Normal Coordinates and Primitive Elements in the Hopf Algebra of Renormalization

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 Poincaré 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.

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

Cited by in corpus (13)

• Combinatorial Hopf algebras in quantum field theory I
• Integrable Renormalization II: the general case
• Hopf algebra approach to Feynman diagram calculations
• Rota-Baxter Algebras in Renormalization of Perturbative Quantum Field Theory
• Integrable Renormalization I: the Ladder Case
• Improved Epstein-Glaser renormalization in $x$-space versus differential renormalization
• The uses of Connes and Kreimer's algebraic formulation of renormalization theory
• Smooth rough paths, their geometry and algebraic renormalization
• From Hurwitz numbers to Feynman diagrams: counting rooted trees in log gravity
• Higher Order Measures, Generalized Quantum Mechanics and Hopf Algebras
• The Hopf Algebra of Renormalization, Normal Coordinates and Kontsevich Deformation Quantization
• Branched Itô formula and natural Itô-Stratonovich isomorphism
• Hopf Algebra Primitives in Perturbation Quantum Field Theory