Question

Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$-module divides the dimension of $H$.

In which cases the conjecture is known to be true?

Answer 1

From Shlomo Gelaki research statement (which is nice survey, by the way):

We also proved that the dimension of an irreducible representation of a semisimple Hopf algebra H, which is either quasitriangular or cotriangular, divides the dimension of H. This result partially answers a celebrated conjecture of Kaplansky, which is still open.

Answer 2

Yorck Sommerhäuser has a very nice survey about Kaplansky's conjectures. Section 6 is devoted to Kaplansky's 6th conjecture.

In Sommerhäuser's survey it is mentioned that Richmond and Nichols proved that the conjecture is true if the simple module has dimension two:

Theorem (Nichols & Richmond). The dimension of a semisimple Hopf algebra over $\mathbb{C}$ is even if the Hopf algebra has a simple module of dimension 2.

In Sommerhäuser's survey it is also mentioned that Montgomery and Witherspoon proved that Kaplansky's conjecture holds if it holds for a subalgebra.

In this paper

Cohen, Miriam; Gelaki, Shlomo; Westreich, Sara. Hopf algebras. Handbook of algebra. Vol. 4, 173--239, Handb. Algebr., 4, Elsevier/North-Holland, Amsterdam, 2006. MR2523421 (2010j:16076), link

it is written that Kaplansky's conjecture has been proved

• if $H$ is triangular,

• if $H$ is semisolvable,

• if $H$ is cotriangular,

• if $R(H)$ is central in $H^*$, where $R(H)$ is the span in $H^*$ of all the characters on $H$.