Kaplansky's sixth conjecture: Frobenius-type divisibility for finite-dimensional semisimple Hopf algebras

Kaplansky's sixth conjecture (1975) predicts that every finite-dimensional semisimple Hopf algebra $H$ over an algebraically closed field is of Frobenius type: the dimension of every simple $H$-module divides $\dim H$, generalizing Frobenius's divisibility theorem for complex representations of finite groups. Fifty years of partial results — quasitriangular Hopf algebras, semisolvable and lower solvable Hopf algebras, and weakly group-theoretical representation categories — leave the general characteristic-zero case open, and in positive characteristic the conjecture is likewise open for semisimple Hopf algebras outside the settled classes — for the non-cosemisimple cases that arise when the characteristic divides the dimension and for the cosemisimple cases that lift to the open characteristic-zero conjecture. The conjecture remains open in both characteristic zero and positive characteristic, and no counterexample is known.

Background

A Hopf algebra over a field $k$ is a vector space carrying both an algebra structure and a compatible coalgebra structure: a coproduct $\Delta:H\to H\otimes H$ (a homomorphism encoding how elements 'multiply' under tensoring), a counit, and an antipode $S$. The basic finite-dimensional examples are group algebras $kG$ of finite groups and their duals $(kG)^{*}$, the algebras of $k$-valued functions on $G$. A finite-dimensional algebra is semisimple when every finite-dimensional module over it splits as a direct sum of simple (irreducible) modules. A classical theorem of Frobenius states that the dimension of every irreducible complex representation of a finite group $G$ divides $|G|=\dim kG$; since $\mathbb{C}G$ is the prototype of a semisimple Hopf algebra, one asks whether the same divisibility persists universally. A semisimple Hopf algebra $H$ is of Frobenius type if the $k$-dimension of every simple $H$-module divides $\dim H$. The sixth of the ten conjectures Kaplansky posed in his 1975 lecture notes 'Bialgebras' (as reproduced and discussed in Montgomery's CBMS monograph and in the survey of Dai and Dong) asserts that every finite-dimensional semisimple Hopf algebra over an algebraically closed field is of Frobenius type. The standard modern formulation works over an algebraically closed field of characteristic zero, where it remains open in general; the stronger reading without characteristic restriction is also open, with an added subtlety in positive characteristic: when $\operatorname{char}k$ divides $\dim H$, a semisimple Hopf algebra need not be cosemisimple (its dual algebra need not be semisimple) — for example the function algebra $(kY)^{*}$ is always semisimple but fails to be cosemisimple exactly when $\operatorname{char}k$ divides $|Y|$.

The main partial results frame the difficulty. Montgomery and Witherspoon proved the divisibility for Hopf algebras that are upper or lower semisolvable — those admitting a finite chain of normal Hopf subalgebras whose successive quotient Hopf algebras are commutative or cocommutative, the Hopf-theoretic analogue of solvable groups — provided $\operatorname{char}k$ does not divide $\dim H$; this covers, for instance, all prime-power dimensions. Etingof and Gelaki proved it for quasitriangular Hopf algebras (those equipped with a universal $R$-matrix making the coproduct quasi-cocommutative) in characteristic zero, and for quasitriangular Hopf algebras that are also cosemisimple in any characteristic, via a lifting theorem reducing semisimple-and-cosemisimple Hopf algebras in positive characteristic to characteristic zero. Letzter removed the characteristic restriction for lower solvable Hopf algebras (chains of normal Hopf subalgebras with commutative quotients), giving the first characteristic-free divisibility theorem for a class that includes semisimple Hopf algebras whose dimension is divisible by the characteristic, such as smash products $kX\#(kY)^{*}$ (Hopf-theoretic analogues of semidirect products of the finite groups $X$ and $Y$) with $\operatorname{char}k=p$, $p\mid |Y|$, $p\nmid |X|$, and $X$ solvable. In characteristic zero, the theory of fusion categories — the semisimple rigid monoidal categories of finite-dimensional $H$-modules, with Frobenius–Perron dimensions (the categorical analogue of module dimensions) replacing them — yields the conjecture whenever the representation category is weakly group-theoretical, meaning it is equivalent, in a suitable Morita sense, to a category built from pointed categories of finite groups; by the Burnside-type theorem of Etingof–Nikshych–Ostrik this includes every dimension with at most two distinct prime factors. Finally, an arithmetic reformulation is available over $\mathbb{C}$: Rumynin proved that a semisimple Hopf algebra admitting a weak integral form (a $\mathbb{Z}$-order whose base change recovers $H$ and whose ranks of simples match the dimensions) is of Frobenius type, and Cuadra and Meir proved the converse direction, so the conjecture over $\mathbb{C}$ is equivalent to the existence of a weak order over the integers for every complex semisimple Hopf algebra. None of these methods settles the general case.

Problem Statement

Let $k$ be an algebraically closed field and let $H$ be a finite-dimensional semisimple Hopf algebra over $k$. Is $H$ necessarily of Frobenius type — that is, does the $k$-dimension of every simple left $H$-module divide $\dim_k H$? Prove this divisibility in general, or exhibit a semisimple Hopf algebra over an algebraically closed field together with a simple module whose dimension does not divide the dimension of the algebra.

References

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