C Van Chiên

Gérard H E Bui

Quoc Duchamp

H Hoan

Vincel Ngô

Ngoc Hoang

Christophe Minh

Tollu

V C Bui

G H E Duchamp † ‡

Q H Ngô ‡

V Hoang

Ngoc Minh

C Tollu

(Pure) transcendence bases in ϕ-deformed shuffle bialgebras

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The shuffle product first appeared in 1953 in the work of Eilenberg and Mac Lane [START_REF] Eilenberg | On the groups H(Π, n) I[END_REF]. As soon as 1954, Chen used it to express the product of iterated (path) integrals [START_REF] Chen | Iterated integrals and exponential homomorphisms[END_REF], and Ree, building on Friedrichs' criterion, proved that the non-commutative generating series of iterated integrals are exponentials of Lie polynomials, thus connecting the Lie polynomials with the shuffle product [START_REF] Ree | Lie elements and an algebra associated with shuffles[END_REF]. In 1956, Radford proved that the Lyndon words form a (pure) transcendence basis of the shuffle algebra [START_REF] Radford | A natural ring basis for the shuffle algebra and an application to group schemes[END_REF]. The latter result is now well understood through the duality between bialgebras and enveloping algebras (see for example [START_REF] Reutenauer | Free Lie algebras[END_REF]), of which the construction in 1958 of the Poincaré-Birkhoff-Witt 2 -Lyndon basis by Chen, Fox and Lyndon [START_REF] Chen | -Free differential calculus, IV. The quotient groups of the lower central series[END_REF] and of its dual basis by Schützenberger, via monoidal factorization [START_REF] Schützenberger | Sur une propriété combinatoire des algèbres de Lie libres pouvant être utilisée dans un problème de mathématiques appliquées, Séminaire Dubreil[END_REF][START_REF] Reutenauer | Free Lie algebras[END_REF], gave a striking illustration. This pair of dual bases enabled one to factorize the diagonal series in the shuffle bialgebra and, consequently, to proceed combinatorially with the Dyson series [START_REF] Ngoc | Symbolic integration of meromorphic differential equation via Dirichlet functions[END_REF] or the transport operator [START_REF] Ngoc | Input/Output behaviour of nonlinear control systems: Rational approximations, nilpotent structural approximations[END_REF], which play a leading role in the relations between special functions involved in the theory of quantum groups [START_REF] Kassel | Quantum groups[END_REF] and in number theory [START_REF] Cartier | Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents, Séminaire Bourbaki, 53 e année[END_REF].

In 1973, that is, within twenty years of the introduction of the shuffle product, Knutson defined the quasi-shuffle in [START_REF] Knutson | λ-rings and the representation theory of the symmetric group[END_REF], where it shows up as the inner product of functions on the symmetric groups 3 . This product is very similar to the Rota-Baxter operator introduced by Cartier in 1972, in his study of the so-called Baxter algebras [START_REF] Cartier | On the structure of free Baxter algebras[END_REF]. Although the analogue of Radford's theorem was pointed out by Malvenuto and Reutenauer [START_REF] Malvenuto | Duality between quasi-symmetric functions and the Solomon descent algebra[END_REF], the factorization of the diagonal series in the quasi-shuffle bialgebra, initiated in [START_REF] Ngoc | On a conjecture by Pierre Cartier about a group of associators[END_REF][START_REF] Ngoc | Structure of polyzetas and Lyndon words[END_REF], has not yet been carried over to more general bialgebras.

Schützenberger's factorization 4 [START_REF] Reutenauer | Free Lie algebras[END_REF] and its extensions have since been applied to the renormalization of the associators [START_REF] Ngoc | On a conjecture by Pierre Cartier about a group of associators[END_REF][START_REF] Ngoc | Structure of polyzetas and Lyndon words[END_REF], to which matter they turned out to be central 5 .

The coefficients of these power series are polynomial functions of positive integral multiindices of Riemann's zeta function 6 [START_REF] Lê | Kontsevich's integral for Kauffman polynomial[END_REF][START_REF] Zagier | Values of zeta functions and their applications[END_REF], and they satisfy quadratic relations [START_REF] Cartier | Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents, Séminaire Bourbaki, 53 e année[END_REF] which Lyndon words help explicit and explain. The latter relations can be obtained by identifying the local coordinates on a bridge equation connecting the Cauchy and the Hadamard algebras of polylogarithmic functions, and by using the factorization of the non-commutative generating series of polylogarithms [START_REF] Hoang Ngoc Minh | Computation of the monodromy of generalized polylogarithms[END_REF] and of harmonic sums [START_REF] Ngoc | On a conjecture by Pierre Cartier about a group of associators[END_REF][START_REF] Ngoc | Structure of polyzetas and Lyndon words[END_REF]. This bridge equation is mainly a consequence of the isomorphisms between the algebra of non-commutative generating series of polylogarithms and the shuffle algebra on the one hand, and between the algebra of non-commutative generating series of harmonic sums and the quasi-shuffle algebra on the other hand.

As for the generalization of Schützenberger's factorization to more general bialgebras, the key step, and the main difficulty thereof, is to decompose orthogonally such bialgebras into the Lie algebra generated by its primitive elements and the associated orthogonal ideal, as Ree was able to achieve in the case of the shuffle bialgebra [START_REF] Ree | Lie elements and an algebra associated with shuffles[END_REF], and to construct, whenever possible, the respective bases. In favorable cases, it is to be hoped that those bialgebras are enveloping algebras, so that the Eulerian projectors are convergent and other analytic computations can be performed.

To make that decomposition possible, one first needs to determine the Eulerian projectors by taking the logarithm of the diagonal series and second to insure their convergence. A 2 From now on, Poincaré-Birkhoff-Witt will be abbreviated to PBW.

3

In the present paper, that product will be referred to as the quasi-shuffle or as the stuffle product, indifferently.

4 Also called MSR factorization after the names of Mélançon, Schützenberger and Reutenauer.

5

These associators, which are formal power series in non-commutative variables, were introduced in quantum field theory by Drinfel'd [START_REF] Drinfel'd | -On quasitriangular quasi-Hopf algebra and a group closely connected with Gal(Q/Q)[END_REF]. The explicit coefficients of the universal associator Φ KZ are polyzetas and regularized polyzetas [START_REF] Lê | Kontsevich's integral for Kauffman polynomial[END_REF]. 6 These values are usually referred to as MZV's by Zagier [START_REF] Zagier | Values of zeta functions and their applications[END_REF] and as polyzetas by Cartier [START_REF] Cartier | Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents, Séminaire Bourbaki, 53 e année[END_REF].

key ingredient is the fact that the diagonal series are group-like and give a host of grouplike elements by specialization, so one can use the exponential-logarithm correspondence to compute within a combinatorial Hausdorff group.

To that effect, the present work generalizes the recursive definitions of the shuffle and quasi-shuffle products given by Fliess [START_REF] Fliess | Sur divers produits de séries formelles[END_REF] and Hoffman [START_REF] Hoffman | Quasi-shuffle products[END_REF], respectively, to introduce the ϕ-deformed shuffle product, where ϕ stands for an arbitrary algebra law. Recent studies on these structures can be found in [START_REF] Duchamp | Deformations of algebras: Twisting and perturbations[END_REF][START_REF] Manchon | Nested sums of symbols and renormalised multiple zeta functions[END_REF][START_REF] Novelli | Natural endomorphisms of quasi-shuffle Hopf algebras[END_REF].

These ϕ-shuffle products interpolate between the classical shuffle and quasi-shuffle products (for ϕ ≡ 0 and ϕ ≡ 1, respectively), and allow a classification of the associated bialgebras.

This paper is devoted to the combinatorics of ϕ-deformed shuffle algebras and to the effective constructions of pairs of dual bases. Its organisation is as follows:

• Section 2 is a short reminder of well-known facts about the combinatorics of the qstuffle product [START_REF] Bui | Schützenberger's factorization on the (completed) Hopf algebra of q-stuffle product[END_REF], which encompasses the shuffle [START_REF] Reutenauer | Free Lie algebras[END_REF] and the quasi-shuffle products [START_REF] Ngoc | On a conjecture by Pierre Cartier about a group of associators[END_REF][START_REF] Ngoc | Structure of polyzetas and Lyndon words[END_REF]. • In Section 3, we thoroughly investigate algebraic and combinatorial aspects of the ϕ-deformed shuffle products and explain how to use bases in duality to get a local system of coordinates on the (infinite-dimensional) Lie group of group-like series.

Throughout the paper, we have a particular concern for Lie series and their correspondence with the Hausdorff group.

A SURVEY OF SHUFFLE PRODUCTS

For standard definitions and facts pertaining to the (algebraic) combinatorics on words, we refer the reader to the classical books by Lothaire [START_REF] Lothaire | Combinatorics on words[END_REF] and Reutenauer [START_REF] Reutenauer | Free Lie algebras[END_REF].

Throughout the paper, K stands for a (unital, associative and commutative) Q-algebra containing a parameter q. In this section, we review the known combinatorics of bases in duality and local coordinates on the infinite-dimensional Lie group of group-like series (Hausdorff group). The parameter q allows for a unified treatment between shuffle (q = 0) and quasishuffle (q = 1) products.

Let Y = {y i } i≥1 be an alphabet, totally ordered by y 1 > y

+ = Y * \ {1 Y * }.
The q-stuffle [START_REF] Bui | Schützenberger's factorization on the (completed) Hopf algebra of q-stuffle product[END_REF], which interpolates between the shuffle [START_REF] Ree | Lie elements and an algebra associated with shuffles[END_REF], quasi-shuffle [START_REF] Malvenuto | Duality between quasi-symmetric functions and the Solomon descent algebra[END_REF] (or stuffle) and minus-stuffle products [START_REF] Costermans | Noncommutative algebra, multiple harmonic sums and applications in discrete probability[END_REF], for q = 0, 1, and -1, respectively, is defined as follows:

u q 1 Y * = 1 Y * q u = u, (1)
y s u q y t v = y s (u q y t v) + y t (y s u q v) + q y s+t (u q v), (2)
or its dual co-product, as follows, for any y s , y t ∈ Y and u, v ∈ Y * ,

∆ q (1 Y * ) = 1 Y * ⊗ 1 Y * , (3)
∆ q (y s ) = y s ⊗ 1 Y * + 1 Y * ⊗ y s + q s 1 +s 2 =s y s 1 ⊗ y s 2 . (4)
We now turn to the study of the combinatorial q-stuffle Hopf algebra, which we do by stressing the importance of the Lie elements 7 studied by Ree [START_REF] Ree | Lie elements and an algebra associated with shuffles[END_REF], and show how Schützenberger's factorization extends to this new structure.

The q-stuffle is commutative, associative and unital. With the co-unit defined by (P ) = P | 1 Y * , for P ∈ K Y , we get

H q = (K Y , conc, 1 Y * , ∆ q , ) and
H ∨ q = (K Y , q , 1 Y * , ∆ conc ,
) which are mutually dual bialgebras and, in fact, Hopf algebras because they are N-graded by the weight.

Let D Y be the diagonal series over H q , i.e.,

D Y = w∈Y * w ⊗ w. (5)
Then 8 log

D Y = w∈Y + w ⊗ π 1 (w), (6)
where π 1 is the extended Eulerian projector 9 over H q , defined by (see [START_REF] Bui | Schützenberger's factorization on the (completed) Hopf algebra of q-stuffle product[END_REF])

π 1 (w) = w + k≥2 (-1) k-1 k u 1 ,...,u k ∈Y + w | u 1 q • • • q u k u 1 . . . u k . (7)
Let {Π l } l∈Lyn Y be defined by

Π y = π 1 (y), for y ∈ Y, Π l = [Π s , Π r ], for the standard factorization (s, r) of l ∈ Lyn Y -Y. (8)
Then it forms a basis of the Lie algebra of primitive elements of H q (see [START_REF] Bui | Schützenberger's factorization on the (completed) Hopf algebra of q-stuffle product[END_REF]).

Let {Π w } w∈Y * be defined, for any w ∈ Y * such that w = l i 1 1 . . . l i k k with l 1 > . . . > l k and l 1 . . . , l k ∈ Lyn Y , by

Π w = Π i 1 l 1 . . . Π i k l k . (9)
Then, by the PBW theorem, the set {Π w } w∈Y * is a basis of K Y (see [START_REF] Bui | Schützenberger's factorization on the (completed) Hopf algebra of q-stuffle product[END_REF]).

7

Following Ree [START_REF] Ree | Lie elements and an algebra associated with shuffles[END_REF], the Lie elements contain the non-commutative power series which are Lie series (as the Chen non-commutative generating series of iterated integrals), i.e., they are group-like for the co-product of the shuffle. The diagonal series lives in

K Y * ⊗ Y * (K Y ⊗ K Y ) * . 9
In fact, π 1 is a projector which maps H q onto the space of its primitive elements P rim(H q ), see Lemma 7.

Let {Σ w } w∈Y * be the family dual 10 to {Π w } w∈Y * in the quasi-shuffle algebra. Then {Σ w } w∈Y * freely generates the quasi-shuffle algebra, and the subset {Σ l } l∈Lyn Y forms a transcendence basis of (K Y , q , 1 Y * ). The Σ w can be obtained as follows (see [START_REF] Bui | Schützenberger's factorization on the (completed) Hopf algebra of q-stuffle product[END_REF]):

             Σ y = y, for y ∈ Y, Σ l = (!) q i-1 i! y s k 1 +•••+s k i Σ l 1 •••ln , for l = y s 1 . . . y s k ∈ Lyn Y, Σ w = Σ q i 1 l 1 q • • • q Σ q i k l k i 1 ! • • • i k ! , for w = l i 1 1 . . . l i k k , (10)
and

l 1 lex • • • lex l k ∈ Lyn Y .
In the second expression of [START_REF] Costermans | Noncommutative algebra, multiple harmonic sums and applications in discrete probability[END_REF] ⇐ denotes the transitive closure of the relation on standard sequences, denoted by ⇐ (see [START_REF] Bui | Schützenberger's factorization on the (completed) Hopf algebra of q-stuffle product[END_REF]).

In this case, since {Π w } w∈Y * and {Σ w } w∈Y * are multiplicative, we get the q-extended Schützenberger's factorization as follows (see [START_REF] Bui | Schützenberger's factorization on the (completed) Hopf algebra of q-stuffle product[END_REF]):

D Y = w∈Y * Σ w ⊗ Π w = l∈Lyn Y exp(Σ l ⊗ Π l ). (11)
This series, in the factorized form, encompasses a large part of the combinatorics of Dyson's functional expansions in quantum field theory [START_REF] Dyson | The radiation theories of Tomonaga, Schwinger and Feynman[END_REF][START_REF] Magnus | On the exponential solution of differential equations for a linear operator[END_REF]. It is the infinite-dimensional analogue of the theorem of Wei and Norman [START_REF] Bui | Structure of polyzetas and explicit representation on transcendence bases of shuffle and stuffle algebras[END_REF][START_REF] Wei | Lie algebraic solution of linear differential equations[END_REF][START_REF] Wei | On global representation of the solution of linear differential equations as product of exponentials[END_REF].

ALGEBRAIC ASPECTS OF ϕ-SHUFFLE BIALGEBRAS

From now on, we will work with an alphabet Y = {y i } i∈I with I an arbitrary index set 11 , which needs not be totally ordered unless we write it explicitly.

First properties. Let us consider the following recursion in order to construct a map

Y * × Y * -→ K Y . ( 12
) i) For any w ∈ Y * , (Init) 1 Y * ϕ w = w ϕ 1 Y * = w. (13)
ii

) For any a, b ∈ Y and u, v ∈ Y * , (Rec) au ϕ bv = a(u ϕ bv) + b(au ϕ v) + ϕ(a, b)(u ϕ v), ( 14
)
where ϕ is an arbitrary mapping defined by its structure coefficients

ϕ : Y × Y -→ KY, (15)
(y i , y j ) -→ k∈I γ y k y i ,y j y k . ( 16
)
The following proposition guarantees the existence of a unique bilinear law on K Y satisfying (Init) and (Rec).

10

The duality pairing is given by u

| v = δ u,v , for u, v ∈ Y * .
11

The indexing is one-to-one, i.e., there is no repetition.

Proposition 1 ([14]

). The recursion (Rec) together with the initialization (Init) defines a unique mapping

ϕ : Y * × Y * -→ K Y ,
which can, at once, be extended by linearity as a law

ϕ : K Y ⊗ K Y -→ K Y .
The space K Y endowed with the law ϕ is an algebra (with unit 1 K Y by definition). It will be called the ϕ-shuffle algebra. In full generality, this algebra need not be associative or commutative if ϕ is not so. In the next example, we give a table of well known laws which can be defined according to this pattern (in which ϕ is reasonable).

Example 1. Below, a summary table of ϕ-deformed cases found in the literature is given. The last case (infiltration product) comes from computer science (see [START_REF] Ochsenschläger | Binomialkoeffizenten und Shuffle-Zahlen[END_REF][START_REF] Ochsenschläger | Eine Klasse von Thue-Folgen[END_REF][START_REF] Duchamp | Direct and dual laws for automata with multiplicities[END_REF])

Name (recursion) Formula ϕ Shuffle au bv = a(u bv) + b(au v) ϕ ≡ 0 Quasi-shuffle x i u x j v = x i (u x j v) + x j (x i u v) ϕ(x i , x j ) = x i+j Stuffle + x i+j (u v) Min-shuffle x i u x j v = x i (u x j v) + x j (x i u v) ϕ(x i , x j ) = -x i+j -x i+j (u v) Muffle x i u a x j v = x i (u a x j v) + x j (x i u a v) ϕ(x i , x j ) = x i×j + x i×j (u a v) q-stuffle x i u q x j v = x i (u q x j v) + x j (x i u q v) ϕ(x i , x j ) = qx i+j + qx i+j (u v) q-stuffle x i u q x j v = x i (u q x j v) + x j (x i u q v) ϕ(x i , x j ) = q i×j x i+j (character) + q i×j x i+j (u v) LDIAG(1, q s ) non-crossed, au * bv = a(u * bv) + b(au * v) ϕ(a, b) = q |a||b| s (a.b) non-shifted + q |a||b| s (a.b)(u * v) (a.b) assoc. B-shuffle au B bv = a(u B bv) + b(au B v) ϕ(a, b) = a, b + a, b (u B v) = b, a Semigroup-x t u ⊥ x s v = x t (u ⊥ x s v) + x s (x t u ⊥ v) ϕ(x t , x s ) = x t⊥s -shuffle + x t⊥s (u ⊥ v) q-Infiltration au ↑ bv = a(u ↑ bv) + b(au ↑ v) ϕ(a, b) = qδ a,b a + qδ a,b a(u ↑ v)
Now we set forth the first properties of ϕ (see [START_REF] Enjalbert | Combinatorial study of Hurwitz colored polyzêtas[END_REF]): associativity, commutativity and dualizability.

Definition 1 ([14]). A law

µ : K Y ⊗ K Y → K Y is said to be dualizable if there exists a (linear) mapping ∆ µ : K Y -→ K Y ⊗ K Y
(necessarily unique) such that the dual mapping

K Y ⊗ K Y * -→ K Y restricts 12 to µ. Or, equivalently, for all u, v, w ∈ Y * : µ(u ⊗ v) | w = u ⊗ v | ∆ µ (w) ⊗2 .
Theorem 1 ( [START_REF] Duchamp | Combinatorics of ϕdeformed stuffle Hopf algebras[END_REF]). We have:

(1) The law ϕ is associative (respectively commutative) if and only if the extension

ϕ : KY ⊗ KY -→ KY is so. (2) Let γ z
x,y := ϕ(x, y) | z be the structure constants of ϕ. Then ϕ is dualizable if and only if ϕ is also dualizable, that is to say, there exists a map δ : KY -→ KY ⊗ KY such that for all x, y, z ∈ X we have

ϕ(x, y) | z = x ⊗ y | δ(z) .
This map δ is given by 13

δ(z) = x,y∈Y γ z x,y x ⊗ y .
For the proof of the theorem we need the following auxiliary result.

Lemma 1 ([14]

). Let ∆ be the morphism K Y -→ A Y * ⊗ Y * defined on the letters by 14

∆(y s ) = y s ⊗ 1 + 1 ⊗ y s + n,m∈I γ ys yn,ym y n ⊗ y m . Then (1) for all u, v, w ∈ Y * , u ϕ v | w = u ⊗ v | ∆(w) ⊗2 . (2) for all w ∈ Y + , ∆(w) = w ⊗ 1 + 1 ⊗ w + u,v∈Y + ∆(w) | u ⊗ v u ⊗ v.
Proof of Theorem 1 (sketch). The theorem follows by application of items ( 1) and (2) in Lemma 1.

If ϕ is associative (which is fulfilled in all cases of Table 1), we extend ϕ to Y + by the universal property of the free semigroup Y + ,

ϕ(x) = x, for x ∈ Y, ϕ(xw) = ϕ(x, ϕ(w)), for x ∈ Y and w ∈ Y + , (17)
and we extend the definition of the structure constants accordingly: for

x 1 . . . x l ∈ Y + , γ y x 1 ...x l = y | ϕ(x 1 . . . x l ) = t 1 ,...,t l-2 ∈Y γ y x 1 ,t 1 γ t 1 x 2 ,t 2 . . . γ t l-2 x l-1 ,x l . ( 18
)
Note that the fact that ϕ is dualizable can be rephrased as:

for all y ∈ Y : {w ∈ Y 2 |γ y w = 0} is finite. ( 19
)
12 through the pairings -| -.

13 Note that all these conditions are equivalent to the fact that (γ z x,y ) x,y,z∈Y satisfies: for all z ∈ Y : #{(x, y) ∈ Y 2 |γ z

x,y = 0} < +∞ .

14 If ϕ is dualizable, this expression can be written

∆(y s ) = y s ⊗ 1 + 1 ⊗ y s + δ(y s ) .
In this case, it can be checked immediately that, for an arbitrarily fixed N ≥ 1,

for all y ∈ Y : {w ∈ Y N |γ y w = 0} is finite, (20)
but, by no means, we have in general that for all y ∈ Y :

{w ∈ Y + |γ y w = 0} is finite. ( 21
)
Remark 1. i) Condition ( 21) is strictly stronger than [START_REF] Duchamp | Noncommutative symmetric functions III: Deformations of Cauchy and convolution algebras[END_REF] as the example of any group law on Y , with |Y | ≥ 2 and finite, shows.

ii) Non-dualizable laws occur with the alphabet Y Z = {y j } j∈Z and the stuffle on it (ϕ(y i , y j ) = y i+j ). This alphabet naturally appears in the theory of polylogarithms at negative integers in [START_REF] Duchamp | Ngô-Harmonic sums and polylogarithms at negative multi-indices[END_REF] where another non-dualizable law (called ) arises. See also Example 2 below. Definition 2. An associative law ϕ on KY will be said to be moderate if and only if it fulfils condition [START_REF] Enjalbert | Combinatorial study of Hurwitz colored polyzêtas[END_REF].

Let us now state the structure theorem from [START_REF] Duchamp | Combinatorics of ϕdeformed stuffle Hopf algebras[END_REF].

Theorem 2 ([14]

). Let us suppose that ϕ is dualizable and associative. We still denote its dual co-multiplication by

∆ ϕ : K Y -→ K Y ⊗ K Y . Then B ϕ = (K Y , conc, 1 Y * , ∆ ϕ , ε) is a bialgebra. If,
moreover, ϕ is commutative, the following conditions are equivalent:

(1) B ϕ is an enveloping bialgebra.

(2) B ϕ is isomorphic to (K Y , conc, 1 Y * , ∆ , ) as a bialgebra.

(3) For all y ∈ Y , the following series is a polynomial

(P ) y + l≥2 (-1) l-1 l x 1 ,...,x l ∈Y y | ϕ(x 1 . . . x l ) x 1 . . . x l .
(4) ϕ is moderate.

Proof (sketch). 4 =⇒ 3) Obvious.

3 =⇒ 2) One first constructs an endomorphism of (K Y , conc, 1 Y * ) sending each letter y ∈ Y to the polynomial form (P ) and then proves that it is an automorphism of AAU 15 which sends

(K Y , conc, 1 Y * , ∆ ϕ , ε) to (K Y , conc, 1 Y * , ∆ , ). 2 =⇒ 1) Because (K Y , conc, 1 Y * , ∆ , ) is an enveloping bialgebra.
1 =⇒ 4) Observe that, for each letter y ∈ Y , we have

∆ (n-1) ϕ (y) | x 1 ⊗ x 2 ⊗ • • • ⊗ x n = γ y x 1 ...x l . Example 2.
(1) The muffle product (see Table 1), which determines the product of Hurwitz polyzetas with rational centers and correspond to ϕ(x i , x j ) = x i.j for i, j ∈ Q * + , is not dualizable (γ 1 n,1/n = 1 for all n ≥ 1).

(2) The q-infiltration bialgebra (see again Table 1) has its origin in computer science [START_REF] Ochsenschläger | Binomialkoeffizenten und Shuffle-Zahlen[END_REF][START_REF] Ochsenschläger | Eine Klasse von Thue-Folgen[END_REF] and appears as a generic solution in [START_REF] Duchamp | Direct and dual laws for automata with multiplicities[END_REF]. It provides a bialgebra

H q-inflitr = (K Y , conc, 1 X * , ∆ ↑q , )
(q ∈ K) based on a ϕ which is an associative, commutative and dualizable law, but, if Y = ∅ this law is moderate only if and only if q is nilpotent in the Q-algebra K. Indeed, for all x ∈ Y , (1 + qx) is group-like and it has an inverse in K X if and only if q is nilpotent. In this case the antipode is the involutive antiautomorphism defined on the letters by

S(x) = -x 1 + qx .

Structural properties.

Here, we only assume that ϕ is associative.

The bialgebra

H ∨ ϕ = (K Y , ϕ , 1 Y * , ∆ conc , ) (22)
is a Hopf algebra because it is co-nilpotent 16 . Its antipode can be computed by a(1

Y * ) = 1 and, for w ∈ Y + , a ϕ (w) = k≥1 (-1) -k u 1 ,...,u k ∈Y + u 1 ...u k =w u 1 ϕ • • • ϕ u k . (23)
Due to the finite number of decompositions of any word u 1 . . . u k = w ∈ Y + into factors u 1 , . . . , u k ∈ Y + , we can, at this very early stage, define an endomorphism Φ(S) of K Y as follows:

Φ(S)[w] = k≥1 a k u 1 ,...,u k ∈Y + u 1 ...u k =w u 1 ϕ • • • ϕ u k , (24)
associated to any univariate formal power series

S = a 1 X + a 2 X 2 + a 3 X 3 + • • • . The case of log(1 + X) = k≥1 (-1) k-1 k X k (25)
will be of particular importance. It reads here in the style of formula [START_REF] Ngoc | Input/Output behaviour of nonlinear control systems: Rational approximations, nilpotent structural approximations[END_REF],

π1 (w) = k≥1 (-1) k-1 k u 1 ,...,u k ∈Y + u 1 ...u k =w u 1 ϕ • • • ϕ u k . ( 26
)
16

The law ∆ conc , dual to the concatenation is, of course, defined by

∆ conc (w) = uv=w u ⊗ v.
The corresponding n-fold ∆ + conc (∆ + = ∆ minus the primitive part) reads

∆ +(n-1) conc (w) = u1u2•••un=w ui∈Y + u 1 ⊗ u 2 ⊗ • • • ⊗ u n , from which it is clear that ∆ +(n-1) conc (w) = 0 for n > |w|.
This π1 ∈ End(K Y ) has an adjoint π 1 ∈ End(K Y ) which reads

π 1 (S) = w∈Y * S | π1 (w) w (27) = k≥1 (-1) k-1 k u 1 ,...,u k ∈Y + S | u 1 ϕ • • • ϕ u k u 1 . . . u k . (28)
It is an easy exercise to check that the family in the sums of ( 27) is summable 17 . It is easy to check that the dominant term of all terms in a ϕ product is the corresponding product. This explains why we still have the theorem of Radford.

Theorem 3 (RADFORD'S THEOREM). When ϕ is commutative, the associative and commutative algebra with unit (K Y , ϕ , 1 Y * ) is a polynomial algebra. More precisely, the morphism

β : K[Lyn Y ] → (K Y , ϕ , 1 Y * ) defined by β(l) = l for l ∈ Lyn Y is an isomorphism.
In other words, the family

l ϕi1 1 ϕ • • • ϕ l ϕik k k≥0, {l 1 ,l 2 ,...,l k }⊂Lyn Y (i 1 ,i 2 ,...,i k )∈(N + ) k is a linear basis of K Y .
Proof. One checks that

l ϕi1 1 ϕ • • • ϕ l ϕik k = l i 1 1 . . . l i k k + |v|< 1≤j≤k i j |l j | c v v.

The result follows.

The theorem of Radford is important in the classical cases because it is the left-hand side of Schützenberger's factorization in which we have the move PBW → Radford; see [START_REF] Deneufchâtel | Radford bases and Schützenberger's factorizations[END_REF] for a discussion of the converse.

Lemma 2 (ϕ-EXTENDED FRIEDRICHS' CRITERION). We denote 18 by

∆ ϕ : K Y → K Y * ⊗ Y *
the dual of ϕ applied to series, i.e., defined by

∆ ϕ (S) = u,v∈Y * S | u ϕ v u ⊗ v.
Let now S ∈ K Y . Then we have:

(1) 19 A family of (simple, double, etc.) series is summable if it is locally finite (see [START_REF] Duchamp | Combinatorics of ϕdeformed stuffle Hopf algebras[END_REF] for a complete development). [START_REF] Dyson | The radiation theories of Tomonaga, Schwinger and Feynman[END_REF] As in the classical case, ∆ ϕ is a conc-morphism as can be seen by transposition of the fact that ∆ conc is a ϕ -morphism. [START_REF] Duchamp | Noncommutative symmetric functions III: Deformations of Cauchy and convolution algebras[END_REF] Tensor products of linear forms.

If S | 1 Y * = 0 then S is primitive (i.e., ∆ ϕ (S) = S ⊗ 1 Y * + 1 Y * ⊗ S)

idem

Proof. The expected equivalences are due to the following facts:

∆ ϕ (S) = S ⊗ 1 Y * + 1 Y * ⊗ S -S | 1 Y * 1 Y * ⊗ 1 Y * + u,v∈Y + S | u ϕ v u ⊗ v, ∆ ϕ (S) = u,v∈Y * S | u ϕ v u ⊗ v and S ⊗ S = u,v∈Y * S | u S | v u ⊗ v. Lemma 3. Let S ∈ K Y be such that S | 1 Y * = 1. Then S is
∆ ϕ (log S) = log S ⊗ 1 Y * + 1 Y * ⊗ log S,
and, since log S ⊗ 1 Y * and 1 Y * ⊗ log S commute, we get successively

∆ ϕ (S) = ∆ ϕ (exp(log S)) = exp(∆ ϕ (log S)) = exp(log S ⊗ 1 Y * ) exp(1 Y * ⊗ log S) = (exp(log S) ⊗ 1 Y * )(1 Y * ⊗ exp(log S)) = S ⊗ S.
This means, together with S | 1 Y * , that S is group-like. The converse can be obtained in the same way.

Remark 2. i) In fact, Lemma 3 establishes a nice log-exp correspondence for the Lie group of group-like series.

ii) Through the canonical pairing -| -: K Y ⊗ K Y → K, we have K Y (K Y ) * . Group-like (respectively primitive) series are in bijection with characters (respectively infinitesimal characters) of the algebra (K Y , ϕ , 1 Y * ).

Lemma 4.

(1) Group-like series form a group (for concatenation).

(2) The space Prim(K Y ) is a Lie algebra (for the bracket derived from concatenation).

Proof. As in the classical case.

We extend the transposition process in the same way as in Lemma 2 and note, for n ≥ 1, that

∆ (n-1) ϕ : K Y → K (Y * ) ⊗ n , (29)
the dual of (n-1) ϕ applied to series, i.e., defined by

∆ (n-1) ϕ (S) = u 1 ,u 2 ,...un∈Y * S | u 1 ϕ • • • ϕ u n u 1 ⊗ • • • ⊗ u n . ( 30
) 21 For any h ∈ K Y , if h | 1 Y * = 0, we define log(1 Y * + h) = n≥1 (-1) n-1 n h n and exp(h) = n≥0 h n n! ,
and we have the usual formulas log(exp(h)) = h and exp(log(1

Y * + h)) = 1 Y * + h.
We will use the following lemma several times, which gives the combinatorics of products of primitive series (and the polynomials).

Lemma 5 (HIGHER ORDER CO-MULTIPLICATIONS OF PRODUCTS). Let us consider the language M over the alphabet A = {a 1 , a 2 , . . . , , a m },

M = {w ∈ A * | w = a j 1 . . . a j |w| , j 1 < • • • < j |w| , 1 ≤ |w| ≤ m},
and the morphism

µ : K A -→ K Y , a i -→ S i ,
where S 1 , . . . , S m are primitive series in K Y . Then

∆ (n-1) ϕ (S 1 . . . S m ) = w 1 ,...,wn∈M |w 1 |+•••+|wn|=m a 1 •••am∈supp(w 1 ... wn) µ(w 1 ) ⊗ • • • ⊗ µ(w n ).
Proof (sketch). Let S = (S 1 , . . . , S m ) be this family of primitive series and, for

I = {i 1 , . . . , i k } ⊂ [1 • • • m] in increasing order, let us write S[I] for the product S i 1 • • • S i k . Then we have ∆ (n-1) ϕ (S 1 . . . S m ) = I 1 +•••+In=[1•••m] S[I 1 ] ⊗ • • • ⊗ S[I n ].
Setting w i = (a 1 a 2 . . . a m )[I], one gets the expected result.

Lemma 6 (PAIRING OF PRODUCTS). Let S 1 , . . . , S m be primitive series in K Y , and let P 1 , . . . , P n be proper 22 polynomials in K Y . Then one has in general

P 1 ϕ • • • ϕ P n | S 1 . . . S m = w 1 ,...,wn∈M |w 1 |+•••+|wn|=m a 1 •••am∈supp(w 1 ... wn) n i=1 P i | µ(w i ) .
In particular, we have the following exhaustive list of circumstances:

(1) If n > m, then P 1 ϕ • • • ϕ P n | S 1 . . . S m = 0. ( 2
) If n = m, then P 1 ϕ • • • ϕ P n | S 1 . . . S n = σ∈Sn n i=1 P i | S σ(i) .
(3) If n < m, then one has the general form in which every product in the sum contains at least a factor

P i | µ(w i ) with |w i | ≥ 2.
Proof. It is a consequence of Lemma 5 through the equality i.e., polynomials without constant term; see [START_REF] Berstel | Rational series and their languages[END_REF].

P 1 ϕ • • • ϕ P n | S 1 . . . S n = P 1 ⊗ • • • ⊗ P n | ∆ (n-1) ϕ (S 1 . . . S n ) .
In the sequel, we assume that ϕ is associative and dualizable. Now, we have the following two structures:

H ϕ = (K Y , conc, 1 Y * , ∆ ϕ , ), (31)
H ∨ ϕ = (K Y , ϕ , 1 Y * , ∆ conc , ), (32)
which are mutually dual 23

I := span K {u ϕ v} u,v∈Y + , (33)
K + Y := {P ∈ K Y | P | 1 Y * = 0}, (34)
P := Prim(H ϕ ) = {P ∈ K Y | ∆ + ϕ (P ) = 0}, (35)
where

∆ + ϕ (P ) = ∆ ϕ (P ) -(P ⊗ 1 Y * + 1 Y * ⊗ P ) + P | 1 Y * 1 Y * ⊗ 1 Y * . ( 36
)
Remark 3. At this stage (ϕ not necessarily moderate), it can happen that Prim(H ϕ ) = {0}. This is for example the case with the q-infiltration bialgebra on one letter at q = 1,

H ϕ = (K[x], conc, 1 x * , ∆ ↑ 1 , ),
and, more generally, when q is not nilpotent 24 .

We can also endow End(K Y ), the K-module of endomorphisms of K Y , with the convolution product defined by for all f, g ∈ End(K Y ),

f g = conc • (f ⊗ g) • ∆ ϕ , (37)
i.e., for all P ∈ K Y ,

(f g)(P ) = u,v∈Y * P | u ϕ v f (u)g(v). ( 38
)
Then End(K Y ) becomes a K-associative algebra with unity (AAU), its unit being e = 1 K Y • . It is convenient to represent every f ∈ End(K Y ) by its graph, a double series which reads

Γ(f ) = w∈Y * w ⊗ f (w). ( 39
)
This representation is faithful and, by direct computation, one gets

Γ(f )Γ(g) = Γ(f g), (40)
where the multiplication of double series is performed by the stuffle on the left and the concatenation on the right. From now on, we assume that ϕ is associative, commutative and dualizable.

23

This duality is separating; see [START_REF] Bourbaki | Topological vector spaces[END_REF].

24

Recall that q is an element of the ring K (see example 2.2).

Lemma 7 (π 1 IS A PROJECTOR ON THE PRIMITIVE SERIES).

The endomorphism π 1 is a projector, the image of which is exactly the space of primitive series, Prim(K Y ).

Proof (sketch). The proof follows the lines of [START_REF] Reutenauer | Free Lie algebras[END_REF] with the difference that π 1 (w) might not be a polynomial and the operator defined in Lemma 2 is not a genuine co-product. The diagonal series D Y (when considered as a series in (K Y ) Y , the coefficient ring, K Y , being endowed with the ϕ product) is group-like in the sense of Lemma 2. Then, using

log(D Y ) = w∈Y * w ⊗ π 1 (w)
(which can be established by summability of the family (w ⊗ π 1 (w)) w∈Y * ; but remember that the π 1 (w) are, in general, series 25 ), one gets that π 1 (w) is a primitive series for all w. Now, from π 1 (S) =

w∈Y * S | w π 1 (w), one has π 1 (S) Prim(K Y ). Conversely, from Friedrichs' criterion, one gets π 1 (S) = S if S ∈ Prim(K Y ).
In the remainder of the paper, we suppose that ϕ is moderate (and still dualizable, associative and commutative). Definition 4 (PROJECTORS, [START_REF] Reutenauer | Free Lie algebras[END_REF]). Let I + : K Y -→ K Y be the linear mapping defined by

I + (1 Y * ) = 0,
I + n := conc n-1 • I + ⊗n •∆ (n-1) ϕ .
It follows immediately that

exp(π 1 ) = e + n≥1 1 n! π n 1 = n≥0 π n , (41)
where e = 1 K Y • is the orthogonal complement of I + and neutral for the convolution product. The π n so obtained is called the n-th Eulerian projector.

Lemma 8. The endomorphism π1 defined in [START_REF] Ngoc | On a conjecture by Pierre Cartier about a group of associators[END_REF] is the adjoint of π 1 . One has

π1 = n≥1 (-1) n-1 n (n-1) ϕ • I + ⊗n •∆ (n-1) conc .
Proof. Immediate.

25

In greater detail, this equality amounts to checking the summability of the family

(-1) k-1 k w ⊗ w | u 1 ϕ • • • ϕ u k u 1 • • • u k w∈Y * , k≥1 u1,...,u k ∈Y +
(which is immediate) and rearranging the sums.

26

The series below are summable because the family (I + n ) n≥0 is locally nilpotent (see [START_REF] Duchamp | Combinatorics of ϕdeformed stuffle Hopf algebras[END_REF] for complete proofs). Note that this definition gives the same result as the computation of the adjoint of π1 given in [START_REF] Ngoc | Structure of polyzetas and Lyndon words[END_REF].

Proposition 2. (GRAPH OF π 1 , VALUES AND ITS EXPONENTIAL AS RESOLUTION OF UNITY).

(1) For all Y and ϕ (moderate, associative, commutative and dualizable), one has

log D Y = w∈Y + w ⊗ π 1 (w) = w∈Y + π1 (w) ⊗ w.
(2) Let P ∈ K Y be a primitive polynomial, for ∆ ϕ . Then π 1 (P ) = P, for all k, n ∈ N + , π n (P k ) = δ k,n P k .

(3) One has

Id K Y = e + I + = n≥0 π n . (42)
Equation ( 42) is a resolution of identity with mutually orthogonal summands. (4) We have

K + Y = P ⊥ ⊕I = P ⊕ n≥2 π n (K Y ) .
Proof. The only statement which cannot be proved through an isomorphism with the shuffle algebra is the first equality of the point (4). The fact that P ∩ I = {0} comes from Friedrichs' criterion, and P + I = K + Y is a consequence of the fact (seen again through any isomorphism with the shuffle algebra) that

(H ϕ ) + = span K ( n≥1 (P 1 ϕ • • • ϕ P n ) P i ∈Prim(H ϕ ) ). Remark 4.
(1) The first equality of Proposition 2.(4), i.e.,

K + Y = P ⊥ ⊕I,
is known as the theorem of Ree [START_REF] Ree | Lie elements and an algebra associated with shuffles[END_REF].

(2) The projector on P parallel to I is not in general in the descent algebra (see [START_REF] Duchamp | Noncommutative symmetric functions III: Deformations of Cauchy and convolution algebras[END_REF]).

This proves that, although they are isomorphic, the spaces I and n≥2 π n (K Y ) are, in general, not identical.

Proposition 2.(1) leads to the following corollary.

Corollary 1. We have π 1 (1 Y * ) = π1 (1 Y * ) = 0 and, for all w ∈ Y + , π 1 (w) = w + k≥2 (-1) k-1 k u 1 ,...,u k ∈Y + w | u 1 ϕ • • • ϕ u k u 1 . . . u k , π1 (w) = w + k≥2 (-1) k-1 k u 1 ,...,u k ∈Y + w | u 1 . . . u k u 1 ϕ • • • ϕ u k .
In particular, π 1 (1 Y * ) = π1 (1 Y * ) = 0, for any y ∈ Y , π1 (y) = y, and

π 1 (y) = y + l≥2 (-1) l-1 l x 1 ...x l ∈Y * γ y x 1 ,...,x l x 1 . . . x l .
Remark 5. We already knew that, as soon as ϕ is associative, π1 (w) is a polynomial. Here, because ϕ is moderate, dualizable, and associative, π 1 (w) is also a polynomial, and because ϕ is commutative, it is primitive.

Proposition 3. We have:

(1) The expression of σ Y (t) is given by

σ Y (t) = n≥0 t n w∈Y * w ⊗ π n (w) = n≥0 t n w∈Y * πn (w) ⊗ w,
where πn is the adjoint of π n . These are given by π n (1

Y * ) = πn (1 Y * ) = δ 0,n and, for all w ∈ Y + , π n (w) = 1 n! u 1 ,...,un∈Y + w | π1 (u 1 ) ϕ • • • ϕ π1 (u n ) π 1 (u 1 ) . . . π 1 (u n ), πn (w) = 1 n! u 1 ,...,un∈Y + w | π 1 (u 1 ) . . . π 1 (u n ) π1 (u 1 ) ϕ • • • ϕ π1 (u n )).
(2) For any w ∈ Y * , we have

w = k≥0 1 k! u 1 ,...,u k ∈Y + w | u 1 ϕ • • • ϕ u k π 1 (u 1 ) . . . π 1 (u k ) = k≥0 1 k! u 1 ,...,u k ∈Y + w | u 1 . . . u k π1 (u 1 ) ϕ • • • ϕ π1 (u k ).
In particular, for any y s ∈ Y , we have y s = π1 (y s ) and

y s = k≥1 1 k! ys 1 ,...,ys k ∈Y γ ys ys 1 ,...,ys k π 1 (y s 1 ) . . . π 1 (y s k ).
Proof. Direct computation.

Applying the tensor product 27 of isomorphisms of algebras 28 α ⊗ Id Y to the diagonal series D Y , we obtain a group-like element, and then computing the logarithm of this element (or equivalently, applying α ⊗ Id Y to Haus Y ) we obtain S which is, by Lemma 3, primitive:

S = w∈Y * α(w) π 1 (w) = w∈Y * α • π1 (w) w. (43)
Lemma 9. For any w ∈ Y + , one has π 1 (w) ∈ Prim(K Y ).

Proof. Immediate from Lemma 7.

A primitive projector, π : K Y -→ K Y , is defined in the same way as a Lie projector by the three following conditions:

π • π = π, π(1 Y * ) = 0, π(K Y ) = Prim(K Y ) = P. ( 44
)
For example, π 1 defined in Definition 4 (see also Proposition 2) is a primitive projector which will be used to construct bases of P and its enveloping algebra (see Theorem 5 below).

Another example of a primitive projector is the orthogonal projector on P attached to the decomposition in Remark 4. Let us then equip K Y and K Y 1 with • (the concatenation so denoted to be distinguished from the concatenation within Y + ) and (or equivalently by ∆ • and ∆ ).

Thus, the Hopf algebras

(K Y , •, 1 Y * , ∆ , Y * ) and (K Y 1 , •, 1 Y * , ∆ , Y * 1 )
are connected, N-graded, non-commutative and co-commutative bialgebras, and hence enveloping bialgebras (in fact, they are free algebras but specially indexed to match our purpose). Now we can state the following Theorem 4 (NEW LETTERS AS IMAGES). Let π : K Y -→ K Y be a primitive projector. Let ψ π be the conc-morphism defined by

ψ π : K Y -→ K Y , y w -→ ψ π (y w ) = π(w).
Then ψ π is surjective and a Hopf morphism. Moreover, ker ψ π = J = J 1 + J 2 , where J 1 and J 2 are the two-sided ideals of K Y generated by

S 1 = {y u -y π(u) } u∈Y + and S 2 = {y u • y v -y v • y u -y [π(u),π(v)] } u,v∈Y + ,
respectively, where the indexing of the alphabet has been extended by linearity to polynomials, i.e., y P :=

w∈Y + P | w y w .
Proof. The fact that ψ π is surjective is due to π(K Y ) = P and to the fact that any enveloping algebra (here H ϕ ) is generated by its primitive elements. The fact that ψ π is a Hopf morphism is due to a general property of enveloping algebras: if a morphism of AAU between two enveloping algebras sends the primitive elements of the first to primitive elements of the second, then it is a Hopf morphism.

Let now (p i ) i∈J be an ordered (J is endowed with a total ordering ≺ J ) basis 29 of P = Prim(K Y ), and let us recall that J = J 1 + J 2 denotes the two sided ideal generated by the elements J i (itself generated by S i , i = 1, 2).

First, we observe that the elements of S 1 ∪S 2 are in the kernel of ψ π 1 , and then J ⊂ ker ψ π 1 .

On the other hand, for u 1 , u 2 , . . . , u n ∈ Y + , one has

y u 1 • y u 2 • • • • • y un ≡ y π(u 1 ) • y π(u 2 ) • • • • • y π(un) mod J (45)
(in fact they are even equivalent mod J 1 ), which amounts to saying that K Y = J + P , where P is the space "generated by P", in fact, generated by

n≥0 {y p i 1 • • • • • y p in } i j ∈J .
Now, by recurrence over the number of inversions, one can show, using S 2 , that

y p i 1 • • • • • y p in ≡ y p σ(i 1 ) • • • • • y p σ(in) mod J , (46)
where σ ∈ S n is such that σ(i 1 ) J σ(i 2 ) J • • • J σ(i n ) (large order reordering).

29

With the properties of ϕ here, the bialgebra

(K Y , conc, 1 Y * , ∆ ϕ , ) is isomorphic to (K Y , conc, 1 Y * , ∆ , )
in which the module of primitive elements is free, thus P = Prim(K Y ) is free.

Let C be the space generated by the elements

{y p j 1 • • • • • y p jn } j 1 J j 2 J ••• J jn n≥0 . (47)
By ( 45) and (46), we get J + C = K Y . Now, due to the PBW theorem, the family of images

Φ π 1 (y p j 1 • y p j 2 • • • • • y p jn ) j 1 J j 2 J •••jn n≥0 (48)
is a basis of K Y , which proves that Φ π 1 | C : C → K Y is an isomorphism and completely proves the claim.

We now suppose that the alphabet Y is totally ordered.

Definition 5.

(1) Let {Π l } l∈Lyn Y and {Π w } w∈Y * be the families of elements of P and K Y , respectively, obtained as follows:

Π y k = π 1 (y k ), for k ≥ 1, Π l = [Π s , Π r ],
for l ∈ Lyn X, standard factorization of l = (s, r),

Π w = Π i 1 l 1 . . . Π i k l k , for w = l i 1 1 . . . l i k k , l 1 lex • • • lex l k , l 1 . . . , l k ∈ Lyn Y.
(2) Let {Σ w } w∈Y * be the family of the ϕ-deformed quasi-shuffle algebra obtained by duality with {Π w } w∈Y * :

for all u, v ∈ Y * , Σ v | Π u = δ u,v .
A priori, the {Σ w } w∈Y * could be series. We prove first that, in this context, they are polynomials.

Proposition 4 (ADJOINT OF φ π 1 ). Let φ π 1 , be the conc-endomorphism of algebras defined on the letters as follows:

φ π 1 : K Y -→ K Y , y k -→ φ π 1 (y k ) = π 1 (y k ).
Then φ π 1 is an automorphism with the following properties:

(1) This automorphism is such that, for every l ∈ Lyn Y ,

φ π 1 (P l ) = Π l ,
where P l are the polynomials calculated with the mechanism of Definition 5, setting ϕ ≡ 0 (or, equivalently, by (8) with q = 0), i.e., within the shuffle algebra

(K Y , conc, 1 Y * , ∆ , ). (2
) This automorphism has an adjoint φ ∨ π 1 within K Y which reads, on the words w ∈

Y * , φ ∨ π 1 (w) = k≥0 y i 1 •••y i k ∈Y w | π 1 (y i 1 ) • • • π 1 (y i k ) y i 1 y i 2 • • • y i k .
(3) In the style of Definition 4, one has

φ π 1 = e + k≥1 conc (k-1) • (π 1 • I 1 ) ⊗k • ∆ (k-1) conc , φ ∨ π 1 = e + k≥1 conc (k-1) • (I 1 • π1 ) ⊗k • ∆ (k-1) conc ,
where I 1 is the projector on KY parallel to n =1 (K Y ) n . (4) For all w ∈ Y * , Σ w = (φ ∨ π 1 ) -1 (S w ). Proof (sketch). It was proved in Theorem 2 that the endomorphism φ π 1 is an isomorphism. The recursions used to construct Π l and P l prove that φ π 1 (P l ) = Π l , and then φ πw (P l ) = Π w for every word w. Now the expression of φ π 1 is a direct consequence of the definition of φ π 1 . This implies at once the expression of φ ∨ π 1 and the fact that φ ∨ π 1 ∈ End(K Y ). The last equality comes from the following

δ u,v = Π u | Σ v = φ π 1 (P u ) | Σ v = P u | φ ∨ π 1 (Σ v )
, which shows that, for all w ∈ Y * , φ ∨ π 1 (Σ w ) = S w and the claim follows. We can now state the following result. The first terms of these families, for the q-stuffle (see [START_REF] Cartier | On the structure of free Baxter algebras[END_REF] and [START_REF] Costermans | Noncommutative algebra, multiple harmonic sums and applications in discrete probability[END_REF]) can be found in [START_REF] Bui | Développement asymptotique des sommes harmoniques[END_REF].

3.3.

Local coordinates by ϕ-extended Schützenberger factorization. We have observed very early (ϕ needs only to be associative) that the set of group-like series (for ∆ ϕ ) forms a (infinite-dimensional Lie) group (see Lemmas 3 and 4), its Lie algebra is the (Lie) algebra of Lie series, and we have a nice log-exp correspondence (see Lemma 3). We will see in this paragraph that, when ϕ possesses all the "good" properties (moderate, dualizable, associative and commutative), we have an analogue of the Wei-Norman theorem [START_REF] Bui | Structure of polyzetas and explicit representation on transcendence bases of shuffle and stuffle algebras[END_REF][START_REF] Wei | Lie algebraic solution of linear differential equations[END_REF][START_REF] Wei | On global representation of the solution of linear differential equations as product of exponentials[END_REF] which gives a system of local coordinates for every finite-dimensional (real or complex) Lie group. Let us recall it here.

Theorem 6 ([2, 43, 44]). Given a (finite-dimensional) Lie group G (real k = R or complex k = C), its Lie algebra g, and a basis B = (b i ) 1≤i≤n of g, there exists a neighbourhood W of 1 G (in G) and n local coordinate analytic functions W → k, (f i ) 1≤i≤n such that, for all g ∈ W , we have g = → 1≤i≤n e t i (g)b i = e t 1 (g)b 1 e t 2 (g)b 2 . . . e tn(g)bn . Now, we have seen that, if ϕ is moderate, dualizable, associative and commutative,

H ϕ = (K Y , conc, 1 Y * , ∆ ϕ , ) (49)
is isomorphic to the shuffle bialgebra algebra (K Y , conc, 1 Y * , ∆ , ), therefore one can construct bases {Π w } w∈Y * ; {Σ w } w∈Y * of K Y with the following properties:

(1) the restricted family {Π l } l∈Lyn Y is a basis of P = Prim(K Y );

(2) the whole basis is constructed by decreasing concatenation (see Definition 5) and hence of type PBW; (3) they are in duality Π u | Σ v = δ u,v ; (4) due to these three properties, we have

Σ w = Σ i 1 l 1 • • • Σ i k l k i 1 ! • • • i k ! , for w = l i 1 1 . . . l i k k . (50)
Now, within the algebra of double series (whose support is K Y * ⊗Y * ) endowed with the law ϕ ⊗conc, M.-P. Schützenberger (see [START_REF] Reutenauer | Free Lie algebras[END_REF]) gave the beautiful formula

w∈Y * w ⊗ w = l∈Lyn Y e Σ l ⊗P l , (51)
which can be used to provide a system of local coordinates on the Hausdorff group, i.e., the group of series in K Y which are group-like for ∆ ϕ . Indeed, due to the fact that for a group-like S, (S ⊗Id) is compatible with the law of the double algebra, we get 30 , applying the operator (S ⊗Id) to (51), S = (S ⊗Id)(

w∈Y * w ⊗w) = l∈Lyn Y e S|Σ l P l , (52)
which is the perfect analogue of the theorem of Wei and Norman for the Hausdorff group (group of group-like series).

CONCLUSION

In this paper, we have systematically studied the deformations of the shuffle product by addition of a superposition term. Fortunately, this study provides necessary and sufficient conditions for the objects (antipode, Ree ideal, bases in duality) and operators (infinite convolutional series, primitive projectors) to exist together with their consequences. We have established a local system of coordinates for the (infinite-dimensional) Lie group of group-like series. This system is the perfect analogue of the well-known theorem of Wei and Norman which holds for every finite-dimensional Lie group.

8

 8

  if and only if we have S | u ϕ v = 0 for any u and v ∈ Y + . (2) If S | 1 Y * = 1, then S is group-like (i.e., ∆ ϕ (S) = S ⊗ S) 20 if and only if we have S | u ϕ v = S | u S | v for any u and v ∈ Y * .

17

 17

22

 22

Definition 3 .

 3 Here t is a real parameter. Let us define defineD Y := Γ(Id K Y ) = w∈Y * w ⊗ w, Haus Y := log D Y , σ Y (t) := exp(t Haus Y ).

  and for all w ∈ Y + , I + (w) = w. One defines 26 π 1 := log(e + I + ) = n≥1 (-1) n-1 n I + n , where

27 Extended to series. 28 In

 2728 order to clarify the ideas at this point, the reader can also take the alphabet duplication isomorphism for all ȳ ∈ Ȳ , ȳ = α(y), and use {w} w∈ Ȳ * as a basis for K Ȳ . Now, for the remainder of the paper, let Y = {y w } w∈Y + (respectively Y 1 = {y x } x∈Y ) be a copy of Y + (respectively Y ).

Theorem 5 . ( 1 ) 2 ) 3 ) 4 )

 51234 The family {Π l } l∈Lyn Y forms a basis of P. (The family {Π w } w∈Y * is a linear basis of K Y . (The family {Σ w } w∈Y * is a linear basis of the ϕ-shuffle algebra. (The family {Σ l } l∈Lyn Y forms a pure transcendence basis of (K Y , ϕ , 1 Y * ).

  [START_REF] Bui | Structure of polyzetas and explicit representation on transcendence bases of shuffle and stuffle algebras[END_REF]

  > • • • . The free monoid and the set of Lyndon words over Y are denoted by Y * and Lyn Y , respectively. The unit of Y * is denoted by 1 Y * . We also write Y

  , the sum (!) is taken over all subsequences {k 1 , . . . , k i } ⊂ {1, . . . , k} and all Lyndon words l 1 lex • • • lex l n such that (y s 1 , . . . , y s k ) * ⇐ (y s k 1 , . . . , y s k i , l 1 , . . . , l n ), where *

  group-like if and only if 21 log(S) is primitive. Proof. Since ∆ ϕ and the maps T → T ⊗ 1 Y * , T → 1 Y * ⊗ T are continuous homomorphisms, then, if log S is primitive, we have (see Lemma 2(1))

  bialgebras. The bialgebra H ϕ need not be a Hopf algebra, even if ∆ ϕ is cocommutative (see Example 2.2). Now, let us consider

The present work is part of a series of papers devoted to the study of the renormalization of divergent polyzetas (at positive and at non-positive indices) via the factorization of the non-commutative generating series of polylogarithms and of harmonic sums, and via the effective construction of pairs of dual bases in duality in ϕdeformed shuffle algebras. It is a sequel to[START_REF] Duchamp | Combinatorics of ϕdeformed stuffle Hopf algebras[END_REF], and its content was presented in several seminars and meetings, including the 74th Séminaire Lotharingien de Combinatoire.

Abbreviation for associative algebra with unit.