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We discuss a new strategy for the computation of the Hopf-cyclic cohomology of the Connes–Moscovici Hopf algebra \(\mathcal{H}_n\). More precisely, we introduce a multiplicative structure on the Hopf-cyclic complex of \(\mathcal{H}_n\), and we show that the van Est type characteristic homomorphism from the Hopf-cyclic complex of \(\mathcal{H}_n\) to the Gelfand–Fuks cohomology of the Lie algebra \(W_n\) of formal vector fields on \({\mathbb {R}}^n\) respects this multiplicative structure. We then illustrate the machinery for \(n=1\).

Hopf-cyclic cohomology of the Connes–Moscovici Hopf algebras with infinite dimensional coefficients