From left modules to algebras over an operad: application to combinatorial Hopf algebras

The purpose of this paper is two fold: we study the behaviour of the forgetful functor from S-modules to graded vector spaces in the context of algebras over an operad and derive from this theory the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for these Hopf algebras. Let O denote the forgetful functor from S-modules to graded vector spaces. Left modules over an operad P are treated as P-algebras in the category of S-modules. We generalize the results obtained by Patras and Reutenauer in the associative case to any operad P: the functor O sends P-algebras to P-algebras. If P is a Hopf operad then O sends Hopf P-algebras to Hopf P-algebras. If the operad P is regular one gets two different structures of Hopf P-algebras in the category of graded vector spaces. We develop the notion of unital infinitesimal P-bialgebra and prove freeness and cofreeness results for Hopf algebras built from Hopf operads. Finally, we prove that many combinatorial Hopf algebras arise from our theory, as Hopf algebras on the faces of the permutohedra and associahedra.Comment: Section 4.3 removed. To appear in Annales Math\'ematiques Blaise Pasca

The non-symmetric operad pre-Lie is free

We prove that the pre-Lie operad is a free non-symmetric operad.Comment: 12 pages. Details on the trees used added in definition 1.2. The proof of corollary 3.6 has been change

Non-formality of the Swiss-Cheese operad

In this note, we prove that the Swiss-cheese operad is not formal. We also give a criteria in terms of Massey operadic product for the non-formality of a topological operad.Comment: 10 pages, 4 figures. The section containing Massey operadic product has been modified. Accepted version for publication by the Journal of Topolog

Pre-Lie algebras and the rooted trees operad

A Pre-Lie algebra is a vector space L endowed with a bilinear product * : L \times L to L satisfying the relation (x*y)*z-x*(y*z)= (x*z)*y-x*(z*y), for all x,y,z in L. We give an explicit combinatorial description in terms of rooted trees of the operad associated to this type of algebras and prove that it is a Koszul operad.Comment: 13 pages, uses xypic, typos corrected and more explicit description of the free algebr

Lie theory for Hopf operads

The present article takes advantage of the properties of algebras in the category of S-modules (twisted algebras) to investigate further the fine algebraic structure of Hopf operads. We prove that any Hopf operad P carries naturally the structure of twisted Hopf P-algebra. Many properties of classical Hopf algebraic structures are then shown to be encapsulated in the twisted Hopf algebraic structure of the corresponding Hopf operad. In particular, various classical theorems of Lie theory relating Lie polynomials to words (i.e. elements of the tensor algebra) are lifted to arbitrary Hopf operads.Comment: 23 pages. Using xyPi

A combinatorial basis for the free Lie algebra of the labelled rooted trees

The pre-Lie operad can be realized as a space T of labelled rooted trees. A result of F. Chapoton shows that the pre-Lie operad is a free twisted Lie algebra. That is, the S-module T is obtained as the plethysm of the S-module Lie with an S-module F. In the context of species, we construct an explicit basis of F. This allows us to give a new proof of Chapoton's results. Moreover it permits us to show that F forms a sub nonsymmetric operad of the pre-Lie operad T.Comment: 12 pages, uses xypi