On Kaplansky’s sixth conjecture

Abstract

About 39 years ago, Kaplansky conjectured that the dimension of a semisimple Hopf algebra over an algebraically closed eld of characteristic zero is divisible by the dimensions of its simple modules. Although it still remains open, some partial answers to this conjecture play an important role in classifying semisimple Hopf algebras. This paper focuses on the recent development of Kaplansky’s sixth conjecture and its applications in classifying semisimple Hopf algebras.