On the Connes-Moscovici Hopf algebra associated to the diffeomorphisms of a manifold

For our own education, we reconstruct the Hopf algebra of Connes and Moscovici obtained by the action of vector fields on a crossed product of functions by diffeomorphisms. We extend the realization of that Hopf algebra in terms of rooted trees as given by Connes and Kreimer from dimension one to arbitrary dimension of the manifold. In principle there is no modification, but in higher dimension one has to be careful with the order of cuts. The order problem leads us to speculate that in quantum field theory the sum of Feynman graphs which corresponds to an element of the Connes-Moscovici Hopf algebra could have a larger symmetry than the individual graphs.