboring results and the exact remainder future work must settle.",
  "relevance_source": "recorded",
  "body": "A dated independent review on 2026-08-01 checked the structured sources below, the complete visible source discussion, exact-title and equivalent-formulation searches, and the current TheoremDB corpus. On 2026-07-27 all three MathOverflow answers and their comments were checked. They cover particular dimensions and structural classes without resolving the universal statement. Natale's survey, arXiv:1409.2545, lists Kaplansky's sixth conjecture among the central divisibility questions for semisimple Hopf algebras and records known special cases. Meir, arXiv:1506.00314, studies Kaplansky's conjectures through geometric invariant theory. The reformulation and degeneration results do not settle the displayed divisibility assertion. Group algebras satisfy the assertion by the classical degree-divisibility theorem for finite-group representations, and dual group algebras provide another elementary class. A search confined to these families cannot find a counterexample. Trap: the weaker statement that \\(\\dim V\\le\\dim H\\), or divisibility for Frobenius-Perron dimensions in a restricted tensor category, does not settle the integer divisibility required here.\n\nA complete resolution must satisfy: Prove that \\(\\dim_k V\\mid\\dim_k H\\) for every triple \\((k,H,V)\\) satisfying the displayed hypotheses. Alternatively, give explicit Hopf structure maps for a finite-dimensional semisimple counterexample, identify a simple module \\(V\\), prove simplicity, and verify the failed divisibility.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/108404/kaplanskys-6th-conjecture-dimirrep-dimalgebra-for-semi-simple-hopf-alg",
      "locator": "Dataset references and independent 2026-08-01 status search."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/108404/kaplanskys-6th-conjecture-dimirrep-dimalgebra-for-semi-simple-hopf-alg",
    "locator": "Dataset references and independent 2026-08-01 status search."
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R1328",
      "title": "Complete the stated acceptance conditions",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "R1617",
      "title": "Dated status and exact unresolved remainder",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "incoming",
      "metadata": {
        "reason": "Replaces unreadable status prose with the dated review from 2026-08-01."
      }
    },
    {
      "slug": "kaplansky-sixth-semisimple-hopf",
      "title": "kaplansky sixth semisimple hopf",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}