Cyclic cohomology of Hopf algebras

We give a construction of the Connes–Moscovici cyclic cohomology for any Hopf algebra equipped with a character (solving the technical problem raised in Connes and Moscovici (Comm. Math. Phys. 198 (1998) 199–246)). Furthermore, we introduce a non-commutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the Weil complex arises naturally when looking at Hopf algebra actions and invariant higher traces, to give a non-commutative version of the usual Chern–Weil theory.