Lie algebras associated to systems of Dyson-Schwinger equations

We consider systems of combinatorial Dyson-Schwinger equations in the Connes-Kreimer Hopf algebra H-I of rooted trees decorated by a set I. Let H-(S) be the subalgebra of H-I generated by the homogeneous components of the unique solution of this system. If it is a Hopf subalgebra, we describe it as the dual of the enveloping algebra of a Lie algebra g((S)) of one of the following types: 1. g((S)) is an associative algebra of paths associated to a certain oriented graph. 2. Or g((S)) is an iterated extension of the Faa di Bruno Lie algebra. 3. Or g((S)) is an iterated extension of an infinite-dimensional abelian Lie algebra. We also describe the character groups of H-(S). (C) 2010 Elsevier Inc. All rights reserved.