# P2862: Kaplansky's sixth conjecture for semisimple Hopf algebras
- ID: `P2862`
- Reference: `kaplansky-sixth-semisimple-hopf`
- Page: https://theoremdb.org/statements/P2862
- Record maturity: Reviewed problem with recorded work
## 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 \(\dim_k H\)?
### Problem setup
- **Definition.** A Hopf algebra is an algebra equipped with compatible comultiplication, counit, and antipode maps.
- **Definition.** Semisimple means semisimple as an associative \(k\)-algebra.
- **Definition.** A simple left module is a nonzero module with no proper nonzero submodules.
- **Remark.** Divisibility is divisibility of the two positive integers \(\dim_k V\) and \(\dim_k H\).
### What 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.
## Status
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](#reference-3) [2](#reference-2) [1](#reference-1)
## Work
### Evidence for the current status
**Claim 1 (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.
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.
### Background and intake notes
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.
- 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.
- Recorded example: 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: For a one-dimensional simple module, the divisibility assertion is automatic.
### Open directions
- **Route 1** (reported): 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](#reference-1)
### Working on this
Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `kaplansky-sixth-semisimple-hopf`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.
## References
1. MathOverflow question 108404, “Kaplansky's sixth conjecture for semisimple Hopf algebras,” checked 2026-08-01. Original CC0 universal statement written after reading all three answers and comments and checking surveys and geometric reformulations. https://mathoverflow.net/questions/108404/kaplanskys-6th-conjecture-dimirrep-dimalgebra-for-semi-simple-hopf-alg
- 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.
- forum; reference source; checked 2026-08-01
- Source use: citation_only
- 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.
2. Li Dai and Jingcheng Dong, “On Kaplansky's sixth conjecture,” arXiv:1409.2545 (2014). abstract and survey of partial cases of Kaplansky's sixth conjecture https://arxiv.org/abs/1409.2545
- preprint; reference source; arXiv:1409.2545, checked 2026-08-01; checked 2026-08-01
- Source use: citation_only
- 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.
3. Ehud Meir, “Semisimple Hopf Algebras via Geometric Invariant Theory,” arXiv:1506.00314 (2015). abstract and theorem giving algebraic integrality of the invariant family as a sufficient condition for Kaplansky's sixth conjecture https://arxiv.org/abs/1506.00314
- preprint; reference source; arXiv:1506.00314, checked 2026-08-01; checked 2026-08-01
- Source use: citation_only
- 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.