Normal Coordinates and Primitive Elements in the Hopf Algebra of Renormalization

TL;DR

This paper introduces normal coordinates on the Connes-Kreimer group, studies primitive elements in the Hopf algebra of rooted trees, and relates ladder primitives to Schur polynomials, enhancing understanding of renormalization structures.

Contribution

It provides new descriptions of primitive elements using inverse Poincaré lemma and relates ladder primitives to Schur polynomials, advancing the algebraic understanding of renormalization.

Findings

Primitive elements generated by inverse Poincaré lemma

Ladder primitives correspond to Schur polynomials

Lower central series relates to $k$-primitiveness

Abstract

We introduce normal coordinates on the infinite dimensional group $G$ introduced by Connes and Kreimer in their analysis of the Hopf algebra of rooted trees. We study the primitive elements of the algebra and show that they are generated by a simple application of the inverse Poincar\'e lemma, given a closed left invariant 1-form on $G$. For the special case of the ladder primitives, we find a second description that relates them to the Hopf algebra of functionals on power series with the usual product. Either approach shows that the ladder primitives are given by the Schur polynomials. The relevance of the lower central series of the dual Lie algebra in the process of renormalization is also discussed, leading to a natural concept of $k$-primitiveness, which is shown to be equivalent to the one already in the literature.