Characters of Hopf algebras

Abstract

In the development of the character theory of finite groups, one of the key facts used in proving the orthogonality relations is the invariance property of cg E kG. (See, e.g., the proofs of Lemma 5.1.3 in [2] and of Eq. (31.8) in [l].) Since finite-dimensional Hopf algebras [5] and the dual algebras of certain infinite-dimensional Hopf algebras [7] contain elements with properties analogous to those of Cg, it is reasonable to expect that an orthogonality relation will hold for characters of such Hopf algebras. In Section 2 of this paper we prove such an orthogonality relation. In order to include a character theory for infinite dimensional Hopf algebras, we must consider characters as elements of the Hopf algebra which are associated with comodules over the Hopf algebra, rather than as functionals on the Hopf algebra which are associated with modules over the Hopf algebra. This point of view allows a simultaneous treatment of the characters of compact Lie groups and completely reducible affine algebraic groups (taking as the Hopf algebra the algebra of representative functions), of the characters of semisimple Lie algebras over an algebraically closed field of characteristic 0 (taking U(L)” as the Hopf algebra; the characters studied here differ from those of Expose 18 of [6] by a scalar multiple), and of the characters of finite groups (taking as the Hopf algebra the dual Hopf algebra to the group algebra). We then use our results to prove that the dimension of a simple comodule of an involutory cosemisimple Hopf algebra over an algebraically closed field is not divisible by the characteristic of the field. In Section 3 we prove that the antipode y of a cosemisimple Hopf algebra is bijective, and that y2 maps each simple subcoalgebra onto itself. In Section 4 we give a generalization of Maschke’s Theorem: If the characteristic of the field does not divide the dimension of a finite dimensional involutory Hopf algebra, then the Hopf algebra and its dual are semisimple; if the characteristic does divide the