Colored trees and noncommutative symmetric functions

Let CRFS denote the category of S-colored rooted forests,and HCRFS denote its Ringel-Hall algebra as introduced in [6]. We construct a homomorphism from a K-0(+) (CRFS)-graded version of the Hopf algebra of non commutative symmetric functions to HCRFS. Dualizing,we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a K-0(+) (CRFS)-graded version of the algebra of quasisymmetric functions. This homomorphism is are finement of one considered by W.Zhao in [9].