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.


Introduction
An S-module, also named symmetric sequence, is a graded vector space (V n ) n≥0 together with a right action of the symmetric group S n on V n for each n. The present paper is concerned with the study, in an operadic point of view, of the forgetful functor O from S-modules to graded vector spaces and its applications.
The category S-mod of S-modules is a tensor category. Motivated by the study of homotopy invariants, Barratt introduced the notion of twisted Lie algebras in [3], which are Lie algebras in the category of S-modules or -in the operad contextleft modules over the operad Lie. More precisely, a twisted Lie algebra is an S-module (L n ) n≥0 together with a bilinear operation [, ] satisfying, for any a ∈ L p , b ∈ L q and c ∈ L r the relations [a, [b, c]] + [c, [a, b]] · ζ p+q,r + [b, [c, a]] · ζ p,q+r = 0, where ζ p,q is the permutation of S p+q given by ζ p,q (i) = q + i if 1 ≤ i ≤ p and ζ p,q (i) = i − p if p + 1 ≤ i ≤ p + q. For instance the S-module (Lie(n)) n≥0 is a twisted Lie algebra for the bracket induced by the operadic composition Lie(2) ⊗ Lie(n) ⊗ Lie(m) → Lie(n + m).
There exists also a notion of twisted associative algebras-associative algebras in the tensor category S-mod-and a notion of twisted associative bialgebras -associative bialgebras in the category S-mod. Stover proved in [24] that a Cartier-Milnor-Moore theorem relates the categories of twisted associative bialgebras and twisted Lie algebras. Following an idea lying in [24], Patras and Reutenauer proved in [21] that two associative bialgebras arise naturally from a twisted associative bialgebra (A, m, ∆): the symmetrized bialgebraĀ = (A,m,∆) and the cosymmetrized bialgebraÂ = (A,m,∆). In [22] Patras and Schocker derived from this construction some known combinatorial Hopf algebras. The first part of this paper is the generalization of these constructions to any operad P. This generalization is performed in two steps: the first step will focus on the algebra constructions and the second step on the coalgebra constructions.
Given an operad P, the following notions are the same Twisted P-algebras (see e.g. [12]), Left modules over P (see e.g. [11]) and P-algebras in the category of S-modules. Our first question is the following one: given a P-algebra M in S-mod can one endow the graded vector space O(M ) with a P-algebra structure? This question comes from the observation that ⊕ n P(n) is a P-algebra in S-mod but is not a priori a P-algebra in the category of graded vector spaces. To convice the reader, one can look at the Lie case: if P = Lie then ⊕ n Lie(n) is a twisted Lie algebra, and the bracket is not anti-symmetric since the action of the symmetric group is not trivial. Our first theorem 2.3.1 states that if we apply a symmetrization to the twisted P-algebra structure on M then O(M ) is a P-algebra. If P = As we recover the definition of the symmetrized productm of Patras and Reutenauer. Our second theorem 2.4.3 states that another product can be defined if the operad P is regular, that is, P is obtained from a non-symmetric operad tensored by the regular representation of the symmetric group. Then in case P = As we recover the productm defined by Patras and Reutenauer.
The second step of the construction involves Hopf operads. 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. We develop the analogues of the constructions of Patras and Reutenauer. In theorem 3.1.3 to the operad As and also as the cosymmetrized Hopf algebra associated to the operad Zin defining Zinbiel algebras. It gives yet another proof for the Hopf algebra of Malvenuto and Reutenauer to be free and cofree independent from its self-duality. 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]. We deduce also some freeness results from our theory.
In the second case, we assume that there exists a Hopf multiplicative regular operad P such that H = ⊕ n P(n)/S n . Our theory implies that H is a Hopf P-algebra hence a Hopf algebra and is also a unital infinitesimal P-bialgebra. We prove that the Hopf algebra of planar trees described by Chapoton in [5], and the one of planar binary trees described by Loday and Ronco in [15] arise this way. As a byproduct we obtain freeness results for these Hopf algebras. Any permutation σ ∈ S n is written (σ 1 , . . . , σ n ) with σ i = σ(i). There is a natural injection → σ × τ = (σ 1 , . . . , σ n , τ 1 + n, . . . , τ m + n).
The standardisation of a sequence of distinct integers (a 1 , . . . , a p ) is the unique permutation σ ∈ S p following the conditions σ(i) < σ(j) ⇔ a i < a j , ∀i, j.

1.2.
Graded vector spaces and S-modules.
vector spaces A n indexed by the non-negative integers. One can define also A as A = ⊕ n A n . A map A → B of graded vector spaces is a collection of linear morphisms A n → B n . The category of graded vector spaces is denoted grVect.
An S-module M is a graded vector space together with a right S n -action M n ⊗ k[S n ] → M n for each n ≥ 0. A map M → N of S-modules is a collection M n → N n of morphisms of right S n -modules. The category of S-modules is denoted S-mod.
There is a forgetful functor O : S-mod → grVect which forgets the action of the symmetric group.

Tensor product.
The category grVect is a linear symmetric monoidal category with the following tensor product: The symmetry isomorphism τ : A ⊗ B → B ⊗ A is given by The symmetry isomorphism τ induces a left action of the symmetric group S k on A ⊗k , for A ∈ grVect.
The category S-mod is a linear symmetric monoidal category with the following tensor product: Since a (p, q)-shuffle is uniquely determined by an ordered partition I ⊔ J of [p + q], an element in (M ⊗ N )(p + q) can be written m ⊗ n ⊗ (I, J). For the sequel m ⊗ n denotes the element m ⊗ n ⊗ ([p], p + [q]) of (M ⊗ N )(p + q). The right action of the symmetric group is given by The unit for the tensor product is the S-module 1 given by For any σ ∈ S k , the symmetry isomorphism induces an isomorphism τ σ of S-modules from As a consequence τ σ induces a left S k -action on M ⊗k , for any S-module M.
When it is necessary to distinguish the tensor products, we write ⊗ g for the tensor product in grVect and ⊗ S for the one in S-mod.
The forgetful functor O does not preserve the tensor product. There are two natural transformations, When restricted to the full subcategory of vector spaces (the S-modules concentrated in degree 0), these two natural transformations restrict to the identity.

Endofunctors induced by an S-module.
1.3.1. Endofunctors in S-mod. The category of S-modules is endowed with another monoidal structure (which is not symmetric): the plethysm •.
where S k acts on the left on (N ⊗k ) by formula (2). The left and right unit for the plethysm is the S-module I given by Hence any S-module M defines a functor Endofunctors in grVect. For M ∈ S-mod and A ∈ grVect, one can use the same definition for the plethysm: where the tensor product A ⊗k is taken in grVect. Similarly any S-module M defines a functor Here is an example that emphasizes the fact that the two functors are different even if evaluated at the same underlying vector space. Consider the S-module Com(n) = k with the trivial S n -action. A vector space V is considered either as a graded vector space concentrated in degree 1 or as an S-module concentrated in degree 1. This gives where n i ∈ N (l i ) and (T 1 , . . . , T k ) is an ordered partition of [l 1 + . . . + l k ] with |T i | = l i , defines a natural transformation Since the sum is taken over all ordered partitions (T 1 , . . . , T k ), the image of m · σ ⊗ n 1 ⊗ . . . ⊗ n k is with |U i | = l σ −1 (i) , which is the image of m ⊗ n σ −1 (1) ⊗ . . . ⊗ n σ −1 (k) . As a consequence the map is well defined. The known formula yields the commutativity of the diagram.

Algebras over an operad
In this section we give the definitions of operads and algebras over an operad and we refer to Fresse [11] for more general theory on operads. We further state the main results of the section: the underlying graded vector space of an algebra over an operad P in S-mod is always a P-algebra and when P is regular there exists a second P-algebra structure.

2.1.
Operads. An operad is a monoid in the category of S-modules with respect to the plethysm. Namely, an operad is an S-module P together with a product µ P : P • P → P and a unit η P : I → P satisfying As a consequence the functors F P and F g P are monads in the category S-mod and grVect.
The product µ P is expressed in terms of maps called compositions which are morphisms of right S l 1 +...+ln -modules and which factors through the quotient by the action of the symmetric group S n .

2.2.
Algebras over an operad. Let C denotes either the category of Smodules or the category of graded vector spaces. For any S-module M , the functor F C M denotes the functor F M or F g M . Let P be an operad. A P-algebra or an algebra over P is an algebra over the monad F C P , that is an object M of C together with a product µ M : F C P (M ) → M such that the following diagrams commute: For p ∈ P(n) and m 1 , . . . , m n ∈ M the product µ M (p ⊗ m 1 ⊗ . . . ⊗ m n ) is usually written p(m 1 , . . . , m n ) ∈ M.
In the category of graded vector spaces one gets the usual definition of an algebra over an operad. In the category of S-modules, P-algebras are also -Left modules over P in the terminology of Fresse [11], -If P = As or P = Lie, twisted associative or twisted Lie algebras in the terminology of Barratt [3], -Twisted P-algebras in the terminology of Livernet and Patras [12].
In the sequel we dedicate the word twisted to the only case P = As: a twisted algebra is an algebra over the operad As in the category S-mod.
Any free P-algebra in the category C writes F C P (M ) for some M ∈ C. As a consequence P is the free P-algebra in S-mod generated by the S-module I.

2.3.
Relating P-algebras in S-mod and in grVect.
Proof. One has to prove the commutativity of the following two diagrams: The second diagram is commutative because ψ N is functorial in N , so The commutativity of the first diagram is a consequence of the computation As pointed out in section 2.2 any free P-algebra in S-mod is a P-algebra and satisfies the conditions of the theorem. Hence any free P-algebra in S-mod gives rise to a P-algebra in grVect. In particular the graded vector space ⊕ n≥0 P(n) is a P-algebra.

2.3.2.
Example. We apply formula (4) for the examples of the commutative operad and the associative operad.
The commutative operad Com is the trivial S n -module k for all n. Let e n be a generator of Com(n). The composition is since the number of (n, m)-shuffles is n+m n .
The associative operad was defined in section 2.1. The associative product on the space ⊕ n k[S n ] induced by the composition µ As is where σ ∈ S p and τ ∈ S q . This is the product defined by Malvenuto and Reutenauer in [18].

2.3.3.
Remark. Let P be an operad and M be a P-algebra in S-mod such that the action of S n on M (n) is trivial. There is another P-algebra structure on O(M ) given by since the formula (2) together with the trivial action imply the S n -invariance. If P = Com then k[X] is a commutative algebra for the product In characteristic 0 the two commutative products on k[X] are isomorphic but it is no more the case in characteristic p.

Regular operads.
In this section we prove that any algebra over a regular operad P gives rise to two structures of P-algebra on its underlying graded vector space. This is the generalization to operads of the result of Patras and Reutenauer in [21] in the associative case. Note that this generalization holds only for regular operads.

2.4.1.
Definition. The forgetful functor O : S-mod → grVect has a left adjoint, the symmetrization functor S : grVect → S-mod which associates to a graded vector space (V n ) n the S-module (V n ⊗ k[S n ]) n , where the action of the symmetric group is the right multiplication. An S-module M is regular if there exists a graded vector spaceM such that M = SM . For instance, the S-module I is regular, since I = SI. Let S-mod r be the subcategory of S-mod of regular modules (and regular morphisms). A regular operad P = SP is an operad in the category S-mod r , i.e. µ(ν 1 , . . . , ν k ) ∈P as soon as µ, ν i ∈P.
Indeed, there is also a plethysm in the category grVect: A non-symmetric operad is a monoid in the category grVect with respect to the plethysm • g . The operad SP is regular if and only ifP is a nonsymmetric operad.

2.4.2.
Proposition. Let M = SM be a regular module and N be an S-module.
It is straightforward to verify the statement with this definition of ψ r M . Adapting the proof of theorem 2.3.1 by using ψ r P in place of ψ P , we prove the following theorem: 2.4.3. Theorem. Let M ∈ S-mod be an algebra over a regular operad P. The graded vector space O(M ) is a P-algebra for the productμ O(M ) given by the composition that is, for all p ∈P(k) and m 1 , . . . , m k ∈ M , .
. Since any free P-algebra is a P-algebra, the theorem holds for any free Palgebra F P (M ). In particular the graded vector space ⊕ n≥0 P(n) is endowed with two structures of P-algebra.
2.5. Multiplicative operads. An operad P is multiplicative if there exists a morphism of operads As → P. Any algebra in S-mod over a multiplicative operad P is a twisted algebra and thus its underlying graded vector space is endowed with two associative products. It holds in particular for the graded vector space ⊕ n P(n). For instance Com is a multiplicative operad and we recover the two associative (and commutative) structures found in example 2.3.2 and remark 2.3.3.

Hopf algebras over a Hopf operad
In this section, we generalize the results of Patras and Reutenauer in [21] obtained in the associative case to any Hopf operad. We prove that any Hopf P-algebra in S-mod yields a Hopf P-algebra in grVect and two Hopf P-algebras in grVect if P is regular.

3.1.
Hopf operads-the general case. From now on C is either the category grVect or the category S-mod. The definitions and propositions related to Hopf operads and Hopf algebras over a Hopf operad can be found in [12]. Here we recall only what is needed for our purpose.
3.1.1. Definition. Let Coalg be the category of coassociative counital coalgebras, that is vector spaces V endowed with a coassociative coproduct ∆ : V → V ⊗ V and a counit ǫ : V → k. A Hopf operad P is an operad in Coalg, i.e. µ P and η P are morphisms of coassociative counital coalgebras. A Hopf operad amounts to the following data: for each n a coproduct δ(n) : P(n) → P(n) ⊗ P(n) and a counit ǫ(n) : P(n) → k preserving the operadic composition and the action of the symmetric group . We use Sweedler's notation, that is, One has maps in S-mod and in grVect and As a consequence if M and N are P-algebras in the category C, then M ⊗ N is a P-algebra for the following product which are morphisms of P-algebras. For P = As, a Hopf As-algebra is named a twisted bialgebra.

3.1.3.
Theorem. The underlying graded vector space of any Hopf P-algebra M in S-mod is a Hopf P-algebra in grVect. More precisely, the P-algebra product on O(M ) isμ O(M ) and the coproduct is a morphism of P-algebras. This Hopf P-algebra is denotedM and named the symmetrized Hopf P-algebra associated to M .
In particular, if for m ∈ M (p) one writes Proof. One has to prove that the following diagram is commutative: The functoriality and naturality of π O and ψ P implȳ . Therefore, the commutativity of the previous diagram follows from the commutativity of the following diagram where the sum is taken over all ordered partitions of [l 1 + r 1 + . . .
where the sum is taken over all ordered partitions of [l 1 +r 1 +.
By the definition of τ (see (8)), where For P = As one recovers the definition given in section 1.1 for the symmetric group.
Recall from [12] that if P is a connected Hopf operad then P is a Hopf P-algebra in S-mod for the coproduct given by Indeed the map ∆ is the unique P-algebra morphism such that ∆( It happens in many examples that P is not connected and P(0) = 0. Nevertheless it is sometimes possible to define a P-algebra structure on (P + ⊗ P + ) − where P + (0) = k P + (n) = P(n), n > 0 and (P + ⊗ P + ) − (0) = 0 (P + ⊗ P + ) − (n) = (P + ⊗ P + )(n), n > 0.
If P is connected, one has two examples of Hopf P-algebras in grVect: • F g P (V ) for V in grVect where the product is given by F g µ P (V ) and the coproduct is given by F g ∆ (V ). • For any S-module M the free P-algebra F P (M ) is a Hopf P-algebra in S-mod. The symmetrized Hopf P-algebra F P (M ) is a Hopf Palgebra in grVect by theorem 3.1.3.
3.2. Regular Hopf operads. Let P = SP be a regular operad. Assume (P, δ) is a Hopf operad. The operad P is a regular Hopf operad if δ(P) ⊂P ⊗ P. For instance As is a regular Hopf operad. We prove that for any regular Hopf operad a Hopf P-algebra in the category S-mod gives rise to two structures of Hopf P-algebra in the category grVect. This is a generalization to regular operads of a theorem announced by Stover in [24] and proved by Patras and Reutenauer in [21] in the context of twisted bialgebras. Note that the hypothesis regular is needed to obtain two structures of Hopf P-algebras.
Proof. The proof is similar to the proof of theorem 3.1.3. The commutativity of the diagram is a consequence of the commutativity of the diagram and st(U 1 , . . . , U k ) = Id and st(V 1 , . . . , V k ) = Id .
Thus R(X) = L(X), ∀X and the diagram is commutative.
As a consequence, if P is a regular Hopf operad then any Hopf P-algebra M in S-mod gives rise to two structures of Hopf P-algebra in grVect. In particular this result holds for ⊕ n P(n) and for the underlying graded vector space of any free P-algebra.

Application to multiplicative Hopf operads.
In a first step we establish that the corresponding Hopf structures in case P = As coincide with the ones discovered by Stover [24] and proved by Patras and Reutenauer in [21]. In a second step we apply the above results to multiplicative Hopf operads.
3.3.1. The associative case. Recall that the operad As is a regular Hopf operad. Hence the underlying graded vector space of a twisted bialgebra is endowed with two structures of Hopf algebra. Let M be a twisted bialgebra with product m and coproduct ∆.
, which is the symmetrized bialgebra associated to the twisted bialgebra M as in [21, proposition 15]. which is the cosymmetrized bialgebra associated to the twisted bialgebra M as in [21, definition 8].
A multiplicative Hopf operad is a Hopf operad P together with an operad morphism As → P which commutes with the Hopf structure. As a consequence any Hopf P-algebra is a Hopf As-algebra. The result below is a consequence of the previous sections.

3.3.2.
Corollary. Let P be a multiplicative Hopf operad. The underlying graded vector space of any Hopf P-algebra is endowed with two different structures of Hopf algebra.

Unital infinitesimal P-bialgebras
In this section we give some comparison between∆ andμ when the operad is regular, in view of generalizing the theory of unital infinitesimal bialgebra developed by Loday and Ronco in [17]. This yields the definition of unital infinitesimal P-bialgebras. As a consequence we prove that any Hopf algebra over a multiplicative Hopf operad is isomorphic to a cofree coassociative algebra. Moreover, if P is regular then this isomorphism respects the Palgebra structure. We study the associative case in detail.
From now on a connected Hopf operad P is given.

Unital infinitesimal P-bialgebras.
In this section, We prove that the underlying graded vector space of a Hopf P-algebra is a unital infinitesimal P-bialgebra (theorem 4.1.2). We prove also in theorem 4.1.3 that the same result holds for F g P (V ) when V is a graded vector space such that V (0) = 0.
A connected coalgebra M in S-mod or grVect is a coalgebra such that M (0) = k and such that the counit ǫ : k = M (0) → k is the identity isomorphism. That is for M ∈ S-mod the coproduct writes µ(m 1 ⊗ 1, . . . , m j ⊗ 1, 1 ⊗ m j+1 , . . . , 1 ⊗ m k ), (12) for µ ∈P(k). Note that the operad needs to be regular since the infinitesimal relation is not S k -equivariant.
For instance if P = As, a unital infinitesimal As-bialgebra is the definition of Loday and Ronco in [17] of a unital infinitesimal bialgebra since the previous relation amounts to Let M be a Hopf P-algebra in S-mod with P regular. Theorems 3.1.3 and 3.2.1 assert that the underlying graded vector space of M is endowed with two structures of Hopf P-algebras in grVect. One is given by (μ,∆) and the other one by (μ,∆). The next theorem explores the relation between µ and∆.   (2) = 1⊗ m. As a consequence , m j+1 , . . . , m k ) On the other hand let us compute the right hand side of the equation (12): .
Thus the left and right hand sides of the equation (12) are equal and the theorem is proved.

4.1.3.
Theorem. Let P = SP be a connected regular Hopf operad. Let V be a graded vector space with V (0) = 0. The free P-algebra in grVect F g P (V ) is a unital infinitesimal P-bialgebra. The product is given by the usual product on free P-algebras and the coproduct is given for Proof. When P is regular F g P (V ) = ⊕ nP (n) ⊗ V ⊗gn , hence it is enough to prove the formula (12) for∆ µ(ν 1 ⊗v 1 , . . . , ν k ⊗v k ) with µ, ν i ∈P andv i ∈ V ⊗l i . The computation is straightforward.

4.2.
Rigidity for twisted bialgebras. Loday and Ronco proved a theorem of rigidity for unital infinitesimal bialgebras. Recall from [17] that the fundamental example of a unital infinitesimal bialgebra is given by where V is a graded vector space concentrated in degree 1 and where the product is given by the concatenation and the coproduct is given by the deconcatenation. Recall also that for a connected coalgebra C, with a coproduct ∆ and a counit ǫ, the space of primitive elements is defined by Here is the statement of the theorem: 4.2.1. Theorem [17]. Any connected unital infinitesimal Hopf bialgebra H is isomorphic to T f c (Prim(H)).  (Prim∆(A)). HenceÂ is a free associative algebra andĀ is a cofree coassociative coalgebra. Assume furthermore that k is of characteristic 0 and ∆ is cocommutative. ThenÂ is a cocommutative Hopf algebra, and by the theorem of Cartier-Milnor-Moore, it is the universal enveloping algebra of its primitive elements. If each A n is finite dimensional, sinceÂ is free as an associative algebra, by lemma 22 in [21] the space of primitive elements is a free Lie algebra.
These results are summed up in the following theorem:

4.2.2.
Theorem. Let (A, m, ∆) be a connected twisted bialgebra. The associated symmetrized bialgebraĀ is a cofree coassociative algebra. The associated cosymmetrized bialgebraÂ is a free associative algebra. If k is of characteristic 0, if ∆ is cocommutative and if A n is finite dimensional for all n, there is an isomorphism of Lie algebras

This isomorphism is functorial in A.
Using the results of Loday and Ronco we have improved the results of Patras and Reutenauer. Furthermore, if P is a connected multiplicative Hopf operad then it provides connected twisted bialgebras: indeed, any Hopf P-algebra in S-mod is a twisted bialgebra. For instance P and more generally F P (M ) with M an S-module such that M (0) = 0 are connected twisted bialgebras. Remark. If (A, m, ∆) is a connected twisted bialgebra then (A,m,m,∆) is a connected 2-associative bialgebra in the terminology of Loday and Ronco in [17], that is (A,m,∆) is a Hopf algebra and (A,m,∆) is a unital infinitesimal bialgebra. By the structure theorem in [17], one gets that Prim∆(A) is a B ∞ -algebra and A is the enveloping 2-as bialgebra of its primitive elements.

4.2.3.
Assume P and V satisfy the conditions of theorem 4.1.3. Assume P is multiplicative and A = F g P (V ) is finite dimentional in each degree. Then A2 = (A * , t ∆, t∆ , t m) where m is the associative product induced by the multiplicative structure of P is also a 2-associative bialgebra. If A2 is connected then it is the enveloping 2-as bialgebra of its primitive elements.

Application to combinatorial Hopf algebras
In this section, we would like to apply our previous results to combinatorial Hopf algebras. The idea is the following: given a graded vector space H = ⊕ n H(n), how does a Hopf algebra structure arise on H? We present two cases coming from the two examples detailed in section 3.1.4. Case 1. The space H(n) is endowed with a right S n -action. We denote by H S the associated S-module. Assume there exists a connected multiplicative Hopf operad structure on P H = H S . From section 3.1.4, we obtain our first result: there exists a P H -algebra product µ and a coalgebra coproduct ∆ such that (H S , µ, ∆) is a Hopf P H -algebra. The graded vector space H has a Hopf P H -algebra structure which is the symmetrized Hopf P H -algebra H S by theorem 3.1.3.
The second result is a direct consequence of theorem 4.2.2: since the operad P H is multiplicative, there is a twisted product m : H S ⊗ H S → H S . As a consequence (H S , m, ∆) is a twisted bialgebra. The associated symmetrized Hopf algebra (H,m,∆) is cofree and the associated cosymmetrized Hopf algebraH = (H,m,∆) is free. In case ∆ is cocommutative, under the hypothesis of theorem 4.2.2,H is the enveloping algebra of the free Lie algebra generated by Prim∆(H). Furthermore, by remark 4.2.3 the 2-associative bialgebra (H,m,m,∆) is the 2-associative enveloping bialgebra of its primitive elements: Prim∆(H) is endowed with a B ∞ -structure.
Case 1 applies also when H is a free P-algebra in S-mod generated by an S-module M , with P a multiplicative Hopf operad and M (0) = 0.
Case 2. Assume P r H = SH is a connected regular Hopf operad. The graded vector space H is the free graded P r H -algebra generated by the graded vector space I. As a consequence, the graded vector space (H, µ, ∆) is a Hopf P r Halgebra, where µ is the P r H -product and where comes from the regular Hopf operad P r H . Also (H, µ,∆) with is a unital infinitesimal P r H -bialgebra by theorem 4.1.3. Again, by remark 4.2.3, if H n is finite dimensional, then (H * , t ∆, t∆ , t m) in which m is the associative product, is a 2-associative bialgebra, which is the 2-associative enveloping bialgebra of its primitive elements.
Case 2 applies also when H is a free P-algebra in grVect generated by a graded vector space V , with P a Hopf regular operad and V (0) = 0.
We illustrate by some examples that many combinatorial Hopf algebras arise either from case 1 or from case 2.

The Hopf algebra T (V ). Let us apply
where V is considered as an S-module concentrated in degree 1. That is H = T (V ). As a twisted bialgebra, T (V ) is endowed with the concatenation product and with the following coproduct It is cocommutative. The symmetrized Hopf algebra structure on T (V ) is the shuffle product together with the deconcatenation, whereas the cosymmetrized Hopf algebra structure on T (V ) is the dual structure: the product is the concatenation and the coproduct is the unshuffle coproduct. In characteristic 0 it is the enveloping algebra of the free Lie algebra generated by V .

The Malvenuto-Reutenauer
Hopf algebra. This Hopf algebra, denoted H M R has been extensively studied in [18], in [9] under the name of free quasisymmetric functions or in [1]. The graded vector space considered is A = ⊕ n k[S n ]. It is the underlying graded vector space of the operad As and Case 1 applies. Recall that the operad As gives rise to a cocommutative twisted bialgebra: The Hopf algebra H M R is the symmetrized Hopf algebra (A,m,∆). That is, for σ ∈ S n , τ ∈ S m m(σ, τ ) = ξ∈Shp,q (σ × τ ) · ξ, which is not commutative nor cocommutative.
The cosymmetrized Hopf algebraÂ = (A,m,∆) is given bȳ The latter Hopf algebra is different from the former one or its dual since it is a cocommutative Hopf algebra. From Case 1 we get that H M R is cofree and thatÂ is free as an associative algebra: it is generated by the connected permutations, the ones which don't write σ ×τ for σ ∈ S n , τ ∈ S m , n, m > 0. In characteristic 0,Â is isomorphic to the enveloping algebra of the free Lie algebra generated by the connected permutations (compare with theorems 20 and 21 in [21]). Furthermore, H M R together withm is a 2-associative bialgebra and it is isomorphic to the 2associative enveloping bialgebra generated by the connected permutations: in [9] and in [1] a basis of the space of primitive elements of H M R , indexed by the connected permutations is given.
In paragraph 5.3.4 we prove that H M R is free as an associative algebra, without using the self-duality of H M R .

Hopf algebra structures on the faces of the permutohedron.
Recall that Com is a Hopf operad. Let Com(n) = Com(n) if n > 0 0 if n = 0 .
Chapoton described some Hopf algebra structures on the graded vector space O(Comp) in [5] and in [6], whereas Patras and Schocker described a twisted bialgebra structure on Comp in [22]. Chapoton described a (differential graded) operad structure on Comp in [8] and Loday described a (filtered) one in [14]. The aim of this section is to apply our operadic point of view Case 1 in order to relate these structures.

5.3.1.
Operad structures on the faces of the permutohedron. Both operads built by Loday in [14] and Chapoton in [8] are quadratic binary operads. They are generated by the commutative operation represented by the set composition (12) and by the operation represented by the set composition (1,2) in Comp (2). Let w f = (12)+(1, 2)+(2, 1) and w g = (1, 2)+(2, 1). The composition in the operad CTD described by Loday is given by the following inductive formula The operad CTD is filtered by the degree of set compositions but not graded. It is not regular and algebras (in the category of vector spaces) over this operad are named commutative tridendriform algebras by Loday, that is vector spaces endowed with a product ≺ and a commutative product · satisfying the relations The composition in the operad Π described by Chapoton has the same definition except that w f is replaced by w g . It is graded by the degree of set compositions. Algebras (in the category of graded vector spaces) over Π are described in [8].
These operads are not connected in the strict sense, since the composition with ∅ ∈ Comp(0) is not always defined. The equalities involving the emptyset above, are needed for an inductive definition and are needed in order to build a coproduct Comp → Comp ⊗ Comp, as was explained in the paragraph 3.1.4 on connected operads.
The S-module Comp is a Π-Hopf algebra for the same coproduct.
Proof. Let X denote either the operad CTD or the operad Π. Let w denote either w f ∈ CTD(2) or w g ∈ Π(2). Note that for any set composition P w(P, ∅) = w(∅, P ) = P.
One can choose for σ the shuffle associated to the set composition P. The coproduct ∆ is a morphism of S-modules. One gets the conclusion with formula (1).
In [12] we proved that the space of primitive elements with respect to ∆ is a suboperad of the initial operad. The space of primitive elements is clearly the vector space generated by the set compositions (n), for n > 0. Then Prim ∆ (CTD) is the operad Com (compare with [14]).

5.3.3.
Twisted bialgebras associated to the faces of the permutohedron. The operation w f (resp. w g ) is associative and commutative. As a consequence, the operads CTD and Π are Hopf multiplicative operads and give rise to twisted connected commutative (non cocommutative) bialgebras H f = (Comp, w f , ∆) and H g = (Comp, w g , ∆).
i) The twisted bialgebra H f . Patras and Schocker [22] defined a twisted bialgebra structure on Comp denoted T = (Comp, ⋆, δ) which is the following. The product * is the concatenation of set compositions and the coproduct δ is defined for a set composition P of [n] by The dual of the twisted bialgebra defined by Patras and Schocker is the free commutative tridendriform algebra on one generator in the category S-mod.
Applying Case 1 one gets that the symmetrized Hopf algebraT associated to T is cofree, and that the associated cosymmetrized Hopf algebraT is free generated by reduced set compositions: a set composition which is non reduced is the concatenation of two non trivial set compositions. For instance (13,24,6,5) is non reduced since it is the concatenation of (13,24) and (2,1). Moreover if the field k is of characteristic 0, thenT is isomorphic to the enveloping algebra of the free Lie algebra generated by reduced set compositions. (Compare with proposition 10 and corollary 13 in [22]). Applying remark 4.2.3 one gets that (T ,⋆,⋆,δ) is the 2-associative enveloping bialgebra on its primitive elements.
The Hopf algebra structure given by Chapoton in [5] is (Comp,w f ,∆) which is the dual of the symmetrized Hopf algebra (T ,⋆,δ). It is also the Hopf algebra N CQSym of Bergeron et Zabrocky in [4] and we recover that it is a free algebra.
ii) The twisted bialgebra H g . This twisted bialgebra gives rise to two Hopf algebras, which are (Comp,ŵ g ,∆) and (Comp,w g ,∆). One can check that the latter Hopf algebra is the one described by Chapoton in [6]. Again it is a free associative algebra because (Comp,w g ,∆) is a unital infinitesimal bialgebra. The space of primitive elements Prim∆(Comp) is generated by reduced set compositions. One can check by an inductive argument, that the Hopf algebra described by Chapoton is a free associative algebra generated by reduced set compositions.

5.3.4.
From set compositions to permutations. Let Comp 0 be the sub Smodule of Comp of set compositions of degree 0. The vector space Comp 0 (n) is isomorphic to k[S n ] but the right S n -action is given by σ · τ = τ −1 σ. The S-module Comp 0 is a sub-operad of Π. It is the operad Zin, as noticed by Chapoton in [8]. In the category of vector spaces, an algebra over Zin is a Zinbiel algebra, that is, a vector space Z together with a product ≺ satisfying the relation As a consequence there are surjective morphisms of Hopf operads The operad Zin is consequently a multiplicative operad and Comp 0 is a commutative twisted bialgebra. The product and coproduct are given, for σ ∈ Comp 0 (p) and τ ∈ Comp 0 (q) by The morphisms above induce surjective morphisms of twisted bialgebras The cosymmetrized Hopf algebra associated to Comp 0 is clearly H M R , and since it is a cosymmetrized algebra associated to a twisted bialgebra it is free on Prim∆ Z (Comp 0 ). But∆ Z (σ) = ρ×τ =σ ρ ⊗ τ. As a consequence, H M R is free generated by the connected permutations and cofree (see section 5.2). We recover the results obtained in e.g. [23], [9] and [1].
Considering the graded linear duals, one has an embedding of cocommutative twisted bialgebras The symmetrized Hopf algebra associated to (Comp 0 ) * is the dual of the Malvenuto-Reutenauer Hopf algebra H * M R (which is isomorphic to H M R ). By functoriality in theorem 4.2.2, we obtain in characteristic 0 an embedding of enveloping algebras at the level of associated cosymmetrized algebras (compare with Theorem 17 in [22]).

5.4.
Hopf algebra structures on the faces of the associahedron. In his thesis, Chapoton considered various Hopf algebra structures on the faces of the associahedra, or Stasheff polytopes, filtered in [5], graded in [6]. He considered also filtered and graded operad structures on these objects in [7]. The filtered operadic structure coincides with the one defined by Loday and Ronco in [16], under the name of tridendriform operad. In this section, we apply Case 2 to obtain Hopf algebra structures on the faces of the associahedra. 5.4.1. Planar trees. The set of planar trees with n+1 leaves is denoted by T n . The set T n = ∪ n−1 k=0 T n,k is graded by k where n − k is the number of internal vertices. For instance T n,0 is the set of planar binary trees. The Stasheff polytope of dimension n − 1 has its faces of dimension 0 ≤ k ≤ n − 1 indexed by T n,k . The aim of this section is to provide the vector space ⊕ k[T n ] with Hopf structures.
Given some planar trees t 1 , . . . , t k the planar tree ∨(t 1 , . . . , t k ) is the one obtained by joining the roots of the trees t 1 , . . . , t k to an extra root, from left to right. If t i has degree l i then ∨(t 1 , . . . , t k ) has degree l 1 + . . . + l k + k − 1.
One can label the n sectors delimited by a tree t in T n from left to right as in the following example:
As a consequence a basis of Prim∆(T ree) is given by the planar trees of type ∨(t 1 , . . . , t k−1 , |).
The Hopf structure defined by Chapoton in [5] is essentially the same: the product is the same and the coproduct is τ ∆, where τ is the symmetry isomorphism. Hence it is a free associative algebra spanned by the set of trees of the form ∨(|, t 2 , . . . , t k ).
The graded linear dual of T ree is a 2-associative bialgebra: it is free as an associative algebra for the product t∆ , cofree as a coalgebra for t * and it is the enveloping 2-as bialgebra of its primitive elements (with respect to t * ). The product of two trees t∆ (t, s) is the tree obtained by gluing the tree s on the right most leave of t. it is usually denoted t\s. For instance \ = .
5.4.5. Some operad morphisms. There are morphisms of Hopf operads T riDend The vertical maps are projection onto cells of degree 0. The map π CTD has been explained in paragraph 5.3.4. The map π T riDend is the projection onto the dendriform operad, which is a regular operad generated by planar binary trees (see e.g. [15]). The morphism ψ sends ≺ to the set composition (1, 2), ≻ to the set composition (2, 1) and · to the set composition (12). Indeed we can describe ψ at the level of trees. There is a map φ from set compositions to trees, described by induction as follows. Let P = (P 1 , . . . , P k ) be a set composition of [n]. If P 1 = {l 1 < . . . < l j } then it splits [n] into j + 1 intervals I s possibly empty: for 0 ≤ s ≤ j, I s =]l s , l s+1 [ with l 0 = 0 and l j+1 = n + 1. The map φ is defined by φ(∅) = |, φ(P ) = ∨(φ(P ∩ I 0 ), . . . , φ(P ∩ I j )).

5.4.7.
Conclusion. For the last decade, many results of freeness and cofreeness of combinatorial Hopf algebras have appeared in the litterature (see the references cited throughout the paper and recently [2], [10], [19]). The present paper illustrates that these freeness results are a consequence of an operadic structure on the Hopf algebra H itself or its symmetrization SH. Namely, either the Hopf algebra H is an S-module and one can find an operad structure on H in order to apply Case 1; or the Hopf algebra is not an S-module and one can find an operad structure on SH in order to apply Case 2.