Some properties of factorizable Hopf algebras

Abstract

Let k be an algebraically closed field of characteristic 0. Itis well known that the degree of an irreducible representation ofa finite group over k divides the order of the group. I.Kaplansky conjectured in 1975 that this holds for finitedimensional semisimple Hopf algebras over k. P. Etingof and S.Gelaki have solved this conjecture in case the Hopf algebra has aquasitriangular structure [Math. Res. Lett. 5 (1998), no. 1 2,191 197; MR \Cite{Etingof98:properties:191--197}[99e:16050]]. Infact, they proved: Theorem A. Let H be a finite dimensionalsemisimple Hopf algebra over k. If V is a simple module overthe Drinfeld double D(H), then \dim V divides\dim H.\par The Kaplansky conjecture for finite dimensionalquasitriangular semisimple Hopf algebras follows easily fromthis. A finite dimensional quasitriangular Hopf algebra (H,R)is called factorizable if the Drinfeld mapF\colon H\sp \ast\to H is an isomorphism. The Drinfeld doubleD(H) is factorizable. The above theorem is easily generalizedas follows: Theorem B. Let (H,R) be a factorizablequasitriangular semisimple Hopf algebra. If V is a simple Hmodule, then (\dim V)\sp 2 divides \dim H.\par To proveTheorem A (or B), Etingof and Gelaki used the theory of modularcategories developed by V. Turaev. The main purpose ofSchneider's paper is to prove Theorem B directly without usingthe theory of modular categories. Instead of modular categories,Schneider uses the class equation result of Kac and Zhu. This isstated as follows: Theorem C. Let H be a finite dimensionalsemisimple Hopf algebra over k. If e is a primitiveidempotent in the character algebra C(H), then\dim H\sp \ast e divides \dim H.\par Recently, the reviewerhas obtained another elementary proof of Theorem B motivated bySchneider's work. His approach is nearer to the original proof ofEtingof and Gelaki. He deduces the necessary identities whichappear in the theory of modular categories from well knownclassical identities of Hopf characters. This result is containedin [M. Takeuchi, J. Algebra 243 (2001), no. 2, 631 643\refcno 1850651\endrefcno ]. Schneider's paper contains twointeresting new results on factorizable quasitriangular Hopfalgebras