Character Theory for Semisimple Hopf Algebras

We study the induction and restriction functor from a Hopf subalgebra of a semisimple Hopf algebra. The image of the induction functor is described when the Hopf subalgebra is normal. In this situation, at the level of characters this image is isomorphic to the image of the restriction functor. A criterion for subcoalgebras to be invariant under the adjoint action is given generalizing Masuoka’s result for normal Hopf algebras. Introduction The representation and character theory of semisimple Hopf algebras over an algebraically closed field of characteristic 0 has been developed in the last thirty years. Many results analogous to the classical theory of finite groups were obtained. One of the most important results was Zhu’s proof that the character ring of a semisimple Hopf algebra is a semisimple ring [13]. The character theory for coalgebras started in [5] and then continued in [11], [12], as well as in other papers. An important updated reference on results on character theory and its applications in classification of finite Hopf algebras can be found in [10]. In this paper we study the induction and restriction functor from a Hopf subalgebra of a semisimple Hopf algebra. Important results are obtained when the Hopf subalgebra is normal. In this situation, at the level of characters, the images of the induction map and restriction map are isomorphic as algebras. Recently was proven that a Hopf subalgebra is normal if and only if it is a depth two subalgebra [3]. One proof of this result uses the character theory for normal Hopf subalgebras developed in [2]. This paper continues the study began in [2]. Our approach uses the commuting pair described in [14]. We show that the Fourier transform of a semisimple Hopf algebra H is a morphism of D(H)-modules. As in [14] an important role is played by Date: November 12, 2008. 1