HOPF CYCLIC COHOMOLOGY AND BIDERIVATIONS

Hopf cyclic cohomology HC((delta,sigma))*(H) for a Hopf algebra H with respect to a modular pair in involution (delta, sigma) was introduced by Connes and Moscovici. By a biderivation D on a Hopf algebra H we shall mean a linear map that satisfies the axioms for both a derivation and a coderivation on H. Given a biderivation D on a Hopf algebra, we define, under certain conditions, a map L(D) : HC((delta,sigma))*(.) We give examples of such maps for the quantized universal enveloping algebra U(h) (9) of a simple Lie algebra g. When H is irreducible, cocommutative and equipped with a character delta such that (delta, 1) is a modular pair in involution, we define "inner biderivations" and use these to produce a left H-module structure on HC((delta,1))* (H). Finally, we show that every morphism L(D) : HC((delta,1))* (H) -> HC((delta,1))* (H) biderivation D on such a. Hopf algebra. H can be realized as a morphism induced by an inner biderivation by embedding H into a larger Hopf algebra H[D].