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 character. This provides the key construction for allowing the extension of cyclic cohomology to Hopf algebras in the nonunimodular case and, further, to developing a theory of characteristic classes for actions of Hopf algebras compatible 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 groups and their duals.