β-character Algebra and a Commuting Pair in Hopf Algebras

Let K be a field. Let H be a finite-dimensional semisimple and cosemisimple K-Hopf algebra. In this paper, we introduce a notion of β-character algebra C _β (H) for each group-like element β in H*. We prove that Radford's action of the Drinfel'd double D(H) on H _β (see Radford, J. Algebra, 270:670-695, 2003) and the right hit action of the β-character algebra C _β (H) on H _β form a commuting pair. This generalizes an earlier result of Zhu (Proc. Amer. Math. Soc., 125(10):2847-2851, 1997). A K-basis of C _β (H) is given when H is split semisimple. Finally, as an example, we explicitly construct all the simple modules for the Drinfel'd double of the unique 8-dimensional non-commutative and non-cocommutative semisimple Hopf algebra.