On Kaplansky's sixth conjecture

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.

17 pages, final version was accepted for publication in Rendiconti del Seminario Matematico della Universita di Padova (European Mathematical Society). arXiv admin note: text overlap with arXiv:0809.3031 by other authors