(Journal of Pure and Applied Algebra, to appear)
We give a construction of Connes-Moscovici's cyclic cohomology for any Hopf algebra equipped with a character. 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. The approach is inspired by the classical construction of characteristic classes for foliations (a la Bott-Haefliger, using the truncations of the Weil complex), and it extends/explains Quillen's construction of cyclic cocycles associated to higher traces (which we re-discover when the Hopf algebra is the trivial unital Hopf algebra, i.e. complex numbers). Hence we describe various complexes, all computing the cyclic cohomology of the given Hopf algebra, and we show their relevance in the construction of various characteristic maps. In particular, for each positive integer n, we describe three such complexes (relevant in the presence of even/odd higer traces, and equivariant cycles, respectively), while one of our theorems shows that Connes-Moscovici complex is obtained for n=0.