For a finite group GG, a classical theorem of Frobenius (1896) states that the degree of every irreducible complex representation of GG divides the order of GG. In his 1975 lecture notes 'Bialgebras', Irving Kaplansky conjectured the analogue for Hopf algebras: if HH is a finite-dimensional semisimple Hopf algebra over an algebraically closed field of characteristic 0, then the dimension of every irreducible HH-module divides dim⁡H\dim H (the 'Frobenius divisibility' or FD property; this is the sixth of his ten conjectures). The conjecture remains open in this unrestricted form. It is known for large classes: quasitriangular and factorizable semisimple Hopf algebras, upper and lower semisolvable ones (hence all prime-power dimensions), all weakly group-theoretical ones, all semisimple Hopf algebras of dimension below 120, and odd dimensions below 600. It admits arithmetic reformulations — FD holds for a complex semisimple HH if and only if HH admits a weak order over Z\mathbb{Z} in the sense of Rumynin and Lorenz (Cuadra–Meir), and if and only if a canonical Casimir element is integral over Z\mathbb{Z} — but none of these has settled the general case. The smallest possible counterexample has dimension at least 120.

Background

A Hopf algebra is an algebra HH equipped with a compatible coalgebra structure (comultiplication Δ:H→H⊗H\Delta:H\to H\otimes H and counit) and an antipode S:H→HS:H\to H, an antihomomorphism substituting for inverses; the defining examples are group algebras kGkG with Δ(g)=g⊗g\Delta(g)=g\otimes g and S(g)=g−1S(g)=g^{-1}, and their duals, the function algebras kGk^{G}. The tensor product of two HH-modules is again an HH-module via Δ\Delta, so the finite-dimensional representations of HH form a rigid tensor category (a tensor category in which every object has a left and right dual). A Hopf algebra is called semisimple if its underlying algebra is semisimple (equivalently, every module is a direct sum of irreducibles); by a theorem of Larson and Radford, over an algebraically closed field of characteristic 0 this is equivalent to cosemisimplicity and to the antipode being involutory, i.e. S2=idS^{2}=\mathrm{id}. For finite groups, Frobenius proved in 1896 that the dimension of every irreducible representation of GG over such a field divides the finite group order ∣G∣|G|. Kaplansky's sixth conjecture (1975) predicts that the same divisibility, called the Frobenius type or Frobenius divisibility (FD) property, holds for every finite-dimensional semisimple Hopf algebra; the characteristic-0 reading is the standard one adopted by the secondary literature, since the originally published wording omitted the hypothesis on the field. A first structural analogue, the Nichols–Zoeller freeness theorem (1989) — a Hopf algebra is free over any Hopf subalgebra, mirroring Lagrange's theorem — gives divisibility information about sub-Hopf-algebras but does not imply FD for irreducible module dimensions. Partial results accumulate along two axes. By representation-theoretic techniques on the character algebra (the span of the characters of irreducible modules inside the dual Hopf algebra H∗H^{*}): Nichols and Richmond showed that a 2-dimensional simple module forces dim⁡H\dim H even, and Kashina, Sommerhäuser and Zhu extended this (a self-dual or even-dimensional simple module forces dim⁡H\dim H even); Etingof and Gelaki proved that dimensions of simple modules over the Drinfeld double D(H)D(H) (the Hopf algebra built on H∗⊗HH^{*}\otimes H that universally encodes braided HH-module structures) divide dim⁡H\dim H, hence FD holds for quasitriangular semisimple Hopf algebras (those carrying a universal RR-matrix, a distinguished invertible element of H⊗HH\otimes H that equips the module category with a braiding, by virtue of which every simple HH-module becomes a D(H)D(H)-module), and Schneider extended this to factorizable ones (a strengthening of quasitriangularity); Montgomery and Witherspoon proved FD is inherited by crossed products, giving the conjecture for upper and lower semisolvable Hopf algebras (those filtered by normal Hopf subalgebras with commutative or cocommutative quotients), in particular for all dimensions pnp^{n}. By fusion-categorical methods: Rep HH is a fusion category (a semisimple rigid tensor category with finitely many simple objects), and Etingof, Nikshych and Ostrik proved that every weakly group-theoretical fusion category (one Morita equivalent to a nilpotent category, a class containing all group-theoretical and solvable categories) has the strong Frobenius property (the divisibility together with the same property for the dual category); since every known fusion category of integer Frobenius–Perron dimension — the categorical size invariant computed from the tensor-product fusion rules, equal to dim⁡H\dim H for Rep H\mathrm{Rep}\ H — is weakly group-theoretical, this covers all known examples. Along the dimension axis, Natale and Kashina independently established FD for all semisimple Hopf algebras of dimension below 60, and Dong, Natale and Vendramin proved that every weakly integral fusion category (one some tensor power of which has integer Frobenius–Perron dimensions of simple objects) of Frobenius–Perron dimension below 120 is of Frobenius type. There are also arithmetic reformulations: Cuadra and Meir proved that a complex semisimple Hopf algebra satisfies FD if and only if it admits a weak order over Z\mathbb{Z} (an order, i.e. a Z\mathbb{Z}-lattice subalgebra, with weak-integral data in the sense of Rumynin and Lorenz), and equivalently if and only if the Casimir element cλc_{\lambda} (the canonical symmetric-algebra invariant determined by the normalized integral λ∈H∗\lambda\in H^{*}) is integral over Z\mathbb{Z}; Rumynin showed that admitting a weak integral form implies Frobenius type. None of these routes settles the unrestricted conjecture.

Problem Statement

Let HH be a finite-dimensional semisimple Hopf algebra over an algebraically closed field of characteristic 00, and let VV range over the irreducible HH-modules. Must dim⁡V\dim V divide dim⁡H\dim H? This is Kaplansky's sixth conjecture (the Frobenius divisibility or Frobenius-type property); equivalently, over C\mathbb{C}, it asks whether every semisimple Hopf algebra admits a weak order over Z\mathbb{Z} in the sense of Rumynin–Lorenz, or whether the canonical Casimir element cλc_{\lambda} associated with the normalized integral is integral over Z\mathbb{Z}. The question is posed with no restriction on dimension, algebra or coalgebra structure, or method: an answer is either a proof of the divisibility for all such HH, or an explicit semisimple Hopf algebra in characteristic 00 with an irreducible module whose dimension does not divide dim⁡H\dim H.

References

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