On Kaplansky's sixth conjecture

TL;DR

This paper reviews recent progress on Kaplansky's sixth conjecture, which posits divisibility relations between the dimensions of semisimple Hopf algebras and their simple modules, and discusses its implications for classifying such algebras.

Contribution

It summarizes recent developments and applications of partial results related to Kaplansky's sixth conjecture in the classification of semisimple Hopf algebras.

Findings

Partial results support the conjecture in specific cases

Applications aid in classifying semisimple Hopf algebras

Progress enhances understanding of algebraic structure

Abstract

About $39$ years ago, Kaplansky conjectured that the dimension of a semisimple Hopf algebra over an algebraically closed field 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.