Normal Coordinates and Primitive Elements in the Hopf Algebra of Renormalization

Details Normal Coordinates and Primitive Elements in the Hopf Algebra of Renormalization Normal Coordinates and Primitive Elements in the Hopf Algebra of Renormalization

2001-05-25

arxiv.org/abs/hep-th/0105259v1

Commun.Math.Phys. 225 (2002) 465-485

Physics

High Energy Physics

High Energy Physics - Theory

Latex, 24 pages. Submitted to Commun. Math. Phys

Scientific paper

10.1007/s002200100588

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.

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