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

## TL;DR

This paper introduces a new method for computing the Hopf-cyclic cohomology of Connes-Moscovici Hopf algebras using a multiplicative structure and a characteristic homomorphism, with an illustration for the case n=1.

## Contribution

It develops a multiplicative structure on the Hopf-cyclic complex and demonstrates its compatibility with the van Est homomorphism for Connes-Moscovici algebras.

## Key findings

- Established a multiplicative structure on the Hopf-cyclic complex.
- Proved the homomorphism respects this structure.
- Illustrated the approach for the case n=1.

## 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 $n=1$.