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

Abstract

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 \(n=1\).