Kaplansky's sixth conjecture (1975) asserts that every finite-dimensional semisimple Hopf algebra HH over an algebraically closed field of characteristic 00 has the Frobenius divisibility property: the dimension of every irreducible HH-module divides dim⁡H\dim H — the exact Hopf-theoretic analogue of Frobenius's theorem on irreducible character degrees of finite groups. The conjecture is the open question recorded in A. Jacoby's note 'Frobenius Divisibility and Hopf Centers' (arXiv:1704.04256), whose Schur-type divisibility theorem is conditional on this property. The conjecture remains open: it is known for all semisimple Hopf algebras of dimension below 126, for quasitriangular ones, and for all weakly group-theoretical (hence all currently known) examples; for complex HH it is equivalent to the existence of a weak order over the integers (Cuadra–Meir). No uniform proof and no counterexample is known.

Background

A Hopf algebra HH over a field kk is a vector space that is simultaneously an algebra (multiplication mm, unit) and a coalgebra (comultiplication Δ:H→H⊗H\Delta:H\to H\otimes H, counit ε\varepsilon) with Δ\Delta and ε\varepsilon compatible with mm, together with an antipode S:H→HS:H\to H that intertwines multiplication and comultiplication as a formal inverse; the group algebra kGkG of a finite group GG, with Δ(g)=g⊗g\Delta(g)=g\otimes g, is the basic example, and the class of Hopf algebras is stable under duality, tensor products, quotients, and Drinfeld twists (deformations of the comultiplication by an invertible element satisfying a cocycle condition). A finite-dimensional Hopf algebra is called semisimple when it is semisimple as an algebra, meaning that every module over it decomposes as a direct sum of irreducible modules; over an algebraically closed field of characteristic 00 this is equivalent, by a theorem of Larson and Radford, to cosemisimplicity (the dual condition on the coalgebra structure), so the dual Hopf algebra H∗H^* is again semisimple. Semisimple Hopf algebras are a central object of the classification program for finite-dimensional Hopf algebras.

For finite groups, a classical theorem of Frobenius states that the degree of every irreducible complex representation of GG divides the order ∣G∣|G|, and Schur strengthened this to ∣G/Z(G)∣|G/Z(G)|, where Z(G)Z(G) is the center. By analogy, a finite-dimensional algebra AA over an algebraically closed field is said to have the Frobenius divisibility property (FD) if the dimension of every irreducible AA-module divides dim⁡kA\dim_k A. A general finite-dimensional algebra fails FD, but Hopf algebras carry enough extra structure for the property to be plausible: the Nichols–Zoeller freeness theorem gives the Lagrange-type half of the group analogy, namely that dim⁡kK\dim_k K divides dim⁡kH\dim_k H for every Hopf subalgebra K⊆HK\subseteq H.

In his 1975 lecture notes Bialgebras, Irving Kaplansky posed ten conjectures about Hopf algebras; the sixth conjectures that FD holds for all finite-dimensional semisimple Hopf algebras over an algebraically closed field of characteristic 00. The published wording omitted the characteristic hypothesis and is false in positive characteristic (the group algebra of SL(2,p)SL(2,p) in characteristic pp already fails, as discussed in the survey of Dai and Dong), so the characteristic-zero form above — used uniformly by Jacoby, Etingof–Gelaki, Cuadra–Meir, and the modern literature — is the standard form of the conjecture. By the Larson–Radford theorem it admits the equivalent dual formulation: the dimension of every simple subcoalgebra of a semisimple Hopf algebra divides dim⁡kH\dim_k H.

The conjecture is known for several large but restricted classes: quasitriangular semisimple Hopf algebras (those equipped with a universal RR-matrix, an invertible element of H⊗HH\otimes H implementing a coherent interchange of tensor factors in the module category; Etingof–Gelaki); dimension pnp^n for a prime pp (Montgomery–Witherspoon); and all dimensions in which semisimple Hopf algebras have been completely classified, namely pnp^n for n≤3n\le 3, pqpq, pq2pq^2, and pqrpqr for primes p,q,rp,q,r, as well as all dimensions below 6060 (Natale, Kashina). A powerful modern viewpoint uses fusion categories — semisimple tensor categories with finitely many simple objects in which every object has a dual; Rep(H)\mathrm{Rep}(H) of a semisimple HH is one, with Frobenius–Perron dimension (a categorical analogue of vector-space dimension defined through the Perron–Frobenius eigenvalues of the fusion matrices) equal to dim⁡kH\dim_k H. The conjecture then says that Rep(H)\mathrm{Rep}(H) is of Frobenius type: FPdim⁡X\operatorname{FPdim} X divides FPdim⁡Rep(H)\operatorname{FPdim}\mathrm{Rep}(H) for every simple object XX. A fusion category is integral when every simple object has integer Frobenius–Perron dimension, and weakly integral when its total Frobenius–Perron dimension is an integer. Etingof–Nikshych–Ostrik proved that every weakly group-theoretical fusion category (one equivalent, in the module-category sense analogous to Morita equivalence of rings, to a category built from representations of finite groups by iterated group-graded extensions) satisfies a strong form of Frobenius divisibility. Every weakly integral fusion category of Frobenius–Perron dimension below 126126 is known to be of Frobenius type; since Rep(H)\mathrm{Rep}(H) is integral (simple HH-modules have integer vector-space dimensions), this verifies the conjecture for all semisimple Hopf algebras of dimension below 126126 and for every HH whose representation category is weakly group-theoretical — which includes every example currently known. In an arithmetic direction, Rumynin proved that a semisimple Hopf algebra admitting a weak integral form is of Frobenius type, and Cuadra and Meir proved that a complex semisimple Hopf algebra satisfies the conjecture if and only if it admits a weak order over the integers — a (weak) integral form or order being a subring of finite rank over Z\mathbb{Z} that spans HH and is compatible with the Hopf operations, weakly up to finite index; since certain Drinfeld twists of group algebras admit no Hopf order over any number ring, the order-theoretic route faces genuine obstructions. No argument covers an arbitrary semisimple Hopf algebra, and no counterexample is known.

Problem Statement

Prove or disprove Kaplansky's sixth conjecture in its standard characteristic-zero form:

Every finite-dimensional semisimple Hopf algebra H over an algebraically closed field k of characteristic 0 satisfies: dim⁡kV divides dim⁡kH for every irreducible H-module V.\text{Every finite-dimensional semisimple Hopf algebra } H \text{ over an algebraically closed field } k \text{ of characteristic } 0 \text{ satisfies: } \dim_k V \text{ divides } \dim_k H \text{ for every irreducible } H\text{-module } V.

By the Larson–Radford theorem the equivalent dual formulation asks the same divisibility for the dimensions of the simple subcoalgebras of HH.

References

  1. Irving Kaplansky, Bialgebras, Lecture Notes in Mathematics, Department of Mathematics, University of Chicago, Chicago, IL, 1975 (Zbl 1311.16029), https://zbmath.org/?q=an%3A1311.16029
  2. Richard G. Larson and David E. Radford, Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple, Journal of Algebra 117 (1988), no. 2, 267–289, DOI 10.1016/0021-8693(88)90107-X, https://doi.org/10.1016/0021-8693(88)90107-X
  3. Warren D. Nichols and M. Bettina Zoeller, A Hopf algebra freeness theorem, American Journal of Mathematics 111 (1989), no. 2, 381–385, DOI 10.2307/2374514, https://doi.org/10.2307/2374514
  4. Susan Montgomery and S. J. Witherspoon, Irreducible representations of crossed products, Journal of Pure and Applied Algebra 129 (1998), no. 3, 315–326, DOI 10.1016/S0022-4049(97)00077-7, https://doi.org/10.1016/S0022-4049(97)00077-7
  5. Pavel Etingof and Shlomo Gelaki, Some properties of finite-dimensional semisimple Hopf algebras, Mathematical Research Letters 5 (1998), no. 2, 191–197, DOI 10.4310/mrl.1998.v5.n2.a5, https://doi.org/10.4310/mrl.1998.v5.n2.a5; preprint arXiv:q-alg/9712033, https://arxiv.org/abs/q-alg/9712033
  6. Pavel Etingof and Shlomo Gelaki, Semisimple Hopf algebras of dimension pq are trivial, Journal of Algebra 210 (1998), no. 2, 664–669, DOI 10.1006/jabr.1998.7568, https://doi.org/10.1006/jabr.1998.7568; preprint arXiv:math/9801129, https://arxiv.org/abs/math/9801129
  7. Yevgenia Kashina, Some results on Hopf algebras of Frobenius type, in Groups, Rings, Lie and Hopf Algebras, 2003, pp. 85–104, DOI 10.1007/978-1-4613-0235-3_6, https://doi.org/10.1007/978-1-4613-0235-3_6
  8. Sonia Natale, Semisolvability of semisimple Hopf algebras of low dimension, Memoirs of the American Mathematical Society 186 (2007), no. 874, DOI 10.1090/memo/0874, https://doi.org/10.1090/memo/0874
  9. Jingcheng Dong, Semisimple Hopf algebras of dimension 9q29q^2 and high-dimensional semisimple Hopf algebras of Frobenius type, preprint arXiv:1103.5117 (2011), https://arxiv.org/abs/1103.5117
  10. Pavel Etingof, Dmitri Nikshych, and Victor Ostrik, Weakly group-theoretical and solvable fusion categories, Advances in Mathematics 226 (2011), no. 1, 176–205, DOI 10.1016/j.aim.2010.06.009, https://doi.org/10.1016/j.aim.2010.06.009
  11. Li Dai and Jingcheng Dong, On Kaplansky's sixth conjecture, Rendiconti del Seminario Matematico della Università di Padova 135 (2016), 1–20, DOI 10.4171/rsmup/135-1, https://doi.org/10.4171/rsmup/135-1; preprint arXiv:1409.2545, https://arxiv.org/abs/1409.2545
  12. Juan Cuadra and Ehud Meir, On the existence of orders in semisimple Hopf algebras, Transactions of the American Mathematical Society 368 (2016), no. 4, 2547–2562, DOI 10.1090/tran/6380, https://doi.org/10.1090/tran/6380; preprint arXiv:1307.3269, https://arxiv.org/abs/1307.3269
  13. Jingcheng Dong, Sonia Natale, and Leandro Vendramin, Frobenius property for fusion categories of small integral dimension, Journal of Algebra and Its Applications 14 (2015), no. 2, 1550011, DOI 10.1142/S0219498815500115, https://doi.org/10.1142/S0219498815500115; preprint arXiv:1209.1726, https://arxiv.org/abs/1209.1726
  14. Dmitriy Rumynin, Weak integral forms and the sixth Kaplansky conjecture, preprint arXiv:1907.02529 (2019; note written 1998), DOI 10.48550/arXiv.1907.02529, https://arxiv.org/abs/1907.02529
  15. Zhengwei Liu, Sebastien Palcoux, and Jinsong Wu, Fusion bialgebras and Fourier analysis, Advances in Mathematics 390 (2021), 107905, DOI 10.1016/j.aim.2021.107905, https://doi.org/10.1016/j.aim.2021.107905; preprint arXiv:1910.12059, https://arxiv.org/abs/1910.12059
  16. Juan Cuadra and Ehud Meir, Non-existence of Hopf orders for a twist of the alternating and symmetric groups, Journal of the London Mathematical Society 100 (2019), no. 1, 137–158, DOI 10.1112/jlms.12203, https://doi.org/10.1112/jlms.12203
  17. Sebastian Burciu, On some Frobenius type divisibility results in a premodular category, preprint arXiv:2209.12530 (2022), DOI 10.48550/arXiv.2209.12530, https://arxiv.org/abs/2209.12530
  18. Adam Jacoby and Martin Lorenz, Frobenius divisibility for Hopf algebras, Contemporary Mathematics 688, American Mathematical Society, 2017, pp. 125–137, DOI 10.1090/conm/688/13831, https://doi.org/10.1090/conm/688/13831; preprint arXiv:1511.02130, https://arxiv.org/abs/1511.02130
  19. Adam Jacoby, Frobenius Divisibility and Hopf Centers, preprint arXiv:1704.04256 (2017), DOI 10.48550/arXiv.1704.04256, https://arxiv.org/abs/1704.04256
  20. Li Dai, Frobenius property for fusion categories of dimension 120, Revista de la Unión Matemática Argentina 68 (2025), no. 2, 395–403, DOI 10.33044/revuma.4390, https://doi.org/10.33044/revuma.4390