Primitive elements in infinitesimal bialgebras.

For any set S, the underlying vector space of the non-symmetric operad A, spanned by a collection {·s}s∈S of binary products satisfying no relation, is the vector space K[PBTS] spanned by the set of all planar rooted binary trees with the internal nodes colored by the elements of S, graded by the number of leaves of a tree. We show that K[PBTS] has a coassociative coalgebra structure which satisfies the unital infinitesimal relation with all products ·s, s ∈ S, and that the coproduct satisfies a similar formula than the ones proved by M. Aguiar and F. Sottile in [3] and [4] for the Hopf algebras of permutations and of planar binary rooted trees, respectively. This formula defines the coproduct in terms of the Moebius basis for the Tamari order, and allows us to construct a basis of the subspace of primitive elements of K[PBTS]. Given a non-symmetric operad P defined by a family of binary products {·s}s∈S, the vector space K[PBTS] acts naturally on any P-algebra A. Whenever the algebra A is equipped with a coassociative coproduct satisfying the unital infinitesimal relation with all the products ·s, our basis acts on its subspace of primitive elements of A. We describe some examples in the last section of the manuscript.