Groups of tree-expanded series

In [Ch. Brouder, A. Frabetti, Renormalization of QED with planar binary trees, Eur. Phys. J. C 19 (2001) 715–741; Ch. Brouder, A. Frabetti, QED Hopf algebras on planar binary trees, J. Algebra 267 (2003) 298–322] we introduced three Hopf algebras on planar binary trees related to the renormalization of quantum electrodynamics. One of them, the algebra , is commutative, and is therefore the ring of coordinate functions of a proalgebraic group Gα. The other two algebras, and , are free non-commutative. Therefore their abelian quotients are the coordinate rings of two proalgebraic groups Ge and Gγ. In this paper we describe explicitly these groups. Using two monoidal structures and a set-operad structure on planar binary trees, we show that these groups can be realized on formal series expanded over trees, and that the group laws are generalizations of the multiplication and the composition of usual series in one variable. Therefore we obtain some new groups of invertible tree-expanded series and of tree-expanded formal diffeomorphisms respectively. The Hopf algebra describing the renormalization of the electric charge corresponds to the subgroup of tree-expanded formal diffeomorphisms formed of the translations, which fix the zero, by some particular tree-expanded series which remind the proper correlation functions in quantum field theory. In turn, the group of tree-expanded formal diffeomorphisms and some of its subgroups give rise to new Hopf algebras on trees. All the constructions are done in a general operad-theoretic setting, and then applied to the specific duplicial operad on trees.