[#P2862] Kaplansky's sixth conjecture for semisimple Hopf algebras

Problem. Let \(k\) be an algebraically closed field of characteristic zero, let \(H\) be a finite-dimensional semisimple Hopf algebra over \(k\), and let \(V\) be a finite-dimensional simple left \(H\)-module. Must \(\dim_k V\) divide \(\dim_k H\)?

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Definitions and notation

1Context

Dimension-by-dimension classifications and exclusions of fusion rules can be retained as bounded evidence. An explicit counterexample would have a finite algebraic certificate once the Hopf identities and semisimplicity are checked.

2Problem setup

Definition 1. A Hopf algebra is an algebra equipped with compatible comultiplication, counit, and antipode maps.

Definition 2. Semisimple means semisimple as an associative \(k\)-algebra.

Definition 3. A simple left module is a nonzero module with no proper nonzero submodules.

Remark 1. Divisibility is divisibility of the two positive integers \(\dim_k V\) and \(\dim_k H\).

3What counts as a solution

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.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: 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. Exact unresolved remainder: 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.^[3][2][1]

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Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: 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. Exact unresolved remainder: 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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: 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. Exact unresolved remainder: 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.^[3][2][1] Relevance to this problem For Kaplansky's sixth conjecture for semisimple Hopf algebras, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 3, Reference 2, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: 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. Exact unresolved remainder: 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. Open full record →
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.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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.^[1] Relevance to this problem For Kaplansky's sixth conjecture for semisimple Hopf algebras, record kaplansky-sixth-semisimple-hopf-next-route-20260801 (“Complete the stated acceptance conditions”) documents a concrete method, search boundary, or failed route. The record states: Prove that \(\dim_k V\mid\dim_k H\) for every triple \((k,H,V)\) satisfying the displayed hypotheses. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Work against the displayed statement and preserve every hypothesis and quantifier. 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. Any computation must retain a replayable witness and a matching exclusion or completeness certificate. Open full record →
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Notes and companion material

Original intake status. 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.

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.
Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.

Recorded example 1. For \(H=k[G]\) with \(G\) finite, irreducible representation degrees divide \(|G|=\dim H\), so the assertion holds in the group-algebra case.

Recorded example 2. For a one-dimensional simple module, the divisibility assertion is automatic.

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“Kaplansky's sixth conjecture for semisimple Hopf algebras.” TheoremDB. P2862. Problem statement; statement text SHA-256 fb245142a94c2e0257e88d60ec3ed6b4a1ea66650d8cfa797f21ce340da35a2f. https://theoremdb.org/statement/?ref=P2862

BibTeX

@misc{theoremdb-problem-fb245142a94c2e0257e88d60ec3ed6b4a1ea66650d8cfa797f21ce340da35a2f,
  title = {{Kaplansky's sixth conjecture for semisimple Hopf algebras}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 fb245142a94c2e0257e88d60ec3ed6b4a1ea66650d8cfa797f21ce340da35a2f},
  url = {https://theoremdb.org/statement/?ref=P2862}
}

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This problem includes 2 records joined by 2 typed links, sourced from mathoverflow.net^[1], current as of August 1, 2026.

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1References

Packet source. MathOverflow question 108404, “Kaplansky's sixth conjecture for semisimple Hopf algebras,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Original CC0 universal statement written after reading all three answers and comments and checking surveys and geometric reformulations.Also cited at Full question, answers, and visible comments concerning Kaplansky's sixth conjecture for semisimple Hopf algebras; checked 2026-08-01.Also cited at Editorial research route recorded 2026-08-01.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Kaplansky's sixth conjecture for semisimple Hopf algebras, the reviewed source scope is Full question, answers, and visible comments concerning Kaplansky's sixth conjecture for semisimple Hopf algebras; checked 2026-08-01.. The packet makes no inference beyond that cited scope.Source named by the research packet.
arXiv preprint 1409.2545, linked primary source for “Kaplansky's sixth conjecture for semisimple Hopf algebras,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:1409.2545, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and survey of partial cases of Kaplansky's sixth conjecture.Source used to assess the problem's recorded status.For Kaplansky's sixth conjecture for semisimple Hopf algebras, this source directly records the general conjecture as open and maps special cases without supplying a general proof.
arXiv preprint 1506.00314, linked primary source for “Kaplansky's sixth conjecture for semisimple Hopf algebras,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:1506.00314, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and theorem giving algebraic integrality of the invariant family as a sufficient condition for Kaplansky's sixth conjecture.Source used to assess the problem's recorded status.For Kaplansky's sixth conjecture for semisimple Hopf algebras, this source supplies a conditional criterion and invariant-theory framework rather than the general divisibility theorem.

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[#P2862] Kaplansky's sixth conjecture for semisimple Hopf algebras

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