Abstract
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 the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for those 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 the functor O sends Hopf P-algebras to Hopf P-algebras. If the operad P is regular one gets two dierent structures of Hopf P-algebras in the category of graded vector spaces. We develop the notion of unital infinitesimal P-bialgebras 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 it is the case for various Hopf algebras defined on the faces of the permutohedra and associahedra.
Highlights
An S-module, named symmetric sequence, is a graded vector space (Vn)n≥0 together with a right action of the symmetric group Sn on Vn for each n
We prove that Hopf algebra structures on the faces of the permutohedra given e.g. by Chapoton in [5], Bergeron and Zabrocky in [4] and Patras and Schocker in [22], arise from the operad CTD of commutative tridendriform algebras defined by Loday in [14]
As a consequence we prove that any Hopf algebra over a multiplicative Hopf operad is isomorphic to a cofree coassociative algebra
Summary
An S-module, named symmetric sequence, is a graded vector space (Vn)n≥0 together with a right action of the symmetric group Sn on Vn for each n. We define the notion of Hopf P-algebras in the category S-mod, so that in case P = As we recover the notion of twisted associative bialgebras. It is shown in theorem 4.1.3 that the graded vector space ⊕nP(n)/Sn has a structure of unital infinitesimal P-bialgebra These results combined with the theorem of Loday and Ronco (see 4.2.1) yield the main theorems of our paper, which have some importance in the study of combinatorial Hopf algebras. We prove that such a structure always exists and when the operad is regular one has an additional structure We compare this result to the one obtained by Patras and Reutenauer [21] in the case of twisted associative algebras. Throughout the paper, the ground field is denoted by k and all vector spaces are k-vector spaces
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