A nondiagrammatic description of the Connes–Kreimer Hopf algebra

We demonstrate that the fundamental algebraic structure underlying the Connes–Kreimer Hopf algebra–the insertion pre-Lie structure on graphs–corresponds directly to the canonical pre-Lie structure of polynomial vector fields. Using this fact, we construct a Hopf algebra built from tensors that is isomorphic to a version of the Connes–Kreimer Hopf algebra that first appeared in the perturbative renormalization of quantum field theories.