Some properties of factorizable Hopf algebras (Q2718947)

A finite dimensional quasitriangular Hopf algebra \((H,R)\) is called factorizable if the Drinfeld map \(F\colon H^*\to H\) is an isomorphism. \textit{P. Etingof} and \textit{S. Gelaki} [Math. Res. Lett. 5, No. 1-2, 191-197 (1998; Zbl 0907.16016)] proved that if \((H,R)\) is a factorizable quasitriangular semisimple Hopf algebra over an algebraically closed field of characteristic zero, and if \(V\) is a simple \(H\)-module, then \((\dim V)^2\) divides \(\dim H\). Since the Drinfeld double is factorizable, this was used to solve Kaplansky's conjecture concerning the dimension of simple modules over a semisimple Hopf algebra in the case where the Hopf algebra has a quasitriangular structure. The aim of the paper under review is to give a different proof of the result of Etingof and Gelaki, without using modular categories. The author uses the Kac-Zhu class equation for semisimple Hopf algebras. Some results on factorizable quasitriangular Hopf algebras are also proved.