Nonsymmetric Operads

Abstract

The natural Hopf algebra N • O of an operad O is a Hopf algebra whose bases are indexed by some words on O. We introduce new bases of these Hopf algebras deriving from free operads via new lattice structures on their basis elements. We construct polynomial realizations of N • O by using alphabets of variables endowed with unary and binary relations. By specializing our polynomial realizations, we discover links between N • O and the Hopf algebra of word quasi-symmetric functions of Hivert, the decorated versions of the Connes-Kreimer Hopf algebra of Foissy, the Faà di Bruno Hopf algebra and its deformations, and the noncommutative multi-symmetric functions Hopf algebras of Novelli and Thibon.