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 $n=1$.