Kaplansky's sixth conjecture (1975) predicts that for every finite-dimensional semisimple Hopf algebra HH over an algebraically closed field of characteristic 00, the dimension of every irreducible HH-module divides dim⁡kH\dim_{\mathbb{k}}H — a direct generalization of Frobenius' 1896 theorem for finite groups. The conjecture remains open: no proof and no counterexample is known, and the general case beyond quasitriangular, factorizable, central-character, group-theoretical, and classified small dimensions is untouched. Equivalent reformulations due to Cuadra and Meir reduce the conjecture to integrality over Z\mathbb{Z} of the canonical Casimir element cλc_{\lambda} of HH, or to the existence of a weak Z\mathbb{Z}-order on HH — making the arithmetic of this single canonical element the precise open core of the problem.

Background

A Hopf algebra HH over a field k\mathbb{k} is simultaneously an algebra and a compatible coalgebra (with comultiplication Δ\Delta and counit ε\varepsilon) equipped with an antipode SS, an antiautomorphism substituting for inverses; the basic examples are group algebras kG\mathbb{k}G and their linear duals. A finite-dimensional Hopf algebra is called semisimple when it is semisimple as an algebra, and involutory when S2=IdS^2=\mathrm{Id}; a theorem of Larson and Radford shows that over characteristic 00 these two conditions are equivalent, so the conjecture below is stated interchangeably for semisimple or involutory Hopf algebras. An irreducible representation (simple module) SS of HH is a simple module over the underlying algebra, and Irr H\mathrm{Irr}\,H denotes a full set of them. The guiding analogy is Frobenius' classical theorem of 1896: for a finite group GG, the degree of every irreducible complex representation divides ∣G∣|G|. Group algebras are the commutative-or-cocommutative tip of the Hopf world, and Kaplansky's list of ten conjectures from his 1975 Chicago lecture notes 'Bialgebras' asked how far group representation theory carries over; the sixth conjecture, known as Frobenius divisibility (FD), asserts that the degrees of all irreducible representations of any finite-dimensional involutory Hopf k\mathbb{k}-algebra HH divide dim⁡kH\dim_{\mathbb{k}}H (over an algebraically closed field of characteristic 00; more generally, over any field of characteristic 00 for which HH is split, meaning the endomorphism algebra of every simple module is k\mathbb{k}). Several families of cases are known: the conjecture holds for group algebras and their duals by Frobenius' theorem itself; Kac (1972) and Yongchang Zhu (1994) proved the arithmetic 'class equation' consequences (an analogue for semisimple Hopf algebras of the conjugacy-class equation of a finite group, forcing divisibility relations among the dimensions of the simple modules); Sheng Lin Zhu (1993) proved divisibility for irreducibles whose character is central in H∗H^*; Etingof and Gelaki (1998) proved it for quasitriangular semisimple Hopf algebras (those equipped with a distinguished invertible element R∈H⊗HR\in H\otimes H, an 'RR-matrix', making the comultiplication quasi-cocommutative); and Schneider (2001) proved the stronger statement that (dim⁡S)2(\dim S)^2 divides dim⁡kH\dim_{\mathbb{k}}H when HH is factorizable (meaning the map Φ ⁣:H∗→H\Phi\colon H^*\to H induced by the quasitriangular structure is bijective). It is also verified case by case in every dimension in which semisimple Hopf algebras have been classified, and no counterexample is known in any dimension. The modern arithmetic reformulation, due to Cuadra and Meir and presented in a unified way by Jacoby and Lorenz, goes as follows. A semisimple Hopf algebra HH is a symmetric algebra: it carries a nondegenerate trace-like linear form, and one takes λ∈H∗\lambda\in H^*, the normalized regular character, which is the unique integral of H∗H^* (a left-and-right invariant functional) with ⟨λ,1⟩=1\langle\lambda,1\rangle=1. The Casimir element of the pair is cλ=S(Λ(1))⊗Λ(2)∈H⊗Hc_{\lambda}=S(\Lambda_{(1)})\otimes\Lambda_{(2)}\in H\otimes H, where Λ\Lambda is the integral of HH normalized by ⟨ε,Λ⟩=dim⁡kH\langle\varepsilon,\Lambda\rangle=\dim_{\mathbb{k}}H and Δ(Λ)=Λ(1)⊗Λ(2)\Delta(\Lambda)=\Lambda_{(1)}\otimes\Lambda_{(2)}. An element of a ring is integral over Z\mathbb{Z} when it satisfies a monic polynomial with integer coefficients. Cuadra and Meir proved that FD holds for a split semisimple Hopf algebra HH over characteristic 00 if and only if cλc_{\lambda} is integral over Z\mathbb{Z}, and equivalently if and only if HH admits a weak order over Z\mathbb{Z} in the sense of Rumynin and Lorenz (a Z\mathbb{Z}-lattice in HH stable under all inner products with elements of a corresponding lattice, a condition weaker than being a Hopf order over a number ring). The known cases all ultimately follow from ring-theoretic properties of the representation ring R(H)\mathscr{R}(H) (the free abelian group on irreducible classes with tensor-product multiplication) forcing this integrality. The conjecture also generalizes categorically: in the language of fusion categories (semisimple rigid monoidal categories with finitely many simple objects, rigidity meaning every object has left and right duals), Rep(H)\mathrm{Rep}(H) is an integral fusion category (every simple object has integer Frobenius–Perron dimension, the categorical size invariant, whose value on a simple object SS is dim⁡kS\dim_{\mathbb{k}}S and whose value on the whole category Rep(H)\mathrm{Rep}(H) is dim⁡kH\dim_{\mathbb{k}}H), and the divisibility question for spherical fusion categories (those equipped with a nondegenerate two-sided trace functional) is, in work through 2025, still described as a central open problem.

Problem Statement

Prove or refute Kaplansky's sixth conjecture (Frobenius divisibility) in its full generality: for every finite-dimensional semisimple — equivalently, involutory — Hopf algebra HH over an algebraically closed field k\mathbb{k} of characteristic 00, and every irreducible HH-module SS,

dim⁡kS divides dim⁡kH.\dim_{\mathbb{k}}S \text{ divides } \dim_{\mathbb{k}}H.

By the theorem of Cuadra and Meir, an equivalent target is: for every split semisimple Hopf algebra HH over a field of characteristic 00, the Casimir element cλ=S(Λ(1))⊗Λ(2)∈H⊗Hc_{\lambda}=S(\Lambda_{(1)})\otimes\Lambda_{(2)}\in H\otimes H attached to the integral Λ∈H\Lambda\in H with ⟨ε,Λ⟩=dim⁡kH\langle\varepsilon,\Lambda\rangle=\dim_{\mathbb{k}}H and the normalized integral λ∈H∗\lambda\in H^* is integral over Z\mathbb{Z} (equivalently, HH admits a weak order over Z\mathbb{Z}). A refutation of the conjecture is an explicit semisimple Hopf algebra over a field of characteristic 00 together with an irreducible module whose dimension does not divide dim⁡kH\dim_{\mathbb{k}}H. Known cases (group algebras and their duals, commutative and cocommutative Hopf algebras, quasitriangular or factorizable semisimple Hopf algebras, irreducibles with central character, and all dimensions in which the classification is complete) do not settle the question. The question is asked without restriction on dimension or structural class.

References

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