The Killing Form And Adjoint Representations Of Hopf Algebras

In this paper, we develop the theory of Killing form over any finite dimensional Hopf algebras, and discuss the deep relations between the non-degeneracy of Killing form and the adjoint representations of Hopf algebras.Firstly, we define the Killing form over any finite dimensional Hopf algebras using the (left) adjoint action, then we investigate the fundamental properties of the Killing form, and obtain a decomposition of Hopf algebras under the adjoint action. In particular, for a finite dimensional semisimple Hopf algebra, we give explicitly the radical of the Killing form. Secondly, for a finite dimensional semisimple Hopf algebra, we prove that the Killing form is an ad-invariant bilinear form, which leads to the further orthogonal relations between the center of Hopf algebras and the kernel of linear mapping induced by a left integral. Finally, for a finite dimensional semisimple Hopf algebra H, we show that every simple module of H is appeared in the adjoint representation of H if and only if the Killing form of H is non-degenerate. In particular, for a finite group G , every simple module of G is appeared in the conjugacy representation of G if and only if the sum of every row in the character table of G is not zero.