Cyclic cohomology and Hopf algebras

We associate canonically a cyclic module to any Hopf algebra endowed with a modular pair in involution, consisting of a group-like element and a chara cter. This provides the key construction for allowing the extension of cycl ic cohomology to Hopf algebras in the nonunimodular case and, further, to d eveloping a theory of characteristic classes for actions of Hopf algebras c ompatible not only with traces but also with the modular theory of weights. This applies to both ribbon and coribbon algebras as well as to quantum gr oups and their duals.