Formulas for the Connes-Moscovici Hopf algebra

TL;DR

This paper provides explicit formulas for the coproduct and antipode in the Connes-Moscovici Hopf algebra by leveraging its isomorphisms with the Faà di Bruno and shuffle Hopf algebras, simplifying complex calculations.

Contribution

It introduces explicit formulas for key Hopf algebra operations in the Connes-Moscovici algebra using isomorphisms with well-understood algebraic structures.

Findings

Explicit formulas for coproduct and antipode in $ ext{H}_{ ext{CM}}$

Isomorphisms with Faà di Bruno and shuffle Hopf algebras

Simplified computations using algebraic isomorphisms

Abstract

We give explicit formulas for the coproduct and the antipode in the Connes-Moscovici Hopf algebra $\mathcal{H}_{\tmop{CM}}$. To do so, we first restrict ourselves to a sub-Hopf algebra $\mathcal{H}^1_{\tmop{CM}}$ containing the nontrivial elements, namely those for which the coproduct and the antipode are nontrivial. There are two ways to obtain explicit formulas. On one hand, the algebra $\mathcal{H}^1_{\tmop{CM}}$ is isomorphic to the Fa\`a di Bruno Hopf algebra of coordinates on the group of identity-tangent diffeomorphism and computations become easy using substitution automorphisms rather than diffeomorphisms. On the other hand, the algebra $\mathcal{H}^1_{\tmop{CM}}$ is isomorphic to a sub-Hopf algebra of the classical shuffle Hopf algebra which appears naturally in resummation theory, in the framework of formal and analytic conjugacy of vector fields. Using the very simple structure of the shuffle Hopf algebra, we derive once again explicit formulas for the coproduct and the antipode in $\mathcal{H}^1_{\tmop{CM}}$.