On Cyclic Cohomology of x-Hopf algebras

In this paper we study the cyclic cohomology of certain x-Hopf algebras: universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici x-Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic cohomology. Furthermore, we provide a pairing for the cyclic cohomology of x-Hopf algebras which generalizes the Connes-Moscovici characteristic map to x-Hopf algebras. This enables us to transfer the x-Hopf algebra cyclic cocycles to algebra cyclic cocycles.