Loïc Foissy
email: foissy@univ-littoral.fr

Bialgebras in cointeraction, the antipode and the eulerian idempotent

Keywords: Double bialgebra, monoid of characters, eulerian idempotent. AMS classication. 16T05 16T30 68R15 05A05 05C15

Bialgebras in cointeraction, the antipode and the eulerian idempotent

Introduction

Quite recently, various combinatorial Hopf algebras equipped with a second coproduct appear in the literature: some based on trees [START_REF] Calaque | Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series[END_REF], on several families of graphs [START_REF] Manchon | On bialgebras and Hopf algebras or oriented graphs[END_REF][START_REF]Chromatic polynomials and bialgebras of graphs[END_REF], on nite topologies or posets [START_REF] Mohamed | Doubling bialgebras of nite topologies[END_REF][START_REF] Foissy | Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials[END_REF][START_REF] Fauvet | The Hopf algebra of nite topologies and mould composition[END_REF], noncrossing partitions [START_REF] Ebrahimi-Fard | Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations[END_REF], or on words related to Ecalle's mould calculus [START_REF] Ebrahimi-Fard | A comodule-bialgebra structure for word-series substitution and mould composition[END_REF], for example. These objects play an important role in Bruned, Hairer and Zambotti's study of stochastic PDEs [START_REF] Bruned | Singular KPZ Type Equations[END_REF][START_REF] Bruned | Algebraic renormalisation of regularity structures[END_REF]. Let us give some common properties of these objects. These are families pB, m, ∆, δq such that: pB, m, ∆q is a bialgebra. In most cases, it is a graded and connected Hopf algebra. pB, m, δq is a bialgebra, sharing the same product as pB, m, ∆q. It is generally not a connected coalgebra, as it contains non trivial group-like elements. Moreover, as these elements are not invertible, this is generally not a Hopf algebra.

It turns out that pB, m, ∆q is a bialgebra in the category of right comodules of pB, m, δq, with the right coaction given by δ itself: the product m, the coproduct ∆, the unit map ν and the counit ε ∆ of ∆ are comodule morphisms. It is rather trivial for m and ν, but gives the two interesting following relations for ∆ and its counit ε ∆ :

p∆ Idq m 1,3,24 ¥ pδ δq ¥ ∆, pε ∆ Idq ¥ δ ν ¥ ε ∆ ,
where m 1,3,24 : B 4 ÝÑ B 3 send the tensor a 1 a 2 a 3 a 4 to a 1 a 3 a 2 a 4 . In other words, for such an object, pB, m, ∆q is a right-comodule bialgebra over pB, m, δq, that is to say a bialgebra in the symmetric monoidal category of right comodules over pB, m, δq. For the sake of simplicity, these objects will be called double bialgebras in this text. Considering the associated characters monoids, we obtain two products ¦ and on the same set CharpBq, coming respectively from ∆ and δ, such that: pCharpBq, ¦q is a monoid (in most cases a group). pCharpBq, q is a monoid. pCharpBq, q acts (on the right) on pCharpBq, ¦q by monoid endomorphisms: for any λ 1 , λ 2 and µ CharpBq, pλ 1 ¦ λ 2 q µ pλ 1 µq ¦ pλ 2 µq.

In the particular case where ∆ and δ are cocommutative, we obtain that pCharpBq, ¦, q is in fact a ring.

Our aim in this text is a review of the theoretical consequences of this setting, illustrated by examples based on words with quasishue products and on graphs, with an unexpected application to the eulerian idempotent. We start with general results, with no particular hypothesis on the structure of pB, m, ∆q. We show that, as mentioned before, the monoid of characters pCharpBq, q of pB, m, δq acts on the monoid of characters pCharpBq, ¦q of pB, m, ∆q, but also on the space HompB, V q of linear homomorphisms from B to any vector space V (Proposition 2.5).

If V is an algebra (respectively a bialgebra or a coalgebra), the subset of algebra (respectively bialgebra or coalgebra) morphisms is stable under this action. We also prove that, in the case where pB, m, ∆q is a Hopf algebra, then its antipode S is automatically a comodule morphism (Proposition 2.1), that is to say: δ ¥ S pS Idq ¥ δ.

We also introduce an important tool, the map Θ, which sends a linear form λ on B to the endomorphism Θpλq pλ Idq ¥ δ. We prove in Proposition 2.2 that this map is compatible with both ¦ and : for any λ, µ B ¦ , Θpλ ¦ µq Θpλq ¦ Θpµq, Θpλ µq Θpµq ¥ Θpλq.

As an example of consequence, we give in Corollary 2.3 a criterion for the existence of the antipode for pB, m, ∆q: this is a Hopf algebra, if and only if, the counit ϵ δ of the coproduct δ is invertible for the convolution product ¦ dual of ∆, and then the antipode is Θpϵ ¦¡1 δ q. An immediate consequence is that S is an algebra morphism and an algebra antimorphism, by a very classical result. Consequently, we obtain that pH, mq is commutative. By the way, this explains why no non commutative example of double bialgebra was known.

We then add the assumption that pB, ∆q is a connected coalgebra. This gives the existence of an increasing ltration pB ¤n q nN (the coradical ltration) of B such that for any k, l, n N, mpB ¤k B ¤l q B ¤k l , ∆pB ¤n q n p0 B ¤p B ¤n¡p , and such that B ¤0 K1 B . In this case, for any vector space V , EndpB, V q inherits a distance, making it a complete hypermetric space. When V is an algebra, then EndpB, V q inherits a convolution product, which makes it a complete hypermetric algebra. Moreover, if f EndpB, V q satises f p1 B q 0, we obtain a continuous algebra map from the algebra of formal series KrrT ss to EndpB, V q, which sends T to f (Proposition 3.3): this allows to dene exponential, logarithm, or non integral powers of elements of EndpB, V q. This formalism can be used to prove Takeuchi's formula for the antipode, a universal property for shue coalgebras (Proposition 3.5), or the well-known exp-ln bijection between the Lie algebra of innitesimal characters to the group of characters of pB, m, ∆q (Proposition 3.6).

One of the simplest examples of double bialgebra is the polynomial algebra KrXs, with its two coproducts dened by ∆pXq X 1 1 X, δpXq X X.

We prove in Theorem 3.9 that it is a terminal object in the category of connected double bialgebras: in other words, for any connected double bialgebra pB, m, ∆, δq, there exists a unique double bialgebra morphism from B to KrXs. Moreover, this morphism is

Φ ϵ X δ p1 pϵ δ ¡ ε ∆ qq X V ņ0 XpX ¡ 1q . . . pX ¡ n 1q n! ϵ n δ ¥ ∆pn¡1q ,
with the use of formal series described earlier, and where the maps ∆pn¡1q are the iterated of the reduced coproduct ∆. We also prove that this morphism Φ allows to construct all bialgebra morphisms from pB, m, ∆q to pKrXs, m, ∆q, thanks to the action of the monoid of characters pCharpBq, q, see Corollary 3.12. When applied to the double bialgebra of graphs, this gives the chromatic polynomial (Theorem 3.13). When applied to a quasishue double bialgebra, this gives a morphism involving Hilbert polynomials (Proposition 3.14).

When one works with the enveloping algebra Upgq of a Lie algebra g, the eulerian idempotent is a useful projector on g, see [START_REF] Loday | Série de Hausdor, idempotents eulériens et algèbres de Hopf[END_REF][START_REF] Burgunder | Eulerian idempotent and Kashiwara-Vergne conjecture[END_REF][START_REF] Bandiera | Eulerian idempotent, pre-Lie logarithm and combinatorics of trees[END_REF] for several applications. It is originally dened on the enveloping algebra of a free Lie algebra, in terms of descents of permutations [START_REF] Solomon | On the Poincaré-Birkho-Witt theorem[END_REF]. This can be generalized without any problem to any connected bialgebra, by the formula

ϖ lnpIdq V ķ1 p¡1q k 1 k m pk¡1q ¥ ∆pk¡1q ,
where the m pk¡1q are the iterated products. It is generally not a projector. If B is cocommutative, it is well-known that it is a projector on the Lie algebra of the primitive elements of B. The case of a commutative connected bialgebra is not so well known. We here consider the case of a connected double bialgebra B. An especially interesting innitesimal character is given by the logarithm ϕ of the counit ϵ δ . We prove that: ϕ is closely related to the double bialgebra morphism Φ, see Proposition 4.1: for any x B, ϕpxq Φpxq I p0q. for any innitesimal character µ of B, ϕ µ µ, see Lemma 4.3. Consequently, ϕ ϕ ϕ, which implies that ϖ is a projector, that its kernel is K1 B B 2 (Proposition 4.4), and its image contains the Lie algebra PrimpBq of primitive elements of B (but is not equal, except if B is cocommutative). This result can be extended to any commutative connected bialgebra (Proposition 4.13). In the case of the graph bialgebra, this innitesimal character admits a combinatorial interpretation in term of acyclic orientations with a single xed source (Theorem 4.9). For quasishue bialgebras, the eulerian idempotent is given in Corollary 4.17, in term of descents of surjections. Applications of this projector include that any commutative connected bialgebra can be seen as a subbialgebra of a shue bialgebra (Corollary 4.14), which in turns implies Homan's result [START_REF] Michael | Quasi-shue products[END_REF] that any commutative quasishue bialgebra is isomorphic to a shue algebra and that any commutative connected bialgebra can be embedded in a double bialgebra.

We then make precise the hypothesis on pB, m, ∆q and assume that it is connected and graded: there exists a family of subspaces pB n q nN of B such that

B V à n0 B n ,
and such that for any k, l, n N,

mpB k B l q B k l , ∆pB n q n p0 B p B n¡p ,
and with B 0 K1 B . A natural question is the description of homogeneous morphisms from pB, m, ∆q to pKrXs, m, ∆q (noting that the unique double bialgebra morphism is usually not homogeneous). we obtain that these morphisms are in bijection with the space B ¦ 1 (Proposition 5.2), with explicit formulas (Corollary 5.3). In the case of graphs, taking λ B ¦ 1 dened by λp q 1, we obtain the bialgebra morphism sending any graph G of degree n to X n , and the action of pCharpBq, q allows to recover the interpretation of the coecients of the chromatic polynomials in terms of acyclic orientations of [START_REF] Greene | On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs[END_REF][START_REF] Bishal | Chromatic polynomial and heaps of pieces[END_REF]. Finally, we also consider, following Aguiar, Bergeron and Sottile's result [START_REF] Aguiar | Combinatorial Hopf algebras and generalized Dehn-Sommerville relations[END_REF], that under a homogeneity condition, there exists a unique homogeneous double bialgebra morphism from pB, m, ∆, δq to the double bialgebra of quasisymmetric functions QSym, which is a special case of a quasishue double bialgebra based on a semigroup [START_REF] Ebrahimi-Fard | A comodule-bialgebra structure for word-series substitution and mould composition[END_REF].

This paper is organized as follows: the rst section recalls the denition of double bialgebras, the examples of graphs and of quasishue bialgebras. The second section gives general results on double bialgebras, including the properties of the map Θ and the actions of the monoid of characters. The third part concentrates on the particular case of connected double bialgebras, with the exp-ln bijection between innitesimal characters and characters and the polynomial invariants. The eulerian projector, its properties and their consequences, are studied in the next section, and the last section gives results in the more specic case of graded double bialgebras.

Acknowledgements. The author acknowledges support from the grant ANR-20-CE40-0007 Combinatoire Algébrique, Résurgence, Probabilités Libres et Opérades. Notations 0.1.

K is a commutative eld of characteristic zero. All the vector spaces in this text will be taken over K.

For any k N, we put rks t1, . . . , ku. In particular, r0s r. We denote by KrrT ss the algebra of formal series with coecients in K. If P pTq °an T n KrrT ss, the valuation of P pTq is valpP pTqq mintn N | a n $ 0u. By convention, valp0q V. This induces a distance d on KrrT ss, dened by dpP pTq, QpT qq 2 ¡valpPpTq¡QpTqq , with the convention 2 ¡V 0. Then pKrrTss, dq is a complete metric space. If P pTq °an T n and QpT q KrrT ss with Qp0q 0, the composition of P and Q is

P ¥ QpT q V ņ0 a n QpT q n .
We shall use the classical formal series

e T V ņ0 T n n! , lnp1 T q V ņ1 p¡1q n 1 n T n , p1 T q x V ņ0 xpx ¡ 1q . . . px ¡ n 1q n! T n , for x K.
We recall the classical results:

e T ¡ 1 ¨¥ lnp1 T q lnp1 T q ¥ e T ¡ 1 ¨ T, p1 T q x e x lnp1 T q .
Moreover, for any formal series P pTq and QpT q with no constant terms, e T ¥ pPpTq QpT qq e T ¥ P pTq ¨ e T ¥ QpT q ¨, lnp1 T q ¥ pPpTq QpT q P pTqQpTqq lnp1 T q ¥ P pTq lnp1 T q ¥ QpT q, which implies that for any x, y K, p1 T q x y p1 T q x p1 T q y .

1 Cointeracting bialgebras

Denition

We refer to the references [START_REF] Abe | Hopf algebras[END_REF][START_REF] Cartier | Classical Hopf algebras and their applications[END_REF][START_REF] Sweedler | Hopf algebras[END_REF] for the main denitions on bialgebras and Hopf algebras. Let pB, m, δq be a bialgebra. Its counit will be denoted by ϵ δ . It is well-known that its category of (right) comodules is a monoidal category:

If pM 1 , ρ 1 q and pM 2 , ρ 2 q are two comodules over B, then M 1 M 2 is also a comodule, with the coaction m 1,3,24 ¥ pρ 1 ρ 2 q, with m 1,3,24 :

4

M 1 B M 2 B ÝÑ M 1 M 2 B m 1 b 1 m 2 b 2 ÝÑ m 1 m 2 b 1 b 2 . If f 1 : M 1 ÝÑ M I 1 and f 2 : M 2 ÝÑ M I 2 are comodule morphisms, then f 1 f 2 : M 1 M 2 ÝÑ M I 1 M I 2 is a comodule morphism.
The associativity of m implies that if pM 1 , ρ 1 q, pM 2 , ρ 2 q and pM 3 , ρ 3 q are three comodules over B, then pM 1 M 2 q M 3 and M 1 pM 2 M 3 q are the same comodule.

The unit comodule is K with the coaction dened by dx K, ρpxq x 1 B . The canonical identications of K M and M K with M are comodules isomorphims for any comodule M .

In particular, B is a comodule over itself with the coaction δ. Hence, for any n N, B n is a comodule over B, with the coaction m 1,3,...,2n¡1,24...2n ¥ δ n , where m 1,3,...,2n¡1,24...2n :

4 B 2n ÝÑ B n b 1 . . . b 2n 1 ÝÑ b 1 b 3 . . . b 2n¡1 b 2 b 4 . . . b 2n .
Note that m : BB ÝÑ B is always a comodule morphism, as well as the unit map ν B : K ÝÑ B, which sends x K to x1 B . A double bialgebra is given by a coproduct ∆ on B, making pB, m, ∆q a bialgebra in the category of comodules over pB, m, δq with the coaction δ. In more details: Denition 1.1. A double bialgebra is a family pB, m, ∆, δq such that: pB, m, δq is a bialgebra. Its counit is denoted by ϵ δ . pB, m, ∆q is a bialgebra. Its counit is denoted by ε ∆ .

∆ : B ÝÑ B B is a comodule morphism: p∆ Id B q ¥ δ m 1,3,24 ¥ pδ δq ¥ ∆. ε ∆ : B ÝÑ K is a comodule morphism: pε ∆ Idq ¥ δ ν B ¥ ε ∆ .
Remark 1.1. Let pB, m, ∆, δq be a double bialgebra. Then, as pB, m, δq is a bialgebra, δ ¥ m m 13,24 ¥ pδ δq pm Idq ¥ m 1,3,24 ¥ pδ δq, with the obvious notation m 13,24 . Therefore, m is a comodule morphism from B B to B. Moreover, as δp1 B q 1 B 1 B , the map ν B : K ÝÑ B is a comodule morphism.

Example 1.1. The algebra KrXs is a double bialgebra, with the two multiplicative coproducts dened by ∆pXq X 1 1 X, δpXq X X. In other terms, identifying KrXs KrXs with KrX, Y s through the algebra map 4 KrXs KrXs ÝÑ KrX, Y s P pXq QpXq ÝÑ P pXqQpY q, for any P KrXs, ∆pP pXqq P pX Y q, δpP pXqq P pXY q. The counit ε ∆ sends P KrXs to P p0q and the counit ϵ δ sends it to P p1q.

The example of graphs

We refer to [START_REF] Harary | Graph theory[END_REF] for classical denitions and notations on graphs. In the context of this article, a graph will be a pair G pV pGq, EpGqq, where V pGq is a nite set (maybe empty), called the set of vertices, and EpGq a sets of 2-elements sets of elements of V pGq, called the set of edges of G. The degree of G is the cardinality of V pGq. If G and H are two graphs, an isomorphism from G to H is a bijection f : V pGq ÝÑ V pHq such that for any x $ y V pGq, tx, yu EpGq if, and only if, tfpxq, f pyqu EpHq. We shall denote by G the set of isoclasses of graphs, and for any n N, by Gpnq the set of isoclasses of graphs of degree n. The vector space generated by G will be denoted by

H G . Example 1.2.
Gp0q t1u, Gp1q t u, Gp2q t , u, Gp3q t , , , u, Gp4q t , , , , , , , , , , u.

If G and H are two graphs, their disjoint union is the graph GH dened by

V pGHq V pGq V pHq, EpGHq EpGq EpHq.
This induces a commutative and associative product m on H G , whose units is the empty graph 1.

Let G be a graph and I V pGq. The subgraph G |I is dened by

V pG |I q I, EpG |I q ttx, yu EpGq | x, y Iu.
This notion induces a commutative and coassociative coproduct ∆ on H G given by dG G, ∆pGq

IV pGq G |I G |V pGqzI . Its counit ε ∆ is given by dG G, ε ∆ pGq δ G,1 .
Example 1.3.

∆p q 1 1 , ∆p q 1 1 2 , ∆p q 1 1 3 3 , ∆p q 1 1 2 2 , ∆p q 1 1 4 6 4 , ∆p q 1 1 2 2 4 2 2 , ∆p q 1 1 2 2 2 2 2 , ∆p q 1 1 4 4 2 4 , ∆p q 1 1 3 3 3 3 , ∆p q 1 1 2 2 2 2 2 2 .
Let G be a graph and let be an equivalence relation on V pGq. We dene the contracted graph G{ by V pG{ q V pGq{ , EpG{ q ttπ pxq, π pyqu | tx, yu EpGq, π pxq $ π pyqu, where π : V pGq ÝÑ V pGq{ is the canonical surjection. δp q , δp q , δp q 3 , δp q 2 , δp q 6 p6 4 q, δp q p4 q p2 2 2 q, δp q p 3 q p 2 q, δp q 4 p2 4 q, δp q 3 3 , δp q 3 p 2 q.

Proposition 1.2. [START_REF]Chromatic polynomials and bialgebras of graphs[END_REF] pH G , m, ∆, δq is a double bialgebra. Let pV, ¤q be a commutative algebra (not necessarily unitary). The quasishue bialgebra associated to V is pTpV q, , ∆q, where

dv 1 , . . . , v k l V, v 1 . . . v k v k 1 . . . v k l σQShpk,lq ¤ ¥ ¤ ¹ σpiq1 v i . . . ¤ ¥ ¤ ¹ σpiqmaxpσq v i
, where the symbol ¤ ¹ means that the products are taken in pV, ¤q. For example, if

v 1 , v 2 , v 3 , v 4 V , v 1 v 2 v 3 v 4 v 1 v 2 v 3 v 4 v 2 v 1 v 3 v 4 v 2 v 3 v 1 v 4 v 2 v 3 v 4 v 1 pv 1 ¤ v 2 qv 3 v 4 v 2 pv 1 ¤ v 3 qv 4 v 2 v 3 pv 1 ¤ v 4 q, v 1 v 2 v 3 v 4 v 1 v 2 v 3 v 4 v 1 v 3 v 2 v 4 v 1 v 3 v 4 v 2 v 3 v 1 v 2 v 4 v 3 v 1 v 4 v 2 v 3 v 4 v 1 v 2 pv 1 ¤ v 3 qv 2 v 4 pv 1 ¤ v 3 qv 2 v 4 v 3 pv 1 ¤ v 4 qv 2 v 1 pv 2 ¤ v 3 qv 4 v 1 v 3 pv 2 ¤ v 4 q v 3 v 1 pv 2 ¤ v 4 q pv 1 ¤ v 3 qpv 2 ¤ v 4 q.
The coproduct is the deconcatenation coproduct:

dv 1 , . . . , v n V, ∆pv 1 . . . v n q n ķ0 v 1 . . . v k v k 1 . . . v n .
In the particular case where ¤ 0, we obtain the quasishue algebra pTpV q, ¡,∆q.

When pV, ¤, δ V q is a commutative (not necessarily unitary) bialgebra, then pTpV q, , ∆q inherits a second coproduct δ:

dv 1 , . . . , v n V, δpv 1 . . . v n q 1¤i 1 ... i k n £ ¤ ¹ 0 i¤i 1 v I i . . . £ ¤ ¹ i k i¤n v I i v P 1 . . . v P i 1 . . . v P i k 1 . . . v P n , with Sweedler's notation δ V pvq v I v P for any v V . For example, if v 1 , v 2 , v 3 V , δpv 1 q v I 1 v P 1 , δpv 1 v 2 q v I 1 v I 2 v P 1 v P 2 v I 1 ¤ v I 2 v P 1 v P 2 , δpv 1 v 2 v 3 q v I 1 v I 2 v I 3 v P 1 v P 2 v P 3 pv I 1 ¤ v I 2 qv I 3 v P 1 v P 2 v P 3 v I 1 pv I 2 ¤ v I 3 q v P 1 v P 2 v P 3 pv I 1 ¤ v I 2 ¤ v I 3 q v P 1 v P 2 v P 3 .
Proposition 1.3. If pV, ¤, δ V q is a commutative (not necessarily unitary) bialgebra, then pTpV q, , ∆, δq is a double bialgebra.

Proof. It is quite well-known that pTpV q, , ∆q is a bialgebra [START_REF] Michael | Quasi-shue products[END_REF][START_REF]Quasi-shue algebras and applications[END_REF]. We shall use the following notation: for any w v 1 . . . v n V n , with n ¥ 1,

|w| v 1 ¤ . . . ¤ v n , w I w P v I 1 . . . v I n v P 1 . . . v P n ,
where we used Sweedler's notation δ V pvq v I v P for any v V . Let w V n , with n ¥ 1. Second step. Let us prove that for any x V k , y V l , 13,24 ¥ pδ δqpx yq δ ¥ px yq.

Then

We proceed by induction on n k l. If k 0, we can assume that x 1 and then 13,24 ¥ pδ δqp1 yq δpyq δ ¥ px yq.

The result also holds if l 0: these observations give the cases n 0 and n 1. Let us now assume that k, l ¥ 1 and the result at all ranks n.

p∆ Idq ¥ 13,24 ¥ pδ δqpx yq 14,25,36 ¥ p∆ Id ∆ Idq ¥ pδ δqpx yq 14,25,36 ¥ 1,3,24,5,7,68 ¥ pδ δ δ δq ¥ p∆ ∆qpx yq 15,37,2468 ¥ pδ δ δ δq ¥ p∆ ∆qpx yq, whereas, with Sweedler's notation δpzq z p1q z p2q for any z T pV q, p∆ Idq ¥ δ ¥ px yq 1,3,24 ¥ pδ δq ¥ ∆ ¥ px yq 1,3,24 ¥ pδ δq ¥ 13,24 ¥ p∆ ∆qpx yq 1,3,24 ¥ pδ δq ¥ 13,24 p∆ ∆px yq ¡ x 1 y 1 ¡ 1 x 1 yq px yq p1q 1 px yq p2q 1 px yq p1q px yq p2q

1,3,24 ¥ 15,24,37,68 ¥ pδ δ δ δq p∆ ∆px yq ¡ x 1 y 1 ¡ 1 x 1 yq px yq p1q 1 px yq p2q 1 px yq p1q px yq p2q

15,37,2468 ¥ pδ δ δ δq ¥ p∆ ∆qpx yq ¡ x p1q y p1q 1 x p2q y p2q ¡ 1 x p1q y p1q x p2q y p2q px yq p1q 1 px yq p2q 1 px yq p1q px yq p2q p∆ Idq ¥ 13,24 ¥ pδ δqpx yq ¡ x p1q y p1q 1 x p2q y p2q ¡ 1 x p1q y p1q x p2q y p2q px yq p1q 1 px yq p2q 1 px yq p1q px yq p2q .

We use the induction hypothesis for the fourth equality. We obtain that p ∆ Id Idq ¥ δ ¥ px yq p ∆ Idq ¥ 13,24 ¥ pδ δqpx yq, so δ ¥ px yq ¡ 13,24 ¥ pδ δqpx yq V T pV q. Let π be the canonical projection from T pV q onto V . For any w V n , with n ¥ 1, pπ Idq ¥ δpwq |w I | w P .

Hence, as V is a commutative bialgebra, pπ Idq ¥ δ ¥ px yq |px yq I | px yq P |x I | ¤ |y I | px P y P q pπ Idq ¥ 13,24 ¥ δpx yq.

We obtain that δ ¥ px yq 13,24 ¥ pδ δqpx yq. Third step. Let us prove that pId δq ¥ δpxq pδ Idq ¥ δpxq for any x V n by induction on n. It is obvious if n 0, taking then x 1. Let us assume the result at all ranks n. The x p2q © P pπ Idq ¥ pId δq ¥ δpxq.

rst step implies that p ∆ Idq ¥ δ 1,3,24 ¥ pδ δq ¥ ∆, so p ∆ Id Idq ¥ pδ Idq ¥ δpxq 1,3,24,5 ¥ pδ δ Idq ¥ p ∆ Idq ¥ δpxq 1,3,24,5 ¥ pδ δ Idq ¥ 1,
Finally, pδ Idq ¥ δpxq pId δq ¥ δpxq.

Final step. It is immediate that ε ∆ is a comodule morphism. Let us prove now that δ has a counit. We put, for any v 1 , . . . , v n V , with n ¥ 1,

ϵ δ pv 1 . . . v n q 5 ϵ V pv 1 q if n 1, 0 otherwise. Then, if w v 1 . . . v n , pϵ δ Idq ¥ δpwq ϵ δ p|w I |qw P 0 ϵ V pv I 1 ¤ . . . ¤ v I n qv P 1 . . . v P n ϵ V pv I 1 q . . . ϵ V pv I n qv P 1 . . . v P n v 1 . . . v n , whereas pId ϵ δ q ¥ δpwq v I 1 . . . v I n ϵ V pv P 1 ¤ . . . ¤ v P n q 0 v I 1 . . . v I n ϵ V pv P
Taking Ω pN ¡0 , q, we recover the double bialgebra of quasisymmetric functions QSym, A basis of QSym is given by words in strictly positive integers, which are called compositions. The second coproduct δ is often called the internal coproduct, and is dual of the Kronecker product of noncommutative symmetric functions.

Characters

Notations 1.2. Let pB, m, ∆q be a bialgebra.

B ¦ inherits an algebra structure, with the convolution product ¦ induced by ∆:

dλ, µ B ¦ , λ ¦ µ pλ µq ¥ ∆.
The unit is ε ∆ . The set of the characters of B, that is to say algebra morphisms from B to K, is denoted by CharpBq. It is a monoid for the convolution product ¦.

In the case of a double bialgebra pB, m, ∆, δq, B ¦ inherits a second convolution product, denoted by and coming from δ: dλ, µ B ¦ , λ µ pλ µq ¥ δ. The unit is ϵ δ . Moreover, CharpBq is also a monoid for the convolution product .

The space of innitesimal characters of B, that is to say ε ∆ -derivations from B to K, is denoted by InfCharpBq. In other words, a linear map λ : B ÝÑ K is an innitesimal character of B if for any x, y B, λpxyq ε ∆ pxqλpyq λpxqε ∆ pyq. In other terms, for any λ B ¦ , λ InfCharpBq if and only if λpK1 B B 2 q p0q, where B Kerpε ∆ q is the augmentation ideal of B. If pB, m, ∆q is a bialgebra, we can consider the transpose m ¦ : B ¦ ÝÑ pB Bq ¦ of the product m. Note that B ¦ B ¦ is considered as a subspace of pB Bq ¦ , through the canonical Proof. Immediate.

injection 6 8 7 B ¦ B ¦ ÝÑ pB Bq ¦ λ µ ÝÑ 4 B B ÝÑ K x y ÝÑ λpxqµpyq. (This is not an isomorphism except if B is nite-dimensional). As m is a coalgebra morphism, dually m ¦ : B ¦ ÝÑ
Lemma 1.5. Let pB, m, ∆, δq be a double bialgebra. For any µ B ¦ ,

ε ∆ µ µp1 B qε ∆ .
Proof. As ε ∆ is a comodule morphism,

ε ∆ µ pε ∆ µq ¥ δ µ ¥ pε ∆ Idq ¥ δ µ ¥ ν B ¥ ε ∆ µp1 B qε ∆ .
Proposition 1.6. Let pB, m, ∆, δq be a double bialgebra. Then InfCharpBq B ¦ InfCharpBq. Proof. Let λ InfCharpBq and µ B ¦ . For any x, y B, using Sweedler's notation δpzq °zI z P for δ, λ µpxyq ¸λppxyq I qµppxyq P q ¸¸λpx I y I qµpx P y P q ¸¸λpx I qε ∆ py I qµpx P y P q ¸¸ε ∆ px I qλpy I qµpx P y P q ¸λpx I qε ∆ pyqµpx P 1 B q ¸ε∆ pxqλpy I qµp1 B y P q λ µpxqε ∆ pyq ε ∆ pxqλ µpyq.

Therefore, λ µ InfCharpBq.

2 General results

Compatibility of the antipode with the coaction

Proposition 2.1. Let pB, m, ∆, δq be a double bialgebra, such that pB, m, ∆q is a Hopf algebra of antipode S. Then S is a comodule morphism:

δ ¥ S pS Idq ¥ δ.
Proof. We consider the space HompB, BBq of linear maps from B to BB, with the convolution product ¦ dened by f ¦ g m 13,24 ¥ pf gq ¥ ∆. The unit ι sends any b B to ε ∆ pbq1 B 1 B . Let us show that δ has an inverse in this algebra.

ppS

Idq ¥ δq ¦ δ m 13,24 ¥ pS Id Id Idq ¥ pδ δq ¥ ∆ pm Idq ¥ pS Id Idq ¥ m 1,3,24 ¥ pδ δq ¥ ∆ pm Idq ¥ pS Id Idq ¥ p∆ Idq ¥ δ ppm ¥ pS Idq ¥ ∆q Idq ¥ δ ppν B ¥ ε ∆ q Idq ¥ δ ι, so pS Idq ¥ δ is a left inverse of δ for the convolution product ¦. δ ¦ pδ Sq m 13,24 ¥ pδ δq ¥ pId Sq ¥ ∆ δ ¥ m ¥ pId Sq ¥ ∆ δ ¥ ν B ¥ ε ∆ ι, so δ ¥ S is a right inverse of δ for the convolution product ¦. As ¦ is associative, δ is invertible and its inverse is pS Idq ¥ δ δ ¥ S.

From linear forms to endomorphisms

Notations 2.1. Let pB, m, ∆, δq be a double bialgebra and let pA, m A q be an algebra. Then the space HompB, Aq of linear maps from B to A is given two convolution products ¦ and : for any

f, g HompB, Aq, f ¦ g m A ¥ pf gq ¥ ∆, f g m A ¥ pf gq ¥ δ. The unit of ¦ is ν A ¥ ε ∆ whereas the unit of is ν A ¥ ϵ δ .
Two particular examples are given by A B, which denes ¦ and for EndpBq, and A K, giving back the products ¦ and on B ¦ . Proposition 2.2. Let pB, m, ∆, δq be a double bialgebra. We consider the linear map

Θ : 4 B ¦ ÝÑ EndpBq λ ÝÑ pλ Idq ¥ δ. For any λ, µ B ¦ , Θpλ ¦ µq Θpλq ¦ Θpµq,
Θpλ µq Θpµq ¥ Θpλq. Moreover, Θpε ∆ q ν B ¥ ε ∆ and Θpϵ δ q Id B . The map Θ is injective, with a left inverse given by Θ I :

4 EndpBq ÝÑ B ¦ f ÝÑ ϵ δ ¥ f. Proof. Let λ, µ B ¦ . Θpλ ¦ µq pλ µ Idq ¥ p∆ Idq ¥ δ pλ µ Idq ¥ m 1,3,24 ¥ pδ δq ¥ ∆ m ¥ pλ Id µ Idq ¥ pδ δq ¥ ∆ m ¥ pΘpλq Θpµqq ¥ ∆ Θpλq ¦ Θpµq. Θpλ µq pλ µ Idq ¥ pδ Idq ¥ δ pλ µ Idq ¥ pId δq ¥ δ pµ Idq ¥ δ ¥ pλ Idq ¥ δ Θpµq ¥ Θpλq.
By denition of the counit,

Θpϵ δ q Id B . As ε ∆ is a comodule morphism, Θpε ∆ q ν B ¥ ε ∆ . Let λ B ¦ . Θ I ¥ Θpλq ϵ δ ¥ pλ Idq ¥ δ pλ ϵ δ q ¥ δ λ ϵ δ λ. So Θ I ¥ Θ Id B ¦.
Corollary 2.3. Let pB, m, ∆, δq be a double bialgebra. Then pB, m, ∆q is a Hopf algebra if, and only if, ϵ δ has an inverse in the algebra pB ¦ , ¦q. If this holds, the antipode of pB, m, ∆q is

S pϵ ¦¡1 δ Idq ¥ δ.
Proof. ùñ. If pB, m, ∆q is a Hopf algebra, denoting by S its antipode, the inverse of ϵ δ in pB ¦ , ¦q is ϵ δ ¥ S.

ðù. If so, putting S Θpϵ ¦¡1 δ q, we obtain

S ¦ Id B Θpϵ ¦¡1 δ q ¦ Θpϵ δ q Θpϵ ¦¡1 δ ¦ ϵ δ q Θpε ∆ q ν B ¥ ε ∆ . Similarly, Id B ¦ S ν B ¥ ε ∆ , so pB, m,
pϵ δ ¦ ϵ ¦¡1 δ q ϵ ¦¡1 δ ε ∆ ϵ ¦¡1 δ ϵ ¦¡1 δ p1 B qε ∆ ε ∆ , and pϵ δ ¦ ϵ ¦¡1 δ q ϵ ¦¡1 δ pϵ δ ϵ ¦¡1 δ q ¦ pϵ ¦¡1 δ ϵ ¦¡1 δ q ϵ ¦¡1 δ ¦ pϵ ¦¡1 δ ϵ ¦¡1 δ q.
Hence,

ϵ ¦¡1 δ ¦ pϵ ¦¡1 δ ϵ ¦¡1 δ q ε ∆ , which implies that ϵ ¦¡1 δ ϵ ¦¡1 δ ϵ δ . Applying Θ, we obtain that Θpϵ ¦¡1 δ ϵ ¦¡1 δ q Θpϵ ¦¡1 δ q ¥ Θpϵ ¦¡1 δ q S ¥ S Θpϵ δ q Id,
so S is involutive and therefore, surjective.

Actions of the groups of characters

Proposition 2.5. Let pB, m, ∆, δq be a double bialgebra and V be a vector space. The following map denes a (right) action of the monoid pCharpBq, q on the space HompB, V q of linear maps

from B to V : 4 HompB, V q ¢ CharpBq ÝÑ HompB, V q pf, λq ÝÑ f øλpf λq ¥ δ.
Moreover:

1. If A is an algebra, λ CharpBq and f : B ÝÑ A is an algebra morphism, then f øλ is an algebra morphism.

2. If C is a coalgebra, λ CharpBq and f : B ÝÑ C is a coalgebra morphism, then f ø λ is a coalgebra morphism.

3. If B I is a bialgebra, λ CharpBq and f : B ÝÑ B I is a bialgebra morphism, then f øλ is a bialgebra morphism.

Proof. The fact that this is an action comes from the coassociativity of δ.

1. By composition, pId λq ¥ δ is an algebra morphism.

2. We obtain, as ∆ is a comodule morphism,

∆ ¥ pf øλq∆¥pf λq ¥ δ pf f λq ¥ p∆ Idq ¥ δ pf f λq ¥ m 1,3,24 ¥ pδ δq ¥ ∆ pf λ f λq ¥ pδ δq ¥ ∆ ppf øλqpf øλqq¥∆. As ε ∆ is a comodule morphism, ε ∆ ¥ pf øλqppε ∆ ¥ f q λq ¥ δ λ ¥ ν B ¥ ε ∆ ε ∆ . Therefore, f øλ is a coalgebra morphism.
3. Direct consequence of 1. and 2.

Remark 2.1. Consequently, if V is an algebra (respectively a bialgebra or a coalgebra), then ø denes an action of the monoid pCharpAq, q on the set Hom a pB, V q (respectively Hom b pB, V q or Hom c pB, V q) of morphisms of algebras (respectively bialgebras or coalgebras), from B to V . Proposition 2.6. Let pB, m, ∆, δq be a double bialgebra, V and W be two spaces and f : B ÝÑ V , g : V ÝÑ W be two linear maps. Then pf ¥ gq øλf ¥ pg øλq.

Proof. Indeed, pf ¥ gq øλppf ¥ gq λq ¥ δ f ¥ ppg Idq ¥ δq f ¥ pg øλq. Proposition 2.7. Let pA, m A q be an algebra. For any f, g HompB, Aq, for any λ CharpBq, pf ¦ gq øλpf øλq¦pg øλq.

Proof. Indeed, pf ¦ gq øλm A ¥ pf g λq ¥ p∆ Idq ¥ δ m A ¥ pf g λq m 1,3,24 ¥ pδ δq ¥ ∆ m A ¥ pf λ g λq ¥ pδ δq ¥ ∆ m A ppf øλqpg øλqq¥∆ pf øλq¦pg øλq. Remark 2.2.
In the particular case where V K, then ø. We obtain that for any λ 1 , λ 2 B ¦ , for any µ CharpBq, pλ 1 ¦ λ 2 q µ pλ 1 µq ¦ pλ 2 µq. So pCharpBq, q acts on pB, ¦q by algebra endomorphisms. By restriction, pCharpBq, q acts on pCharpBq, ¦q by monoid endomorphisms.

3 Connected double bialgebras 3.1 Reminders on connected bialgebras Notations 3.1. Let pB, m, ∆q be a bialgebra. We denote by B its augmentation ideal, that is to say the kernel of its counit ε ∆ . We dene a coassociative (non counitary) coproduct

∆ : B ÝÑ B B by dx B , ∆pxq ∆pxq ¡ x 1 ¡ 1 x.
We may extend ∆ to B by putting ∆p1 B q 0. The iterated reduced coproducts ∆pnq : B ÝÑ B pn 1q are inductively dened by

∆pnq 5 Id B if n 0, ¡ ∆pn¡1q Id © ¥ ∆ if n ¥ 1.
In particular, ∆p1q ∆. Recall that a bialgebra pB, m, ∆q is connected if its coradical is reduced to K. This is equivalent to the fact that ∆ is locally nilpotent: for any x B , there exists n N such that ∆pnq pxq 0. In this case, we obtain a ltration of B dened by

B ¤n Ker ¡ ∆pnq © K1 B .
This is called the coradical ltration. In particular, B ¤0 K1 B and B ¤1 B Kerp ∆q PrimpBq, the space of primitive elements of B. The degree associated to this ltration is denoted by deg p dx B, deg p pxq minpn N, x B ¤n q. The coassociativity of ∆ implies that for all n N, ∆pB ¤n q n ķ0 B ¤k B ¤n¡k .

Combined with the connectivity of B, this gives that for any n N,

∆pB ¤n q n¡1 ķ1 B ¤k B ¤n¡k .
The compatibility of ∆ and m implies that for any k, l N, mpB ¤k B ¤l q B ¤k l . Conversely, if B has an increasing ltration pB ¤n q nN (which may not be the coradical ltration), such that for any k, l, n N,

mpB ¤k B ¤l q B ¤k l , ∆pB ¤n q n ķ0 B ¤k B ¤n¡k ,
and such that B ¤0 K1 B , then B is connected, as the coradical of B is necessarily included in B ¤0 . In particular, if k ¡ |V pGq|, ∆pk¡1q pGq 0, so H G is connected.

Let V be a vector space. For any map f : B ÝÑ V , we dene its valuation by valpf q mintn N, f pB ¤n q $ p0qu, with the convention that valp0q V. Note that for any f, g HompB, V q, valpf gq ¥ minpvalpf q, valpgqq.

We therefore obtain a distance on HompB, V q dened by dpf, gq 2 ¡valpf¡gq , with the convention 2 ¡V 0. Note that for any f, g, h HompB, V q, dpf, hq ¤ maxpdpf, gq, dpg, hqq ¤ dpf, gq dpg, hq. Lemma 3.1. For any vector space V , pHompB, V q, dq is a complete metric space.

Proof. Let pf n q nN be a Cauchy sequence of HompB, V q. For any n N, there exists N pnq such that if k, l ¥ N pnq, then pf k q |B¤n pg k q |B¤n . Let us x for any n, a complement B n of B ¤n¡1 in B ¤n . Then for any n N, B ¤n

n à k0 B k ,
and consequently

B V à k0 B k .
Let n N. We dene g pnq : B n ÝÑ V by g pnq pf N pnq q |Bn , and we consider the map

g V à k0
g pkq .

If k ¥ maxpN p0q, . . . , N pnqq, then pf k q |B¤n g |B¤n , so dpf k , gq ¤ 2 ¡n . Hence, pf n q nN converges to g.

Proposition 3.2. Let pB, m, ∆q be a connected bialgebra and let pA, m A q be an algebra. For any f, g HompB, Aq, valpf ¦ gq ¥ valpf q valpgq. Consequently, ¦ : HompB B, Aq ÝÑ HompB, Aq is continuous. Proof. Let n valpf q valpgq. Then f ¦ gpB ¤n q m A ¥ pf gq ¥ ∆pB ¤n q n ķ0 m A pfpB ¤k q gpB ¤n¡k qq p0q, as either k valpf q or n ¡ k valpgq. Hence, valpf ¦ gq ¥ valpf q valpgq. Consequently, if A is an algebra and f : B ÝÑ A is a map such that valpf q ¥ 1, for any n N, valpf ¦n q ¥ n. Let pa n q nN be a sequence of scalars. For any n, p N,

val £ n p ķn a k f ¦k ¥ minpvalpa k f ¦k q, k tn, . . . , n puq ¥ n.
Hence, as pHompB, Aq, dq is complete, the series °ak f ¦k converge in HompB, Aq. We obtain: Proposition 3.3. Let pB, m, ∆q be a connected bialgebra and let A be an algebra. For any f HompB, Aq such that f p1 B q 0, we obtain a continous algebra morphism ev f :

6 9 8 9 7
KrrT ss ÝÑ HompB, Aq

V ķ0 a k T k ÝÑ V ķ0 a k f ¦k .
Moreover, for any P pTq V ķ0 a k T k KrrT ss, ev f pPpTqqp1 B q gp0q1 A and for any x B ,

ev f pPpTqqpxq valpxq ķ1 a k m pk¡1q A ¥ f k ¥ ∆pk¡1q pxq.
Proof. As f p1 B q 0, valpf q ¥ 1: ev f is well-dened. For any k, l N, ev f pT k T l q ev f pT k l q f ¦pk lq f ¦k ¦ f ¦l ev f pT k q ¦ ev f pT l q. By linearity, if P pTq, QpT q KrXs, ev f pPpTqQpTqq ev f pPpTqq ¦ ev f pQpTqq. By continuity and density of KrT s in KrrT ss, this is still true if P pTq, QpT q KrrT ss. As f p1 B q 0, for any x B , f ¦k pxq m pk¡1q

A ¥ f k ¥ ∆ pk¡1q pxq m pk¡1q
A ¥ f k ¥ ∆pk¡1q pxq, which implies the announced formula for ev f pPpTqqpxq. Notations 3.2. We shall write, for any P pTq KrrT ss and f HompB, Aq such that f p1 B q 0, P pfq ev f pPpTqq. Note that for any P pTq, QpT q KrrT ss, P Qpf q P pfq ¦ Qpf q.

In particular, taking A B and ρ the canonical projection on B which vanishes on K1 B , we can consider

S ev ρ ¢ 1 1 X 1 1 ρ V ķ0 p¡1q k ρ ¦k .
Then S is the inverse of ν B ¥ ε ∆ ρ Id for the convolution product: we proved that pB, m, ∆q is a Hopf algebra and recovered Takeuchi's formula [START_REF] Takeuchi | Free Hopf algebras generated by coalgebras[END_REF]: for any x B , Spxq V ķ1 p¡1q k m pk¡1q ¥ ∆pk¡1q pxq. Lemma 3.4. Let pB, m, ∆q be a connected bialgebra and let A be an algebra. For any f HompB, Aq such that f pA B q 0 and for any formal series P, Q KrrT ss, such that Qp0q 0, ev f pP ¥ QpT qq ev ev f pQpTqq pPpTqq. In other words, pP ¥ Qqpf q P pQpfqq. Proof. As Q has no constant term, if valpf q ¥ 1, then valpev f pQpTqqq ¥ 1 and ev ev f pQpTqq exists.

We start with the particular case P X n , for a certain n N. As ev f is an algebra morphism, ev f pX n ¥ Qq ev f pQ n q ev f pQq ¦n ev f pQq pX n q. By linearity of ev f pP ¥Qq and of ev ev f pQq pPq, the equality is still true if P KrXs. By continuity of P ÝÑ ev f pP ¥ Qq and of P ÝÑ ev ev f pQq pPq, as KrT s is dense in KrrT ss, this remains true for any P KrrT ss.

Applications to shue and quasishue bialgebras

Proposition 3.5 (Universal property of shue bialgebras). Let pB, m, ∆q be a connected bialgebra, V be a vector space, ϕ : B ÝÑ V be a linear map such that ϕp1 B q 0. We consider the shue bialgebra pTpV q, ¡,∆q or the quasishue bialgebra pTpV q, , ∆q if V is a (non necessarily unitary) algebra. We equip the tensor coalgebra T pV q with the concatenation product, and the associated convolution on hompB, T pV qq is denoted by ¦. Then Φ 1 1 ¡ ϕ is the unique coalgebra map making the following diagram commuting:

pB, ∆q ϕ & & N N N N N N N N N N N N Φ / / ppTpV q, ∆q π V
where π is the canonical projection onto V . Moreover:

1. Φ is injective if, and only if, ϕ |PrimpBq is injective.

2. Φ is a bialgebra morphism from pB, m, ∆q to pTpV q, ¡,∆q if, and only if, ϕpB 2 q 0, where B is the augmentation ideal of B.

3. If pV, ¤q is an algebra (not necessarily unitary), then Φ is a bialgebra morphism from pB, m, ∆q to pTpV q, , ∆q if, and only if, for any x, y B , ϕpxyq ϕpxq ¤ ϕpyq. Proof. Firstly, observe that as ϕp1 B q 0, valpϕq ¥ 1 and Φ exists. Let us prove that Φ is a coalgebra morphism. Firstly, as Φp1 B q 1 is a group-like,

∆ ¥ Φp1 B q pΦ Φq ¥ ∆p1 B q 1 1. Let x B . ∆ ¥Φpxq ∆ £ V ķ1 f k ¥ ∆pk¡1q pxq V ķ1 k¡1 i1 ¡ f i f pk¡iq © ¥ ∆pk¡1q pxq V ķ1 k¡1 i1 ¡ f i f pk¡iq © ¥ ¡ ∆pi¡1q ∆pk¡i¡1q © ¥ ∆pxq V i,j1 f i f j ¨¥ ¡ ∆pi¡1q ∆pj¡1q © ¥ ∆pxq pΦ Φq ¥ ∆pxq,
so Φ is indeed a coalgebra morphism. Moreover, for any x B , ϖ ¥ Φpxq ϕpxq 0 ϕpxq. As π ¥ Φp1 B q πp1q 0 ϕp1 B q, π ¥ Φ ϕ.

Let Ψ : pB, ∆q ÝÑ pTpV q, ∆q be another coalgebra morphism, such that π ¥ Ψ π ¥ Φ ϕ. As 1 is the unique group-like element of T pV q, Φp1 B q Ψp1 B q 1. Let us assume that Φ $ Ψ. There exists x B , such that Φpxq $ Ψpxq. Let us choose such an x, with deg p pxq n minimal. As ∆pxq B 2 ¤n¡1 , by denition of n, ∆ ¥Φpxq pΦ Φq ¥ ∆pxq pΨ Ψq ¥ ∆pxq ∆ ¥Ψpxq, so Φpxq ¡ Ψpxq Kerp ∆q V . Hence, Φpxq ¡ Ψpxq π ¥ Φpxq ¡ π ¥ Ψpxq 0: this is a contradiction, so Φ Ψ. pΦ Φq ¥ ∆pxq ∆ ¥Φpxq 0. By denition of n, Φ |B ¤n¡1 is injective. As ∆pxq B ¤n¡1 B ¤n¡1 , we obtain that ∆pxq 0, so x PrimpBq. Then Φpxq ϕpxq 0, so ϕ |PrimpBq is not injective. morphism from pB , mq to pV, ¤q.

ðù. Let us consider Φ 1 ¥pΦΦq and Φ 2 Φ¥m. As m and are coalgebra morphisms, by composition both Φ 1 and Φ 2 are coalgebra morphisms. In order to prove that

Φ 1 Φ 2 , it is enough to prove that π ¥ Φ 1 π ¥ Φ 2 . Let x, y B . π ¥ Φ 1 p1 B yq πp1 Φpyqq π ¥ Φpyq ϕpyq, π ¥ Φ 2 p1 B yq π ¥ Φ 2 pyq ϕpyq, so π ¥ Φ 1 p1 B yq π ¥ Φ 2 p1 B yq. Similarly, π ¥ Φ 1 px 1 B q π ¥ Φ 2 px 1 B q. π ¥ Φ 1 px yq πpΦpxq Φpyqq π ¥ Φpxq ¤ π ¥ Φpyq ϕpxq ¤ ϕpyq, π ¥ Φ 2 px yq π ¥ Φpxyq ϕpxyq. By hypothesis, π ¥ Φ 1 px yq π ¥ Φ 2 px yq, which gives π ¥ Φ 1 π ¥ Φ 2 and nally Φ 1 Φ 2 :
Φ is an algebra morphism.

2. From the second point, with ¤ 0.

Innitesimal characters and characters

Proposition 3.6. Let pB, m, ∆q be a connected bialgebra. The following maps are bijections, inverse one from the other: exp : CharpBq ÝÑ InfCharpBq λ ÝÑ lnp1 pλ ¡ ε ∆ qq lnpλq

V ķ0 p¡1q k 1 k pλ ¡ ε ∆ q ¦k .
Proof. We consider the two subsets

B ¦ 0 tλ B ¦ | λp1 B q 0u, B ¦ 1 tλ B ¦ | λp1 B q 1u,
and the maps exp :

4 B ¦ 0 ÝÑ B ¦ 1
λ ÝÑ e λ ev λ pexppTqq, ln :

4 B ¦ 1 ÝÑ B ¦ 0 λ ÝÑ lnpλq ev λ¡ε ∆ plnp1 T qq. If λ B ¦
0 , then valpλq ¥ 1, so ev λ pexppTqq is well-dened. Moreover, for any λ B ¦ 0 , exppλqp1 B q 1, so exp is well-dened. If λ B ¦ 1 , then pλ ¡ ε ∆ qp1 B q 0, so ev λ¡ε ∆ plnp1 T qq is well-dened. Moreover, for any λ B ¦ 0 , lnpλqp1 B q 0, so ln is well-dened. By Lemma 3.4, for any λ B ¦ 0 , ln ¥ exppλq ev ev λ pexppTqq¡ε ∆ plnp1 T qq ev ev λ pexppTq¡1q plnp1 T qq ev λ plnp1 T q ¥ pexppTq ¡ 1qq ev λ pTq λ.

Similarly, if λ B ¦ 1 , exp ¥ lnpλq ev ev λ¡ε ∆ plnp1 T qq pexppTqq ev λ¡ε ∆ pexppTq ¥ lnp1 T qq ev λ¡ε ∆ p1 T q ε ∆ λ ¡ ε ∆ λ,
so exp and ln are bijections, inverse one from the other.

Let λ InfCharpBq. Then λp1 B q 0, so InfCharpBq B ¦ 0 . By denition, CharpBq B ¦ 1 .

It remains to prove that for any λ B ¦ 0 , exppλq CharpBq if, and only if, λ InfCharpBq.

We shall use the transpose m ¦ of the product. As m is a coalgebra morphism, dually, m ¦ is an algebra morphism for the product ¦. Let f B ¦ , of valuation equal to N . Let n N and let x y pB Bq ¤n . We can assume that x B ¤k and y B ¤n¡k , with 0 ¤ k ¤ n. m ¦ pfqpx yq f pxyq f pB ¤k B ¤n¡k q f pB ¤n q p0q, so valpm ¦ pfqq ¥ N : we deduce that m ¦ is continuous. Hence, for any formal series P pTq KrrT ss, m ¦ pPpλqq m ¦ pev λ pPpTqqq ev m ¦ pλq pPpTqq P pm ¦ pλqq. Let us assume that λ InfCharpBq. Then m ¦ pexppλqq m ¦ pe λ q e m ¦ pλq e λλ e pλε ∆ q¦pε ∆ λq e λε ∆ ¦ e ε ∆ λ pe λ ε ∆ q ¦ pε ∆ e λ q e λ e λ exppλq exppλq, as ε ∆ λ and λ ε ∆ commute for the product ¦, ε ∆ being its unit. So exppλq is indeed in

CharpBq.

Let us assume that exppλq µ CharpBq. Then

m ¦ pλq lnp1 m ¦ pµ ¡ ε ∆ qq lnp1 µ µ ¡ ε ∆ ε ∆ q lnp1 pµ ¡ ε ∆ q ε ∆ ε ∆ pµ ¡ ε ∆ q pµ ¡ ε ∆ q pµ ¡ ε ∆ qq lnp1 pµ ¡ ε ∆ q ε ∆ q lnp1 ε ∆ pµ ¡ ε ∆ qq lnp1 µ ¡ ε ∆ q ε ∆ ε ∆ lnp1 µ ¡ ε ∆ q lnpµq ε ∆ ε ∆ lnpµq λ ε ∆ ε ∆ λ, so λ InfCharpBq.
Lemma 3.7. Let pB, m, ∆, δq be a connected double bialgebra. For any n N, ∆pB ¤n q B ¤n B.

Proof. For any x B, we put ρ L pxq x 1 B and ρ R pxq 1 B x. Then, putting δpxq x I x P , m 1,3,24 ¥ pδ δq ¥ ρ L pxq m 1,3,24 px I x P 1 B 1 B q x I 1 B x P pρ L Idq ¥ δpxq, so ρ L : B ÝÑ B B is a comodule morphism. Similarly, ρ R is a comodule morphism. Hence, ∆ ¡ ρ L ¡ ρ R is a comodule morphism. For any x B , ∆pxq ∆pxq ¡ ρ L pxq ¡ ρ R pxq, so ∆ : B ÝÑ B B is a comodule morphism. By composition, for any n N, ∆pnq is a comodule morphism. So its kernel is a sub-comodule of B: for any n N,

∆ ¡ Ker ¡ ∆pnq ©© Ker ¡ ∆pnq © B.
The result then follows immediately.

Proposition 3.8. Let pB, m, ∆, δq be a connected double bialgebra, A an algebra, f : B ÝÑ A a map such that f p1 B q 0 and λ CharpBq. For any P pTq KrrT ss, P pfq øλPpf øλq.

Proof. Firstly, f øλp1 B q f p1 B qλp1 B q 0, so ev f øλ pPpTqq is well-dened. Let us rst consider the case where P pTq T n , with n N.

Then ev f pT n q øλpT ¦n q øλpT øλq ¦n ev f øλ pT n q.

By linearity in P pTq, for any P pTq KrT s, the announced equality is satised.

Let V be a vector space, f HompB, V q and let us denote by N its valuation. By Lemma 3.7, if n N , f øλpB ¤n q pf λq ¥ δpB ¤n q f pB ¤n q λpBq p0q, so valpf øλq¤val pfq. In other words, the following map is continuous: 4 HompB, V q ÝÑ HompB, V q f ÝÑ f øλ.

Therefore, by density of KrT s in KrrT ss, the announced equality is true for any P KrrT ss.

Polynomial invariants

Theorem 3.9. Let pB, m, ∆q be a connected bialgebra and let λ B ¦ , such that λp1 B q 1.

1. There exists a unique coalgebra morphism Φ λ : pB, m, ∆q ÝÑ pKrXs, m, ∆q such that

ϵ δ ¥ Φ λ λ. Moreover, Φ λ λ X is given by Φ λ p1 B q 1 and dx B , Φ λ pxq λ X pxq V ķ1 λ k ¥ ∆pk¡1q pxqH k pXq,
where for any k N, H k pXq is the k-th Hilbert polynomial

H k pXq XpX ¡ 1q . . . pX ¡ k 1q k! .
2. Φ λ is a bialgebra morphism from pB, m, ∆q to pKrXs, m, ∆q if and only if λ CharpBq.

3. Φ λ is a double bialgebra morphism from pB, m, ∆, δq to pKrXs, m, ∆, δq if and only if λ ϵ δ .

Proof. 1. Existence. We extend the scalars to the eld KppXqq of fractions of KrrXss. Then KppXqq B is a double bialgebra over KppXqq. The map λ is extended as a KppXqq-linear map from KppXqq B to KppXqq, which we denote by λ. As λp1 B q ¡ ε ∆ p1 B q 1 ¡ 1 0, we can consider

λ X ev λ¡ε ∆ pp1 T q X q V ķ0 pλ ¡ ε ∆ q k ¥ ∆ pk¡1q pxqH k pXq. Therefore, for any x B K B KppXqq B , λ X pxq V ķ1 λ k ¥ ∆pk¡1q pxqH k pXq V ķ1 λ k ¥ ∆pk¡1q pxqH k pXq KrXs.
Hence, λ X |B λ X takes its values in KrXs.

Identifying KrXs KrXs and KrX, Y s, as p1

T q X Y p1 T q X p1 T q Y , ∆ ¥ λ X λ X Y λ X ¦ λ Y λ X ¦ λ Y pλ X λ X q ¥ ∆, so λ X is a coalgebra morphism. Moreover, ϵ δ ¥ λ X pλ X q |X1 λ.
Unicity. Let Λ : B ÝÑ K be a coalgebra morphism. We put ϵ δ ¥ Λ λ. We consider Λ as an element of B ¦ rrXss, putting ΛpXq

V ņ0 f n X n ,
where for any n ¥ 0, for any x B, f n pxq is the coecient of X n in Λpxq. As ∆ ¥ Λ pΛ Λq, still identifying KrXs KrXs and KrX, Y s, in B ¦ rrX, Y ss,

ΛpX Y q V ņ0 f n pX Y q n ∆ ¥ ΛpXq pΛpXq ΛpXqq ¥ ∆ ΛpXq ¦ ΛpY q.
Derivating according to Y and taking Y 0, we obtain

ΛpXq Λ I p0q ¦ ΛpXq. So ΛpXq C ¦ e Λ I p0qX , for a certain constant C B ¦ . As Λp0q ε ∆ ¥ Λ ε ∆ , Λp0q C ε ∆ ,
so ΛpXq e Λ I p0qX . We put µ e Λ I p0q B ¦ , then ΛpXq µ X . Moreover,

µ ϵ δ ¥ ΛpXq λ, so nally Λ λ X .
2. ùñ. By composition, if λ X is an algebra morphism, then ϵ δ ¥ λ X λ is an algebra morphism, so λ is a character. ðù. Let us assume that λ is a character. We put µ lnpλq. Then µ is an innitesimal character, so Xµ is also an innitesimal character of KppXqq B. As λ X exppXµq, λ X is a character of KppXqq B, so its restriction to B is an algebra mophism from B to KrXs.

3. ùñ. If λ X is a double bialgebra morphism, then λ ϵ δ ¥ λ X ϵ δ .
ðù. By the second point, as ϵ δ is a character, ϵ X δ is a bialgebra morphism from pB, m, ∆q to pKrXs, m, ∆q. We still identify KrXs KrXs and KrX, Y s. For any λ CharpBq, by Proposition 3.8, as ø for B ¦ ,

λ X λ X ¨¥ δ λ X λ Y pλ λ Y q X .
In the particular case λ ϵ δ , unit of the product , pϵ

X δ ϵ X δ q ¥ δ pϵ X δ q Y ϵ XY δ δ ¥ ϵ X δ .
So ϵ X δ is a double bialgebra morphism.

Using the exp and ln bijections, we obtain: Proposition 3.10. Let pB, m, ∆q be a connected bialgebra and let µ B ¦ , such that µp1 B q 0. 1. There exists a unique coalgebra morphism Ψ µ : pB, m, ∆q ÝÑ pKrXs, m, ∆q such that for any x B, Ψ µ pxq I p0q µpxq. Moreover, Ψ µ e µX is given on any x B by

Ψ µ pxq e µX pxq V ķ1 µ k ¥ ∆pk¡1q pxq X k k! . ( 1
)
2. Ψ µ is a bialgebra morphism from pB, m, ∆q to pKrXs, m, ∆q if and only if µ InfCharpBq.

3. Ψ µ is a double bialgebra morphism from pB, m, ∆, δq to pKrXs, m, ∆, δq if and only if µ lnpϵ δ q. Proof. All can be proved by taking λ exppµq and Ψ µ Φ exppµq . Let us now prove (1).

Ψ µ ev exppµq¡ε ∆ pp1 T q X q ev µ ppe T q X q ev µ pe T X q e µX . Therefore, for any x B , as µp1 B q 0,

Ψ µ pxq V ķ0 µ ¦k X k k! V ķ1 µ k ¥ ∆pk¡1q X k k! .

Moreover,

Ψ µ pxq I p0q µ 1 pxq 0 µpxq, so Ψ µ pxq I p0q µpxq. Corollary 3.11. Let pB, m, ∆, δq be a connected double bialgebra. There exists a unique double bialgebra morphism Φ from pB, m, ∆, δq to pKrXs, m, ∆, δq. For any x B ,

Φpxq V ķ1 ϵ k δ ¥ ∆pk¡1q pxqH k pXq.
Moreover, for any λ CharpBq, the unique bialgebra morphism Φ λ from pB, m, ∆q to pKrXs, m, ∆q such that ϵ δ ¥ Φ λ λ is Φ λ Φ øλpΦλq¥δ. Proof. The rst point is a direct reformulation of Theorem 3.9. By Proposition 2.5, Φ ø λ is a bialgebra morphism. Moreover, by Proposition 2.6,

ϵ δ ¥ pΦ øλqpϵ δ ¥ Φq λ ϵ δ λ λ. So Φ øλΦ λ .
Corollary 3.12. Let pB, m, ∆, δq be a connected double bialgebra and let Φ : B ÝÑ KrXs be the unique double bialgebra morphism. We denote by Hom b pB, KrXsq the set of bialgebra morphisms from pB, m, ∆q to pKrXs, m, ∆q. The following maps are bijections, inverse one from the other: Φ chr p q X, Φ chr p q XpX ¡ 1q, Φ chr p q XpX ¡ 1qpX ¡ 2q, Φ chr p q XpX ¡ 1q 2 , Φ chr p q XpX ¡ 1qpX ¡ 2qpX ¡ 3q, Φ chr p q XpX ¡ 1qpX ¡ 2q 2 , Φ chr p q XpX ¡ 1q 2 pX ¡ 2q, Φ chr p q XpX ¡ 1qpX 2 ¡ 3X 3q, Φ chr p q XpX ¡ 1q 3 , Φ chr p q XpX ¡ 1q 3 . Let us now consider the case of quasishue algebras. Let pV, ¤, δ V q be a commutative (not necessarily unitary) bialgebra. We denote by Φ the unique double bialgebra morphism from pTpV q, , ∆, δq to pKrXs, , ∆, δq. For any v 1 , . . . , v n V , with n ¥ 1,

Φpv 1 . . . v n q v1 ...vnw 1 ...w k , w 1 ,...,w k $r ϵ δ pw 1 q . . . ϵ δ pw k qH k pXq ϵ V pv 1 q . . . ϵ V pv n qH n pXq 0.
Therefore: Proposition 3.14. Let pV, ¤, δ V q be a commutative (not necessarily unitary) bialgebra. The unique double bialgebra morphism Φ from pTpV q, , ∆, δq to pKrXs, , ∆, δq sends any word v 1 . . . v n T pV q of length n ¥ 1 to Φpv 1 . . . v n q ϵ V pv 1 q . . . ϵ V pv n qH n pXq.

Remark 3.1. In the particular case of QSym, the unique double bialgebra morphism from pQSym, , ∆, δq to pKrXs, m, ∆, δq sends the composition pk 1 . . . k n q to H n pXq for any n. This morphism is denoted by Φ QSym . [START_REF] Bruned | Singular KPZ Type Equations[END_REF] The eulerian idempotent Notations 4.1. Let pB, m, ∆q be a connected bialgebra. Its eulerian idempotent is

ϖ ev ρ plnp1 T qq lnp1 ρq V ķ1 p¡1q k 1 k ρ ¦k .

Logarithm of the counit and the eulerian idempotent

Let us go back to the map Θ of Proposition 2.2, with V B. By Lemma 3.7, it is a continuous algebra map from B ¦ to EndpBq, as it sends B ¦ ¤n to EndpBq ¤n for any n.

Proposition 4.1. Let pB, m, ∆, δq be a connected double bialgebra. Let us denote by Φ the unique double bialgebra morphism from B to KrXs. We dene an innitesimal character ϕ B ¦ by dx B, ϕpxq Φpxq I p0q, that is to say ϕpxq is the coecient of X in Φpxq. Then ϕ lnpϵ δ q and the eulerian idempotent

ϖ of B is ϖ Θpϕq pϕ Idq ¥ δ.
Proof. By the proof of Proposition 3.10, for any λ CharpBq, Ψ lnpλq Φ λ , and for any x B, Ψ lnpλq pxq I p0q Φ λ pxq I p0q lnpλqpxq. In the particular case where λ ϵ δ , then Φ Φ ϵ δ and we obtain that ϕ lnpϵ δ q. As Θ is a continuous algebra morphism, Θpϕq Θplnpϵ δ qq lnpIdq ϖ.

Example 4.1. In the case of H G , this character is denoted by ϕ chr .

ϕ chr p q 1, ϕ chr p q ¡1, ϕ chr p q 2, ϕ chr p q 1, ϕ chr p q ¡6, ϕ chr p q ¡4, ϕ chr p q ¡2, ϕ chr p q ¡3, ϕ chr p q ¡1, ϕ chr p q ¡1. Proposition 4.2. Let pB, m, ∆, δq be a connected double bialgebra and let λ CharpBq. Then lnpλq ϕ λ.

Proof. By Proposition 3.8 with V K (and then ø),

ϕ λ lnpϵ δ q λ lnpϵ δ λq lnpλq. Lemma 4.3. Let µ InfCharpBq. Then ϕ µ µ. Proof. Let λ 1 , λ 2 B ¦ and µ InfCharpBq. pλ 1 ¦ λ 2 q µ pλ 1 λ 2 µq ¥ p∆ Idq ¥ δ pλ 1 λ 2 µq ¥ m 1,3,24 ¥ pδ δq ¥ ∆ pλ 1 ε ∆ λ 2 µ λ 1 µ λ 2 ε ∆ q ¥ pδ δq ¥ ∆ pλ 1 ε ∆ q ¦ pλ 2 µq pλ 1 µq ¦ pλ 2 ε ∆ q. Hence, for any n ¥ 1, if λ B ¦ , λ ¦n µ n ķ1 pλ ε ∆ q ¦pk¡1q ¦ pλ µq ¦ pλ ε ∆ q ¦pn¡kq . For λ ϵ δ ¡ ε ∆ , pϵ δ ¡ ε ∆ q µ ϵ δ µ ¡ ε ∆ µ ¡ µp1 B qε ∆ µ, whereas pϵ δ ¡ ε ∆ q ε ∆ ϵ δ ε ∆ ¡ ε ∆ ε ∆ ε ∆ ¡ ε ∆ p1qε ∆ 0.
Therefore, for any n ¥ 1,

pϵ δ ¡ ε ∆ q ¦n µ 5 µ if n 1, 0 otherwise.
We nally obtain that

ϕ µ lnp1 ε ∆ ¡ ϵ δ q µ V ķ1 p¡1q k 1 k pε ∆ ¡ ϵ δ q ¦ µ µ.
Proposition 4.4. Let pB, m, ∆, δq be a connected double bialgebra. Then ϖ is a projector, which kernel is

B 2 K1 B .
Proof. Indeed, by Lemma 4.3 with µ ϕ, ϖ ¥ ϖ Θpµq ¥ Θpµq Θpµ µq Θpµq ϖ. So ϖ is a projector. As ϕ is an innitesimal character, ϕpB 2 K1 B q p0q. Moreover, as ε ∆ is a comodule morphism, δpB q B B, which implies that δpB 2 K1 B q pB 2 K1 B q B. Therefore, as ϖ pϕ Idq ¥ δ, B 2 K1 B Kerpϖq.

Let x B . Then ϖpxq V ķ1 p¡1q k 1 k ρ ¦k pxq x V ķ2 p¡1q k 1 k m 12...k ¥ ∆pk¡1q pxq looooooooooooooooooomooooooooooooooooooon B 2 , so x ¡ ϖpxq B 2 K1 B . In particular, if x Kerpϖq, then x B 2 K1 B . Hence, Kerpϖq B 2 K1 B .
If x PrimpBq, then ϖpxq x, so PrimpBq Impϖq. In the cocommutative case, it is an equality:

Corollary 4.5. Let pB, m, ∆, δq be a connected double bialgebra, such that ∆ is cocommutative. Then the eulerian idempotent ϖ is the projector on PrimpBq which vanishes on B If G is not connected, then ϖpGq 0.

Remark 4.1. If pB, m, ∆q is neither a commutative or a cocommutative bialgebra, then ϖ is generally not a projector, and does not vanishes B 2 . To illustrate this, let us consider the bialgebra freely generated by three generators x 1 , x 2 , and y, with the coproduct dened by

∆px 1 q x 1 1 1 x 1 , ∆px 2 q x 2 1 1 x 2 , ∆pyq y 1 1 y x 1 x 2 .
Then ϖpx 1 x 2 q rx 1 , x 2 s 2 $ 0 and ϖpyq y ¡ x 1 x 2

2

. Therefore,

ϖ 2 pyq y ¡ x 1 x 2 2 ¡ rx 1 , x 2 s 4 y ¡ 3x 1 x 2 ¡ x 2 x 1 4 $ ϖpyq.

Chromatic innitesimal character

In the case of graphs, for any innitesimal character µ, if Ψ µ is the associated bialgebra morphism, for any graph G,

Ψ µ pGq V ķ1 V pGqI 1 ...I k µpG |I 1 q . . . µpG |I k q X k k! .
As µ is an innitesimal character, it vanishes on nonconnected graphs, so this is in fact a sum over E c pGq:

Ψ µ pGq EcpGq ¹ CV pGq{ µpG |C qX clpq ,
where clpq is the number of classes of . Denoting by µ the character dened dG G, µpGq

¹ H connected component of G µpHq,
we obtain

Ψ µ pGq EcpGq µpG |qX clpq , (2)
Let ϕ chr be the innitesimal character associated to the morphism Φ chr from H G to KrXs: for any graph G, ϕ chr pGq Φ chr pGq I p0q lnpϵ δ qpGq.

We obtain from (2) that for any graph G,

Φ chr pGq EcpGq ϕ chr pG |qX clpq .
Notations 4.2. We shall use here the notion of acyclic orientation of G. Recall that:

An oriented graph is a pair G pV pGq, ApGqq, where V pGq is a nite set called the set of vertices of G and ApGq is a set of couples of distinct elements of G, such that for any

x, y V pGq, distinct, px, yq ApGq ùñ py, xq ApGq. A walk in G is a sequence of vertices px 1 , . . . , x k q such that for any i rk ¡ 1s, px i , x i 1 q ApGq. A cycle px 1 , . . . , x k q is a walk if k ¥ 2 and if x 1 x k . The oriented graph is acyclic if it does not contain any cycle.

If G is an oriented graph, its support is the graph supppGq dened by

V psupppGqq V pGq, EpsupppGqq ttx, yu | px, yq ApGqu.
If G is an oriented graph, a source of G is a vertex y V pGq such that for any x V pGq, px, yq ApGq. The set of sources of G is denoted by spGq. It is not dicult to show that any non empty acyclic oriented graph has at least one source. Consequently, if G is a non empty acyclic oriented graph, then any of its connected component is also an acyclic oriented graph and so contains at least one source. Therefore, if

G is not connected, |spGq| $ 1.
If G is a graph, we denote by OpGq the set of acyclic oriented graphs H such that supppHq G. If x V pGq, we denote by OpG, xq the set of acyclic oriented graphs H OpGq such that spHq txu.

Let us start by a combinatorial lemma.

Proposition 4.6. Let G be a graph and x, y V pGq. Then OpG, xq and OpG, yq are in bijection.

Proof. We assume that x $ y. Let H OpGq. We dene a partial order ¤ H on V pGq such that for any x, y V pGq, x ¤ H y if there exists an oriented path px x 1 , x 2 , . . . , x k yq in H. As H is acyclic, this relation is antisymmetric. It is obviously reexive and transitive, so it is an order on V pGq. The set of minimal elements of pV pGq, ¤ H q is spHq. Let H OpG, xq. As spHq txu, x is the unique minimal element of pV pGq, ¤ H q, so x ¤ H y.

We consider rx,

ys H tz V pGq | x ¤ H z ¤ H yu.
This is non empty. Let H I be the oriented graph obtained by changing the orientations of all the edges between two vertices of rx, ys H . Let px 1 , . . . , x k x 1 q be a cycle in H I . If none of the vertices of this cycle belongs to rx, ys H , then it is a cycle in H: this is contradiction, as H is acyclic.

Let us assume that at least one of the vertices of this cycle belongs to rx, ys H : up to a permutation, we assume that x 1 x k rx, ys H . Let us prove by induction on i that x ¤ H x i for any i. It is obvious if i 1. Let us assume that x i¡1 ¤ y. Two cases are possible:

If px i¡1 , x i q ApHq, then x ¤ H x i¡1 ¤ H x i , so x ¤ H x i .
If px i , x i¡1 q ApHq, by denition of H I , this implies that x i , x i¡1 rx, ys H , so x ¤ H x i . Let us now prove by induction on i that x k¡1 ¤ H y for any i. It is obvious for i 0, as x k x 1 . Let us assume that x k 1¡i ¤ H y. Two cases are possible.

If px k¡i , x k 1¡i q ApHq, then

x k¡i ¤ H x k 1¡i ¤ H y, so x k¡i ¤ H y.
If px k 1¡i , x k¡i q ApHq, by denition of H I , this implies that x k 1¡i , x k¡i rx, ys H , so x k¡i ¤ H y. We obtain that x 1 , . . . , x k rx, ys H , so px k , x k¡1 , . . . , x 1 q is a cycle of H: this is a contradiction, as H is acyclic.

As a conclusion, H I is acyclic.

Let z V pH I q. If z rx, ys H , then it is not a source of H (as the unique source of H is x,) so there exists t V pHq, such that pt, zq ApHq. Then pt, zq ApH I q and z spH I q. Let z rx, ys H , dierent from y. As z H y, there exists a walk in H from z to y, so there exists t rx, ys H such that pz, tq ApHq. Then pt, zq ApH I q, so z spH I q. Finally, spH I q tyu and, as spH I q $ r, spH I q tyu. We proved that H I OpG, yq. This dene a map f x,y : If px i¡1 , x i q ApHq, then x ¤ H x i¡1 ¤ H x i , so x ¤ H x i . If px i , x i¡1 q ApHq, then by denition of H I , x i¡1 , x i rx, ys H , so x ¤ H x i . Let us now prove by induction on i that x k¡i ¤ H y for any i. It is obvious if k 0, as x ¤ H y. Let us assume that x k 1¡i ¤ H y. Two cases are possible.

If px k¡i , x k 1¡i q ApHq, then x k¡i ¤ H x k 1¡i ¤ H y, so x k¡i ¤ H y.

If px k 1¡i , x k¡i q ApHq, then by denition of H I , x k 1¡i , x k¡i rx, ys H , so x k¡i ¤ H y. We proved that any vertex of px 0 , . . . , x k q belongs to rx, ys H . In particular, z rx, ys H . Therefore, rx, ys H ry, xs H I. As a consequence, f y,x ¥ f x,y Id OpG,xq . So f x,y is a bijection for any x $ y V pGq, of inverse f y,x .

Consequently, we dene: Denition 4.7. For any graph G, choosing any vertex x V pGq, we denote by φpGq the number of acyclic orientations of G, such that spGq txu. By convention, φp1q 0. This denes an innitesimal character of G.

Proof. By the preceding lemma, this does not depend on the choice of x. As any non connected graph G has at least two sources, if G is not connected, then φpGq 0. So φ is an innitesimal character.

Here is a second combinatorial lemma: Lemma 4.8. Let G be a graph and e EpGq. We denote by G{e the graph obtained by contraction of e (and so identifying the two extremities of e) and by Gze the graph obtained by deletion of e. Then Φ chr pGq Φ chr pGzeq ¡ Φ chr pG{eq, ϕ chr pGq ϕ chr pGzeq ¡ ϕ chr pG{eq, φpGq φpGzeq φpG{eq. Proof. We put e tx, yu.

1. Let us give a proof of this classical result. Let n N. Then Φ chr pGzeqpnq is the numbers of colorations f of G such that for any e I tx I , y I u V pGq, e I $ e, f px I q $ f py I q. Moreover, Φ chr pG{eqpnq is the numbers of colorations f of G such that for any e I tx I , y I u V pGq, e I $ e, f px I q $ f py I q, and such that f pxq f pyq. Taking the dierence, Φ chr pGzeqpnq¡Φ chr pG{eqpnq Φ chr pGqpnq for any n N, which gives the rst equality. Lemma 4.10. Let pA, m, ∆q be a commutative or cocommutative bialgebra. The induced convolution product on EndpAq is denoted by ¦. The canonical projection on the augmentation ideal of A is denoted by ρ. There exists a family of scalars pλpk, l, pqq k,l,pN , which does not depend on A, such that for any k, l N, ρ ¦k ¥ ρ ¦l ρ ¦l ¥ ρ ¦k kl p0 λpk, l, pqρ ¦p .

Proof. We shall use Sweedler's notation for ∆pxq x p1q x p2q for any x A. Note that λpk, l, pq 0 if p ¡ kl. As for any k, l, p N, λpk, l, pq λpl, k, pq, ρ ¦k ¥ ρ ¦l ρ ¦l ¥ ρ ¦k . Lemma 4.11. Let f pTq KrrT ss and let ρ be the projection on the augmentation ideal of KrXs.

If f pρq 0, then f 0.

Proof. For any k, n N,

ρ ¦k pX n q ¤ ¦ ¦ ¥ i1 ... i k n, i 1 ,...,i k ¥1 n! i 1 ! . . . i k ! X n .
In particular, ρ ¦k pX k q k!X k $ 0 and ρ ¦k pX n q 0 if n k. Let f KrrT ss, nonzero, and let k valpf q. Then f pρqpX k q a k ρ ¦k pX k q 0 a k k!X k $ 0, so f pρq $ 0. Lemma 4.12. Let p, k, l N. If p k or p l, then λpk, l, pq 0. Proof. We work in the bialgebra pKrXs, m, ∆q. If p l, then ρ ¦k ¥ ρ ¦l pX p q 0, as ρ ¦l pX p q 0. We consider the formal series f pTq kl p0 λpk, l, pqT p KrrT ss.

Then f pρq T ¦k ¥ T ¦l . Let q valpf q. Then f pρqpT q q λpk, l, qqq!X q $ 0, so q ¥ l. By symmetry in k, l of the coecients λpk, l, pq, valpf q ¥ k. Proposition 4.13. Let A be a connected commutative bialgebra. We put

ϖ lnpIdq V ķ1 p¡1q k 1 k ρ ¦k .
Then ϖ is a projection. Its kernel is A 2 K1 A and its image contains PrimpAq.

Proof. By denition of the coecients λpk, l, pq and by the preceding lemma, for any formal series f °ak T k and g °bk T k in KrrT ss,

f pρq ¥ gpρq V p0 ¤ ¥ ķ ,l¤p λpk, l, pqa k b l ρ ¦p .
We consider the case where A pKrXs, m, ∆q. As it is a double bialgebra, in this case, by , so x A 2 . We obtain that Kerpϖq A 2 K1 A . Note that ϖp1 A q 0. Moreover, π ¥ m m ¦ pπq m ¦ plnpIdqq lnpm ¦ pIdqq lnpId Idq lnppId ιq ¦ pι Idqq lnpId ιq lnpι Idq lnpIdq ι ι lnpIdq ϖ ι ι ϖ.

Therefore, if x, y A , ϖpxyq ϖpxqεpyq εpxqϖpyq 0. So A 2 K1 A Kerpϖq. Corollary 4.14. Let pA, m, ∆q be a connected and commutative bialgebra. Then pA, m, ∆q is isomorphic to a subbialgebra of the shue algebra pTpPrimpAqq, ¡,∆q. Proof. By the universal property of pTpPrimpAqq, ¡,∆q, (Proposition 3.5), for any linear map ϕ : A ÝÑ PrimpAq such that ϕp1 A q 0, there exists a unique coalgebra morphism Φ : A ÝÑ T pPrimpAqq such that π ¥Φ ϕ, where Φ : T pPrimpAqq ÝÑ PrimpAq is the canonical projection.

By Proposition 4.13, PrimpAq A 2 p0q. Let us choose ϕ such that ϕ |PrimpAqq Id PrimpAq and ϕpA 2 q p0q. We denote by Φ the corresponding coalgebra morphism from A to T pPrimpAqq. As Φ |PrimpAq is injective, by Proposition 3.5, Φ is injective. As ϕpA 2 q p0q, still by Proposition 3.5, Φ is a bialgebra morphism from pA, m, ∆q to pTpPrimpAqq, ¡,∆q. Corollary 4.15. Let pA, m, ∆q be a connected commutative bialgebra. Then it can be embedded in a double bialgebra pB, m, ∆, δq, with PrimpBq PrimpAq. If A is cofree or if A is cocommutative, then there exists a second coproduct δ on A making it a double bialgebra.

Proof. First step. Let pV, ¤q be a commutative algebra. We can consider the quasishue algebra pTpV q, , ∆q. By Corollary 4.15 and its proof, choosing a convenient ϕ, there exists an injective bialgebra morphism Φ : pTpV q, , ∆q ÝÑ pTpV q, ¡,∆q, such that Φpvq v for any v V .

Moreover, for any

v 1 , . . . , v n V , with n ¥ 1, Φpv 1 . . . v n q 1 1 ¡ ϕ pv 1 . . . v n q n ķ1 v1 ...vnw 1 ...w k , w 1 ,...,w i $1 ϕpw 1 q . . . ϕpw k q loooooooomoooooooon V k v 1 . . . v n words of length n.
An easy triangularity argument proves then that Φ is bijective. Hence, pTpV q, , ∆q and pTpV q, ¡,∆q are isomorphic.

In the particular case where V is a commutative bialgebra, we obtain a second coproduct δ on T pV q, making it a double bialgebra.

Second step. Let pA, m, ∆q be a connected bialgebra. Let us choose any commutative bialgebra structure on PrimpAq. As pTpPrimpAqq, ¡,∆q and pTpPrimpAqq, , ∆q are isomorphic, from Corollary 4.14, there exists an injective bialgebra morphism from A to pTpPrimpAqq, , ∆q, which proves the rst point, as pTpPrimpAqq, , ∆q is a double bialgebra. Last step. If A is cofree, then the injection from A to T pPrimpAqq is a bijection. If A is cocommutative, as it is connected it is primitively generated by Cartier-Quillen-Milnor-Moore's theorem: hence, its image is the subalgebra A I of pTpV q, q generated by PrimpT pV qq V .

As δpV q V V by construction of V , δpA I q A I A I so A I is a double subbialgebra of pTpPrimpAqq, , ∆, δq.

a k T k , N ķ0 b k X k y N ķ0 a k b k .
2. Let g : rns ↠ rls be a surjective map. We put dpgq Uti rn ¡ 1s | gpiq ¥ gpi 1qu, P g pXq X dpgq 1 p1 Xq n¡1¡dpgq KrXs.

The letter d is for descents. Proposition 4.16. Let pV, ¤, δ V q be a commutative, non necessarily unitary bialgebra. We consider the double quasishue bialgebra pTpV q, , ∆, δq. Let QpT q KrrT ss and let λ Qpϵ δ ¡ ε ∆ q T pV q ¦ . For any word

v 1 . . . v n V n , with n ¥ 1, Θpλqpv 1 . . . v n q n ļ1 ģ:rns↠rls xQpTq, P g pXqy ¤ ¥ ¤ ¹ gpiq1 v i . . . ¤ ¥ ¤ ¹ gpiql v i . Proof. For any v 1 . . . v n V n , with n ¥ 1, λpv 1 . . . v n q Qpϵ δ ¡ ε ∆ qpv 1 . . . v n q V ķ1 a k ϵ k δ ¥ ∆pk¡1q pv 1 . . . v n q a n ϵ V pv 1 q . . . ϵ V pv n q,
as ϵ δ vanishes on any word of length ¥ 2. By denition of the coproduct δ, δpv 1 . . . v n q ķ ,l¥1, f :rns↠rks,increasing g:rns↠rls, di,jrns, pi j and f piqfpjqqùñgpiq gpjq

¤ ¥ ¹ f piq1 v I i . . . ¤ ¥ ¹ f piqk v I i ¤ ¥ ¹ gpiq1 v P i . . . ¤ ¥ ¹ gpiql v P i ķ
,l¥1, g:rns↠rls, f :rns↠rks,increasing, di,jrns, pi j and gpiq¥gpjqqùñf piq f pjq

¤ ¥ ¹ f piq1 v I i . . . ¤ ¥ ¹ f piqk v I i ¤ ¥ ¹ gpiq1 v P i . . . ¤ ¥ ¹ gpiql v P i .
For any g : rns ↠ rls, let us put Apgq tf : rns ↠ rks, increasing | di, j rns, pi j and gpiq ¥ gpjqq ùñ f piq f pjqu.

Then, putting QpT q °ak T k , Θpλqpv 1 . . . v n q pλ Idq ¥ δpv 1 . . . v n q ļ¥1 ģ:rns↠rls fApgq

λ ¤ ¥ ¤ ¥ ¹ f piq1 v I i . . . ¤ ¥ ¹ f piqmaxpfq v I i ¤ ¥ ¹ gpiq1 v P i . . . ¤ ¥ ¹ gpiql v P i ļ¥1 ģ:rns↠rls fApgq a maxpf q n ¹ i1 ϵ V pv I i q ¤ ¥ ¹ gpiq1 v P i . . . ¤ ¥ ¹ gpiql v P i ļ¥1 ģ:rns↠rls ¤ ¥ fApgq a maxpf q ¤ ¥ ¹ gpiq1 v i . . . ¤ ¥ ¹ gpiql v i .
For any k N ¡0 , we put A k pgq tf Apgq | maxpf q ku and we put

R g pXq ķ¥1 |A k pgq|X k .
With this denition, we obtain that The result R g pXq P g pXq then comes from an easy induction on n.

Let us apply this formula for QpT q 1 1 T and QpT q lnp1 T q: Corollary 4.17. Let pV, ¤q be a commutative (non necessarily unitary) algebra. In the Hopf algebra pTpV q, , ∆q, the antipode S is given on any non empty word v 1 . . . v n by Spv 1 . . . v n q p¡1q n ļ¥1 ģ:rns↠rls,decreasing

¤ ¥ ¤ ¹ gpiq1 v i . . . ¤ ¥ ¤ ¹ gpiql v i .
The eulerian idempotent is given on any non empty word v 1 . . . v n by ϖpv 1 . . . v n q ļ¥1 ģ:rns↠rls

p¡1q dpgq dpgq!pn ¡ 1 ¡ dpgqq! n! ¤ ¥ ¤ ¹ gpiq1 v i . . . ¤ ¥ ¤ ¹ gpiql v i .
Proof. By functoriality, as any commutative algebra is the quotient of a commutative bialgebra, it is enough to prove it for a commutative bialgebra. For the antipode, we use Proposition 4.16

with QpT q 1 1 T

. For any n N, xQpTq, X n y p¡1q n , so xQpTq, P pXqy P p¡1q for any P pXq KrXs. Therefore, if g : rns ↠ rls, xQpTq, P g pXqy P g p¡1q 5 p¡1q n if dpgq n ¡ 1, 0 otherwise, 5 p¡1q n if g is decreasing, 0 otherwise.

For the eulerian idempotent, we use QpT q lnp1 T q. For any n N ¡0 , xQpTq, X n y p¡1q An easy induction on p, based on an integration by part, proves that for any p, q N, » 0 ¡1 t p p1 tq q dt p¡1q k k!l! pk l 1q! .

The result immediately follows, with p dpgq and q n ¡ 1 ¡ dpgq.

5 Graded double bialgebras 5.1 Reminders Denition 5.1. Let pB, m, ∆q be a bialgebra. We shall say that it is graded if there exists a graduation pB n q nN of B such that: For any k, l N, mpB k B l q B k l .

For any n N, ∆pB n q n ķ0 B k B n¡k .

We shall say that the graduation is connected if B 0 K1 B . Example 5.1. This is the case of KrXs, with KrXs n KX n for any n. The bialgebra QSym is also graded and connected, putting any composition pk 1 . . . k n q homogeneous of degree k 1 . . . k n . The bialgebra of graphs pH G , m, ∆q is also graded by the number of vertices of the graphs.

Proof. By Theorem 3.12, Ψ µ Φ øλ, so Ψ µ øλ ¡1 Φ øpλλ ¡1 q Φ.

For any x B, ϕpxq, coecient of X in Φpxq, is given by ϕpxq ϵ δ ¥ ϖ 1 ¥ Φ, where ϖ 1 : KrXs ÝÑ KX is the canonical projection. By homogeneity of Ψ µ , ϕ ϵ δ ¥ ϖ 1 ¥ pΨ µ λ ¡1 q ¥ δ ϵ δ ¥ pΨ µ λ ¡1 q ¥ pϖ 1 Idq ¥ δ pλ λ ¡1 q ¥ pπ 1 Idq ¥ δ ppλ ¥ π 1 q λ ¡1 q ¥ δ pµ λ ¡1 q ¥ δ µ λ ¡1 .

Example 5.2. Let µ pH G q 1 dened by µp q 1. Then exppµqp q µp1q 1. In the same way as in [START_REF] Foissy | Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials[END_REF]Proposition 11], it is possible to prove that λ exppµq is invertible for the product.

Moreover, for any graph G with n vertices,

λpGq 1 n! V pGqI 1 ...In, |I 1 |...|In|1 n ¹ i1 µpG |I i q n! n! 1.
Let G be a graph, and let E c pGq. If G is not connected, then G{ has at least two vertices, so pπ 1 Idq ¥ δpGq 0, which implies (again) that ϕ chr pGq 0. Let us now assume that G is connected. The unique 0 E c pGq such that |G{ 0 | 1 is the equivalence with only one class, which indeed belongs to E c pGq as G is connected. Moreover, G | 0 G. We obtain ϕ chr pGq µ λ ¡1 pGq µp qλ ¡1 pGq λ ¡1 pGq. Therefore, for any connected graph G, λ ¡1 pGq ϕ chr pGq, which entirely determines the character λ ¡1 . For any graph G,

Φ chr pGq EcpGq Ψ µ pG{ qλ ¡1 pG |q EcpGq ¤ ¥ ¹ CV pGq{ ϕ chr pG |C q X clpq ,
where clpq is the number of classes of .

Morphisms to QSym

Let us recall the following result, due to Aguiar, Bergeron and Sottile [2]: Proposition 5.5. Let pB, m, ∆q be a graded and connected bialgebra, and let λ CharpBq.

There exists a unique homogeneous bialgebra morphism Φ λ : pB, m, ∆q ÝÑ pQSym, , ∆q such that ϵ δ ¥ Φ λ λ. For any x B , Φ λ pxq ņ¥1 ķ1

,...,knN ¡0 λ k ¥ pπ k 1 . . . π kn q ¥ ∆pn¡1q pxqpk 1 . . . k n q.

Proposition 5.6. Let pB, m, ∆, δq be a double bialgebra, such that pB, m, ∆q is a graded and connected bialgebra in the sense of Denition 5.1.

1. If dn N, δpB n q B n B KerpΦ ϵ δ Φ ϵ δ q,

then the unique homogeneous double bialgebra morphism from pB, m, ∆, δq to pQSym, , ∆, δq is Φ ϵ δ . Otherwise, there is no homogeneous double bialgebra morphism from B to QSym.

2. For any λ CharpBq, the unique homogeneous bialgebra morphism Φ λ : pB, m, ∆q ÝÑ pQSym, , ∆q such that ϵ δ ¥ Φ λ λ is Φ øλ.

Proof. 1. Unicity. Let Φ be such a morphism. Then ϵ δ ¥ Φ ϵ δ . By the unicity in Aguiar, Bergeron and Sottile's theorem, Φ Φ ϵ δ .

1. Existence. Let us rst assume that (3) holds, and let us prove that Φ Φ ϵ δ is a double bialgebra morphism. Let us prove that for any x B n , δ ¥ Φpxq pΦ Φq ¥ δpxq by induction on n. If n 0, we can assume that x 1 B , and then δ ¥ Φp1 B q 1 1 pΦ Φq ¥ δp1 B q. Let us assume the result at all ranks n. For any x B n , as Φ : pB, m, ∆q ÝÑ pQSym, , ∆q is a bialgebra morphism, p ∆ Idq ¥ δ ¥ Φpxq Therefore, if k $ n, x k,l KerpΦ Φq and we nally obtain that x B n B KerpΦ Φq.

2. Let λ CharpBq. Then Φ øλ is a bialgebra morphism. For any n N, Φ øλpB n q pΦ λq ¥ δpB n q pΦ λqpB n Bq ΦpB n q QSym n , so Φ øλ is homogeneous. Moreover, ϵ δ ¥ pΦ øλqpϵ δ ¥ Φq λ ϵ δ λ λ. Hence, Φ øλΦ λ . Remark 5.2. Let pB, m, ∆, δq be a connected double bialgebra and let Φ be a double bialgebra morphism from pB, m, ∆, δq to pQSym, m, ∆, δq. Then the unique double bialgebra morphism from pB, m, ∆, δq to pKrXs, m, ∆, δq is Φ QSym ¥ Φ, where Φ QSym is described in Remark 3.1.

Remark 5.3. Non homogeneous double bialgebra morphisms from B to QSym may exist. For example, the algebra morphism Ψ : KrXs ÝÑ QSym sending X to p1q is a double bialgebra morphism. By composition with the double bialgebra morphism from B to KrXs, we obtain non homogeneous double bialgebra morphisms from B to QSym.

Remark 5.4. The hypothesis (3) does not hold if B H G . For example, pΦ ϵ δ Φ ϵ δ q ¥ δp q pΦ ϵ δ Φ ϵ δ qp q p1q 2p11q 2p11q p2p11q p2qq, δ ¥ Φ ϵ δ p q δp2p11qq p2q 2p11q 2p11q p2p11q p2qq.

A way to correct this is to work with decorated graphs, see [START_REF]Chromatic polynomials and bialgebras of graphs[END_REF] for more details.

Example 3 . 1 .

 31 In pH G , m, ∆q, for any graph G, ∆pk¡1q pGq I1 ...I k V pGq, I 1 ,...,I k $r G |I 1 . . . G |I k .

  Let x A. Then Id ¦k pxq x p1q . . . x pkq . Therefore, for any x A, Id ¦k ¥ Id ¦l pxq Id ¦k ¡ x p1q . . . x plq © ¡ x p1q . . . x plq © p1q . . . ¡ x p1q . . . x plq © pkqx p1q x pl 1q . . . x ppk¡1ql 1q . . . x pkq x p2kq . . . x pklq x p1q . . . x pklq Id ¦kl pxq.We use that A is commutative or cocommutative for the fourth equality. Hence, Id ¦k ¥ Id ¦lId ¦kl .Let ι be the unit of ¦. Then ρ Id ¡ ι, and ρ ¦k ¥ ρ ¦l pId ¡ ιq ¦k ¥ pId ¡ ιq ¦l

Proposition 4 . 4 ,p 1 pρ

 441 ϖ is a projection. Hence, ¦p ϖ, so ϖ is a projection. Let x PrimpAq. Then ϖpxq ρpxq 0 x, so x Impϖq. Let x Kerpϖq A . Then

4. 4

 4 Antipode and eulerian idempotent for quasishue algebras Notations 4.3. 1. We identify KrrT ss and the dual of KrXs, with the pairing dened by x V ķ0

  the eulerian idempotent of B is Θpϕq pϕ Idq ¥ δ, see Proposition 4.1. for any character λ of B, lnpλq ϕ λ, see Proposition 4.2.

We dene the restricted graph G | by V pG |q V pGq, EpG |q ttx, yu EpGq | π pxq π pyqu. In other words

  , G | is the disjoint union of the subgraphs G |C , with C V pGq{ . We shall say that E c pGq if for any class C V pGq{ , G |C is a connected graph. We then dene a

	Example 1.4.
	second coproduct δ on H G by
	dG G,	δpGq	EcpGq	G{ G | .
	This coproduct is coassociative, but not cocommutative. Its counit ϵ δ is given by
	dG G,	ϵ δ pGq	5	1 if EpGq r, 0 otherwise.

  1.3 Quasishue algebras Notations 1.1. Let k, l N. A pk, lq-quasishue is a surjective map σ : rk ls ÝÑ rmaxpσqs such that σp1q . . . σpkq and σpk 1q . . . σpk lq. The set of pk, lq-quasishues is denoted by QShpk, lq.

  First step. Let w V n , with n ¥ 1.

	δpwq		ww 1 ...wn,	|w I 1 | . . . |w I n | w P 1 . . . w P
			w 1 ,...,wn$1
	p∆ Idq ¥ δpwq	ww	|w I 1 | . . . |w I k | |w I k 1 | . . . |w P k l | w P 1 . . . w P n
	1 ...w k l , w 1 ,...,w k l $1 w k 1 ,...,w k l $1 w p2q w k 1 ...w k l , w 1 ,...,w k $1, w p1q w 1 ...w k , ww p1q w p2q ,	|w I 1 | . . . |w I k | |w I k 1 | . . . |w P k l | w P 1 . . . w P n
	1,3,24 ¥ pδ δq ¥ ∆pwq.

n .

  3,24 ¥ pδ δq ¥ ∆pxq 1,4,25,36 ¥ pδ Id δ Idq ¥ pδ δq ¥ ∆pxq, By the induction hypothesis, p ∆ Id Idq ¥ pδ Idq ¥ δpxq p ∆ Id Idq ¥ pId δq ¥ δpxq, so pδ Idq ¥ δpxq ¡ pId δq ¥ δpxq V.

	Moreover,
	pπ Idq ¥ pδ Idq ¥ δpxq

whereas p ∆ Id Idq ¥ pId δq ¥ δpxq pId Id ∆q ¥ p ∆ Idq ¥ δpxq pId Id ∆q ¥ 1,3,24 ¥ pδ δq ¥ ∆pxq 1,4,25,36 ¥ pId δ Id δq ¥ pδ δq ¥ ∆pxq. ¡ x p1q © I δ ¢ ¡

  ∆q is a Hopf algebra of antipode S. Corollary 2.4. Let pB, m, ∆, δq be a double bialgebra, such that pB, m, ∆q is a Hopf algebra. Then pB, mq is commutative. Proof.

As pB, m, ∆q is a Hopf algebra, ϵ δ has an inverse for the convolution product ¦, and the antipode of pB, m, ∆q is S pϵ ¦¡1 δ Idq ¥ δ by Corollary 2.3. As ϵ δ is a character of pB, m, ∆q, its inverse ϵ ¦¡1 δ is also a character. By composition, S is an algebra endomorphism of B. By the classical result on the antipode

[START_REF] Abe | Hopf algebras[END_REF][START_REF] Sweedler | Hopf algebras[END_REF]

, it is also a algebra anti-endomorphism. Hence, SpBq is a commutative subalgebra of B. It is enough to prove that S is surjective. By Lemma 1.5,

  1. ùñ. If Φ is injective, by restriction Φ |PrimpBq is injective. If x PrimpBq, then Φpxq PrimpT pV qq V , so Φpxq π ¥ Φpxq ϕpxq: we obtain that ϕ |PrimpBq is injective. ðù. Let us assume that Φ is not injective. Let x KerpΦq B , nonzero, with deg p pxq n

	minimal. Then

4

  CharpBq ÝÑ Hom b pB, KrXsq λ ÝÑ Φ øλ, Hom b pB, KrXsq ÝÑ CharpBq Ψ ÝÑ ϵ δ ¥ Ψ. pG |f ¡1 p1q q . . . ϵ δ pG |f ¡1 pkq qH k pXq. Moreover, by denition of ϵ δ , ϵ δ pG |f ¡1 p1q q . . . ϵ δ pG |f ¡1 pkq q $ 0 if, and only if, for any i, G |f ¡1 piq has no edge. This gives us the well-known concept of a valid coloration: a k-coloration is a map f : V pGq ÝÑ rks; it is packed if f is surjective and it is valid if for any tx, yu EpGq, f pxq $ f pyq. Hence, denoting by PVCpGq the set of packed valid coloration of G, ΦpGq

	Proof. Immediate.
	Example 3.2. Let us consider the case of H G . For any non empty graph G,
	∆pk¡1q pGq	f:V pGq↠rks	G |f ¡1 p1q . . . G |f ¡1 pkq ,
	therefore	V ķ1	f:V
	ΦpGq		ϵ δ
			pGq↠rks

4 fPVCpGq H maxpf q . Consequently, for any k N, ΦpGqpnq is the number of valid n-colorations of G: in other words, ΦpGq is the chromatic polynomial of G [17]. Theorem 3.13. The unique double bialgebra morphism Φ chr from pH G , m, ∆, δq to pKrXs, m, ∆, δq sends any graph G to its chromatic polynomial. Example 3.3.

  2 K1 B . Proof. As pB, m, ∆q is a cocommutative bialgebra, it is primitively generated by Cartier-Quillen-Milnor-Moore's theorem. Hence, B PrimpBq B 2 K1 B . As PrimpBq Impϖq and ϖ vanishes on B 2 K1 B , PrimpBq Impϖq. Example 4.2. Let G be a connected graph. For any EpGq, G{ is connected. Hence, as H G

	is cocommutative,
	ϖpGq

EcpGq

ϕ chr pG{ qG | PrimpH G q.

4

  OpG, xq ÝÑ OpG, yq H ÝÑ H I . Let us consider ry, xs H I. By construction of H I , rx, ys H ry, xs H I. Let z ry, xs H I. There exists a walk px 0 y, . . . , x j z, . . . , x k xq in H I . Let us prove by induction on i that x ¤ H x i for any i. It is obvious if i 0, as x ¤ H y. Let us assume that x ¤ H x i¡1 . Two cases are possible.

  It remains to prove that R g pXq P g pXq. For this, let us now study Apgq for g : rns ↠ rls. For any f : rn ¡ 1s ÝÑ rks, increasing, we put υ 0 pfq : Denoting by g I the standardization of the restriction of g to rn¡1s (that is to say the composition of g |rn¡1s with the unique increasing bijection from gprn¡1sq to rl I s for a well-chosen l I ), we obtain that Apgq 5 tυ pfq | f Apg I qu if gpn ¡ 1q ¥ gpnq, tυ pfq, υ 0 pfq | f Apg I qu if gpn ¡ 1q gpnq.As υ 0 does not change the maximum and υ 1 increases it by 1:If gpn ¡ 1q ¥ gpnq, then dpgq dpg I q 1 and |A k pgq| |A k¡1 pg I q|. Hence, R g pXq XR g IpXq.If gpn ¡ 1q ¡ gpnq, then dpgq dpg I q and |A k pgq| |A k pg I q| |A k¡1 pg I q|. Hence, R g pXq pX 1qR g IpXq.

		Θpv 1 . . . v n q	ļ¥1 ģ:rns↠rls xQpTq, R g pXqy	¤ ¥	¤ ¹	v i	. . .	¤ ¥	¤ ¹	v i	.
				gpiq1			gpiql
	6 8	rns ÝÑ rks	6 8	rns ÝÑ rks
	7			7

i rn ¡ 1s ÝÑ f piq, n ÝÑ f pn ¡ 1q, υ pfq : i rn ¡ 1s ÝÑ f piq, n ÝÑ f pn ¡ 1q 1.

  In particular, if g : rns ↠ rls, xQpTq, P g pXqy

		n 1 n	» 0 ¡1	t n¡1 dt,
	so for any P pXq XKrXs,	xQpTq, P pXqy	» 0 ¡1	P ptq t	dt.
		» 0
		¡1

t dpgq p1 tq n¡1¡dpgq dt.

  1,3,24 ¥ pδ δq ¥ ∆ ¥Φpxq 1,3,24 ¥ pδ δq ¥ pΦ Φq ¥ ∆pxq 1,3,24 ¥ pΦ Φ Φ Φq ¥ pδ δq ¥ ∆pxq pΦ Φ Φq ¥ m 1,3,24 ¥ pδ δq ¥ ∆pxq pΦ Φ Φq ¥ p ∆ Idq ¥ δpxq p ∆ Idq ¥ pΦ Φq ¥ δpxq. We use the induction hypothesis for the third equality, as Hence, δ ¥ Φpxq ¡ pΦ Φq ¥ δpxq PrimpQSymq QSym. Moreover, by homogeneity of Φ, δ ¥ Φpxq δpQSym n q QSym n QSym. Φq ¥ δpxq pΦ ΦqpB n B KerpΦ Φqq ΦpB n q ΦpBq QSym n QSym, so nally δ ¥ Φpxq ¡ pΦ Φq ¥ δpxq PrimpQSymq QSym n QSym Kpnq QSym, and we put δ ¥ Φpxq ¡ pΦ Φq ¥ δpxq pnq y. As ϵ δ ppnqq 1, y pϵ δ Idq ¥ δ ¥ Φpxq ¡ pϵ δ Idq ¥ pΦ Φq ¥ δpxq Φpxq ¡ pϵ δ Φq ¥ δpxq Φpxq ¡ Φpxq 0, so nally δ ¥ Φpxq pΦ Φq ¥ δpxq. Let us now assume that Φ is a double bialgebra morphism. Let n N. For any x B n , as Φ is homogeneous, pΦ Φq ¥ δpxq δ ¥ Φpxq δpQSym n q QSym n QSym. where x k,l B k B l for any k, l. Then, by homogeneity of Φ, pΦ Φq ¥ δpxq

	Let us put	δpxq	ķ ,l¥0	x k,l ,
		ķ ,l¥0	pΦ Φqpx k,l q
		∆pxq	n¡1 à
	By (3),

i1 B i B n¡i . pΦ looooooomooooooon QSym k QSym l QSym n QSym.

q . . . ϵ V pv P n q v 1 . . . v n .The fact that ϵ V is an algebra morphism is left to the reader.A particular case is obtained when V is the bialgebra of a semigroup pΩ, q. In this case, a basis of the quasishue algebra is given by words in Ω. This construction is established in[START_REF] Ebrahimi-Fard | A comodule-bialgebra structure for word-series substitution and mould composition[END_REF], where it is related to Ecalle's mould calculus (product and composition of symmetrel moulds).For example, if k 1 , k

, k

, k

Ω, in this quasishue double bialgebra, pk 1 q pk 2 k 3 k 4 q pk 1 k2 k 3 k 4 q pk 2 k 1 k 3 k 4 q pk 2 k 3 k 1 k 4 k 2 k 3 k 4 k 1 q ppk 1 k 2 qk 3 k 4 q pk 2 pk 1 k 3 qk 4 q k 2 k 3 pk 1 k 4 qq, pk 1 k 2 q pk 3 k 4 q pk 1 k 2 k 3 k 4 q pk 1 k 3 k 2 k 4 q pk 1 k 3 k 4 k 2 q pk 3 k 1 k 2 k 4 q pk 3 k 1 k 4 k 2 q pk 3 k 4 k 1 k 2 q ppk 1 k 3 qk 2 k 4 q ppk 1 k 3 qk 2 k 4 q pk 3 pk 1 k 4 qk 2 q pk 1 pk 2 k 3 qk 4 q pk 1 k 3 pk 2 k 4 qq pk 3 k 1 pk 2 k 4 qq ppk 1 k 3 qpk 2 k 4 qq, ∆ppk 1 k 2 k 3 k 4 qq pk 1 k 2 k 3 k 4 q 1 pk 1 k 2 k 3 q pk 4 q pk 1 k 2 q pk 3 k 4 q pk 1 q pk 2 k 3 k 4 q 1 pk 1 k 2 k 3 k 4 q,δppk 1 qq pk 1 q pk 1 q, δppk 1 k 2 qq pk 1 k 2 q pk 1 q pk 2 q pk 1 k 2 q pk 1 k 2 q, δppk 1 k 2 k 3 qq pk 1 k 2 k 3 q pk 1 q pk 2 q pk 3 q ppk 1 k 2 qk 3 q pk 1 k 2 q pk 3 q pk 1 pk 2 k 3 qq pk 1 q pk 2 k 3 q pk 1 k 2 k 3 q pk 1 k 2 k 3 q.

ùñ. Let us assume that Φ : pB, m, ∆q ÝÑ pTpV q, , ∆q is a bialgebra morphism. As π : pTpV q , q ÝÑ pV, ¤q is an algebra morphism, by composition π ¥ Φ |B ϕ |B is an algebra

Acknowledgments: the author acknowledges support from the grant ANR-20-CE40-0007 Combinatoire Algébrique, Résurgence, Probabilités Libres et Opérades. The author also warmly thanks Darij Grinberg for his careful reading and his helpful comments.

2. Direct consequence of the rst point, as ϕ chr pHq Φ chr pHq I p0q for any graph H. Let H be an orientation of G{e and let H 1 and H 2 be the two orientations of G inducing H: in H 1 , e is oriented from x to y whereas in H 2 , it is oriented from y to x. We assume that H OpG{e, xq. As H is acyclic, H 1 and H 2 are acyclic (as the contraction of a cycle is a cycle). Let z be a source of H 1 or of H 2 . If z $ x, y, it is also a vertex of G{e and is not a source of G{e, so it is not a source of H Summing, this gives the announced formula.

The following result is rstly due to Greene and Zaslavsky [START_REF] Greene | On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs[END_REF], see [START_REF] Gebhard | Sinks in acyclic orientations of graphs[END_REF] for several proofs of dierent natures: Theorem 4.9. For any graph G, ϕ chr pGq p¡1q |V pGq| 1 φpGq. Proof. We proceed by induction on the number n of edges of G. If EpGq r, then G n for a certain n N. Then Φ chr pGq X n , so

Let us assume the result at all ranks n. Let us choose any edge e of G. As G{e and Gze has strictly less than n edges, ϕ chr pGq ϕ chr pGzeq ¡ ϕ chr pG{eq p¡1q |V pGq| 1 ϕpGzeq ¡ p¡1q |V pGq| ϕpG{eq p¡1q |V pGq| 1 pϕpGzeq ϕpG{eqq p¡1q |V pGq| 1 ϕpGq.

Generalization to commutative connected bialgebras

We proved in Proposition 4.4 that in the case of a connected double bialgebra, the eulerian idempotent ϖ is a projector. We now extend this result to any commutative connected bialgebra.

Remark 5.1. If pB, m, ∆q is a graded and connected bialgebra, then for any n ¥ 1,

Inductively, for any n, k ¥ 1,

In particular, if x B n , for any k ¡ n, ∆pk¡1q pxq 0: B is connected in the sense of the preceding section.

Homogeneous polynomial invariants

If pB, m, ∆, δq is a graded connected bialgebra, a natural question is to nd all the homogeneous bialgebra morphisms from B to KrXs. For this, we identify B ¦

Note that as B is connected, B ¦ 1 InfCharpBq. We obtain: Proposition 5.2. Let µ InfCharpAq and let Ψ µ : pB, m, ∆q ÝÑ pKrXs, m, ∆q associated to µ by Proposition 3.10. Then Ψ µ is homogeneous if, and only if, µ B ¦ 1 .

Proof. ðù. Let us assume that µ B ¦ 1 . Let n ¥ 1 and x B n . Then

ùñ. Let us assume that µ B ¦ 1 . As µp1 B q 0, there exists n ¥ 2 and x B n such that µpxq $ 0. Then the coecient of X in Ψ µ pxq is µpxq $ 0, so Ψ µ pxq is not homogeneous of degree n and Ψ µ is not homogeneous. The second formula comes immediately.

Proposition 5.4. Let µ B ¦ 1 , such that λ exppµq is invertible for the product . Then Φ Ψ µ øλ ¡1 , ϕ µ λ ¡1 .