Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson–Schwinger equations

Abstract

We consider the combinatorial Dyson–Schwinger equation X = B + ( P ( X ) ) in the non-commutative Connes–Kreimer Hopf algebra of planar rooted trees H NCK , where B + is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra A N , P of H NCK . We describe all the formal series P such that A N , P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of H NCK , organized into three isomorphism classes: a first one, restricted to a polynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions; a last (infinite) one, which gives a non-commutative version of the Faà di Bruno Hopf algebra. By taking the quotient, the last class gives an infinite set of embeddings of the Faà di Bruno algebra into the Connes–Kreimer Hopf algebra of rooted trees. Moreover, we give an embedding of the free Faà di Bruno Hopf algebra on D variables into a Hopf algebra of decorated rooted trees, together with a non-commutative version of this embedding.