Lucas Delage (Potsdam)
It is well known that the grafting operator B_+, defined on the vector space of forest by gluing every tree of a forest to a common root, obeys the following universal property : for any commutative Hopf algebra H, and any endomorphism L of this Hopf algebra such that \Delta \circ L = L \otimes 1 + (Id \otimes L) \circ \Delta (with \Delta the coproduct of H), then there is a unique morphism of Hopf algebra \rho from the Connes Kreimer Hopf algebra to H such that L \circ \rho = \rho \circ B_+. Furthermore It was shown that one can define a Pre Lie algebra on rooted trees which is isomoprhic to the free Pre Lie Algebra. We will inspect the link between these two universal property.
