Anne Micheli

Lionel Pournin

Christophe Tollu

Joseph Ben Geloun

Karol Penson

Dao Quang Duc

Combinatoire de l'élimination de Lazard et Interactions Combinatorics of Lazard Elimination and Interactions

This memoir is all about rewriting of inversions in some product structures, reordering and their combinatorial counterparts for partition of alphabets i.e. Lazard's elimination (LE) of generators and associated formulas (in particular their quotients).

(LE) theorems provide uniform formulas for every alphabet and have similar schemes for groups, monoids, Lie algebras and unital associative algebras. These tools give rise to many implementable algorithms. The most celebrated form of (LE) is on the category of Lie k-algebras (k being a unitary ring), we concentrate on monoids and Lie algebras and provide examples on iterated "smash-products", where the rewriting on words ("string rewriting") plays a crucial rôle to understand the normal forms and how one converges to them. The end of this thesis focuses on additional uses of (LE) and word indexing in the context of half-shuffle, Zinbiel algebras and Magnus duality.

Furthermore, it delves into the topics of hyperlogarithms and character theory.

-Thanks to my family (ba Hien, ma Chi, anh hai Bao) and my girlfriend (Thanh Van).

Although words are never enough, thank you for your support and belief in me.

Contents

CONTENTS

Introduction

This memoir is all about rewriting of inversions in some product structures, reordering and their combinatorial counterparts for partition of alphabets i.e. (Lazard's) elimination of generators and formulas of the type ST RU CT ⟨x 1 , x 2 , . . . , x n ⟩ ∼ = N ICE⟨x 1 , x 2 , . . . , x n ⟩ ⋄ ST RU CT 1 ⟨x 1 , . . . , x n-1 ⟩ (0.1)

where N ICE et ST RU CT 1 stand for algebraic structures generated (sometimes freely)

by generators x i . The diamond symbol being, according to the situation, a tensor product, a semi-direct product or a plain (unique) factorization. For example, with the symmetric group S n and the pure braid group PB n [5] :

S n ∼ = Z / nZ ⋄ S n-1 and PB n ∼ = F n-1 ⋄ PB n-1 .
Here, in the first case, ⋄ is only a product of permutable subgroups and the iterated decomposition helps building the infinite chain of embeddings or construct a basis of Q[S n ] adapted to the calculation needs of Dynkin's projector [START_REF] Duchamp | Orthogonal projection onto the free Lie Algebra[END_REF]. In the second case we have a semi-direct product (where F n-1 is the Free Group with n -1 generators).

Let us firstly see the case of two permutable subgroups 1 (where the ⋄ is a symbol of (unique) factorization), we have

G = G 1 G 2 = G 2 G 1 (and it is required that G = G 1 .G 2
be of unique factorization). Then, at the level of the terms, the rewriting reads

g 2 g 1 -→ l(g 1 , g 2 )r(g 1 , g 2 ) (0.2)
and, in the case when r(g 1 , g 2 ) = g 2 identically, we have a semi-direct product i.e. for every (g 1 , g 2 ) ∈ G 1 × G 2 , g 2 g 1 g 2 -1 ∈ G 1 , so that we only need to know the factor l(g 1 , g 2 ).

1 A common occurrence in solvability.

CONTENTS

The categories covered here will be 1. Set, the category of sets.

2. Mon, the category of monoids.

3. Grp, the category of groups.

4. k-Lie, the category of Lie k-algebras (where k is a given commutative ring).

5. k-AAU, the category of unital associative k-algebras (where k is again a given commutative ring).

and (forgetful) functors between the are as follows

Mon(2) Grp (3)
Set [START_REF] Berndt | Ramanujan: Letters and Commentary[END_REF] k-AAU (5) k-Lie(4)

F 12 F 23 F 25 F 45 F 14

CONTENTS

This Ph. D. answers the following questions Q1) What are the expressions of Lazard elimination (LE) in several categories where there are sort of semi-direct products? Q2) Are these universal? (i.e. is every semi-direct product the image of some Lazard elimination? and is LE a free object?2 ) Q3) What is the Combinatorial counterpart of these investigations? (Bases, codes, formulas, etc.) Q4) If possible, is there a deep reason for the similarity of the obtained formulas?

The results of this Ph. D. are 1. Main result: compatibility of a set of generators and relations w.r.t. a partition of the alphabet.

• Application to the Partially Commutative Lie algebra.

• Application to the Drinfeld-Kohno Lie algebra.

2. Every semi-direct product is the image of a Lazard elimination: Boolean gradings.

3. Generalized gradings: in the vein of the enlargement allowed by Wikipedia [START_REF]Graded ring[END_REF] and Bourbaki, the theory of gradings has been enlarged to additive semigroups and applied to the Hilbert series of the Drinfeld-Kohno Lie algebra (and its universal enveloping algebra) with infinite number of generators.

4. The computation of (Id) gen worked out completely for any enveloping algebra 3 (it could be thought as a computation of local coordinates, see Mathoverflow [111]).

5. The infinite product, bases in duality (and Magnus setting) and their compatibility with elimination. CONTENTS 6. Towards a theory of domains for Polylogarithms and Harmonic sums.

Preliminary remark. All results of this memoir (save the Section 3.4 which rests on complex analysis) are of "locally finite" nature and can be reached without topology. In particular the limiting processes of infinite sums and products boil down to the notion of summable families 4 .

From time to time, concepts of general topology (like density and completion) have intentionally been kept because a purely algebraic reformulation would be lengthy, clumsy and poorly expressive.

In the same way, categories are oftentimes used as a unifying concept but many results can be rephrased without using the language of category theory. The chapter contains general facts about "free objects" from the scheme "solutions of universal problems" for categories

Mon, Grp, k-AAU, k-Lie (1.1)
k being a fixed unital commutative ring.

1.1 The (endo)functor Seq.

In the following, we will use the (endo)functor Seq : Set → Set (here all sets of ZFC are admissible whatever their cardinality). Then, a set X being given, Seq(X) is the 1.2. FREE OBJECTS.

set of all sequences of elements of X i.e.

Seq(X)

= n≥0 X {1,••• ,n} (1.2)
where, for E, F sets, F E is the set of all maps E → F , and, for f : X → Y ,

Seq[f ](x 1 , • • • , x n ) := (f (x 1 ), • • • , f (x n )). (1.3)
It is not difficult to check that Seq is a functor.

Remark 1.1. In [START_REF] Flajolet | Analytic Combinatorics[END_REF], P. Flajolet and R. Sedgewick, provide a host of functors and bifunctors with Set (or graded sets1 ), there called "combinatorial classes" as domain (see [START_REF] Flajolet | Analytic Combinatorics[END_REF] Ch 1 §1.2 "Admissible constructions and specifications"). There can be found a host of beautiful combinatorial functors Set → Set as: Seq, Cyc, MSet, ....

Our functor

Seq is then similar to their Seq functor (save a modification of its domain which is the whole sets of ZFC i.e. Set).

1.2 Free objects.

General principle.

In this subsection, we introduce the combinatorial (free) objects we will use throughout the manuscript2 . These objects (call them G(X)) together with a map j X : X → G(X) are all solutions of universal problems. We will recall the definition, notation and terminology about these free objects below (cf. in general Bourbaki [START_REF] Bourbaki | Theory of Sets[END_REF] Ch IV §3 or [START_REF] Lane | Categories for the Working Mathematician[END_REF] and, in particular, [START_REF] Bourbaki | Algebra[END_REF] Ch I §7.1 and Lothaire [START_REF] Lothaire | Combinatorics on words[END_REF] Prop 1.1.1 for free monoids, Bourbaki [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch II §2.2 Prop 1 and Reutenauer [START_REF] Reutenauer | Free Lie Algebras[END_REF] Thm 0.4 for free Lie algebras and Bourbaki [START_REF] Bourbaki | Algebra[END_REF] for enveloping algebras i.e. towards the free associative algebras with unit), but here, we state the general principle.

The scheme is the same for all categories considered in Eq. (1.1) (k being a fixed ring).

CHAPTER 1. PREAMBLE All objects of these categories can be considered as sets, we then have a natural "forgetful" functor F such that, A being an object (of one of these categories), F (A) is the set underlying the structure A. We are now in the position of stating the universal problem corresponding to the forgetful functor F .

Universal problem (w.r.t. F ). -For any set X (C being one of the categories as above) does there exist a pair (j X , G(X))

(G(X) being an object of C and j X : X → F (G(X)) a map) such that:

For any map f : X → F (A), there exists a unique f ∈ Hom C (G(X), A) such that

f = F ( f ) • j X .
Remark 1.2. It might happen that G be not defined everywhere as shows the case with C = FinSet, F being the inclusion functor (i.e. F (X) = X for every finite set and F (f ) = f for every set-theoretical map between finite sets).

However a solution of the universal problem (1.4), for all X, provides a free functor

G : Set → C, X → G(X)
which is left-adjoint to the forgetful functor F : C → Set.

The reader must be aware that, in general, the notion of "forgetful functor" (here constructed from algebraic structures and sets) is informal (see [112]).

Set C X A G(X). F f j X f (1.4)

Presented structures.

For any category of the list (1.1), the notion of a "structure defined by generators and relations" is well defined. For X a set and C ∈ {Mon, Grp, k-AAU, k-Lie}, a relator is a set of equalities t

(1) i = t

(2) i (i ∈ I) where t (j) i are elements of the corresponding free structure G(X). We can express the universal problem of presented structures in terms of maps and co-equalization.

Let u, v : I → G(X) such that u(i) = t

(1)

i , v(i) = t (2)
i , then the structure defined "by generators and relations" with X as set of generators and r = {u(i) = v(i)} i∈I as set 1.2. FREE OBJECTS. of relations is noted

M = ⟨ X | r ⟩ C ∈ C (1.5)
This object comes together with an arrow s r : G(X) → M . The pair (s r , M ) is a solution of the universal problem (see diagram (1.6)) such that

• s = s r is a morphism G(X) → M of C such that F (s) • u = F (s) • v 3
(F : C → Set being the "natural" forgetful functor) and

• For every morphism φ : G(X) → N such that F (φ) • u = F (φ) • v4 there exists a unique morphism φ :

M → N such that φ = φ • s
This situation is summarized by the following diagram.

Set C C I G(X) N M F u v φ sr φ (1.6)
Remark 1. 3. When one has underlying groups (additive or multiplicative), expressions like u i = v i can be made equivalent to w i = e (e is the neutral 5 ). In these cases, the list of relations can be replaced by a list of relators. But, in the case of monoids (and other structures like semigroups), no such mechanism exists and we have to stick to a list of relations of type u i = v i .

Free monoids.

Oftentimes, we will use free monoids as monomials (i.e. in this section C = Mon).

Elements of a free monoid G(X) = X * are usually called "words" which is why the generating set X is often called an alphabet (see [START_REF] Lothaire | Combinatorics on words[END_REF]).

CHAPTER 1. PREAMBLE Alphabets and words.

Given a set X, the free monoid on X is the set X * of all words over the alphabet X (i.e.

the empty word 1 X * or non-empty words x 1 x 2 • • • x n , here n ∈ N + and x i ∈ X, ∀i ∈ [1, n]), made into a monoid using concatenation as

x 1 • • • x n .y 1 • • • y m := x 1 • • • x n y 1 • • • y m .
We notice that, for any X ∈ Set, the set underlying X * (i.e. F (X * ) in the perspective of Subsection 1.2.1) is nothing but Seq(X) with another data structure i.e. words instead of lists (i.e. under this identification, one has F (X * ) = Seq(X)).

We will denote by |w| the length of a word w from X * and a partial degree |w| x is the number of occurrences of an element x of X within w.

A word w ∈ X * being given, we will denote by (w) the multidegree of w, that is the family (|w| x ) x∈X which belongs to N (X) , the set of finitely supported families 6 of N X (which is the set of all functions i.e. families (α x ) x∈X from X → N) (cf. [START_REF] Bourbaki | Algebra[END_REF] Ch I §7.7).

The subset of X * of the words of length n is denoted by

X n = {w ∈ X * | |w| = n}. (1.7)
We also consider the subset of the words with a given multidegree α ∈ N (X) , namely

X α = {w ∈ X * | (w) = α}. (1.8)
Commentary 1. i) When an ambient free monoid X * is fixed, for any Y ⊂ X * , the notation Y * stands for the submonoid (of X * ) generated by Y , this set might not be free as a monoid. For example, in a * (the free monoid on one letter) {a 2 , a 5 } * = {1, a 2 , a 4 } ∪ {a n } n≥5 (1.9)

is not free.

ii) Save in Example 1.3, we will only use the regular expression (see [START_REF]Regular expression[END_REF]) B * Z = {uz} u∈B * ,z∈Z . It can be shown that (B * Z) * is free. 6 Families whose support supp((α x ) x∈X ) := {x ∈ X | α x ̸ = 0} is finite.

FREE OBJECTS.

Free monoid and its universal property.

For X a set (viewed as an alphabet of noncommutative variables), the canonical embedding of X 1 (the set of words of length one, which we identify with X) will be denoted by j X : X → X * .

One can prove easily that the constructed pair (j X , X * ) is a solution of the universal problem corresponding to the following diagram (see description in Subsection 1.2.1, here for C = Mon or [START_REF] Reutenauer | Free Lie Algebras[END_REF] Prop 1.1):

Set Mon X M X * . F f j X f (1.10)
It means that for every monoid M and set-theoretical map f : X → M , there exists a unique morphism of monoids f :

X * → M such that f • j X = f .
In Subsections 1.2.4 and 1.2.5, we will recall the construction of free objects for the category of associative k-algebras with unit k-AAU and in the category of Lie k-algebras k-Lie.

Presentation of a monoid.

Following the general scheme, the monoid presented by a set of generators X and relations r = ((u i = v i )) i∈I i.e.

M = ⟨ X | ((u i = v i )) i∈I ⟩ Mon = X * ≡ r (1.11)
together with the natural arrow s r : X * → M is a solution of the universal problem (see diagram (1.12)) of existence of a pair (s, M ) such that

• s is a morphism of monoids X * → M such that F (s)

• u = F (s) • v 7
(F : Mon → Set being the forgetful functor as above) and

• For all morphisms of monoids φ : (1.12)

X * → N such that F (φ) • u = F (φ) • v
Note that, due to intersection properties of congruences ( [START_REF] Eilenberg | Automata, Languages and Machines[END_REF] Ch 1 §5, see below), the arrow s r is onto as this is the case of the analogue arrows for each category C of the list (1.1).

In this situation, with all set of generators X and relations r = {(u i = v i )} i∈I (u i , v i ∈ X * ), the pair (s r , M ) exists. In fact, M is the quotient of X * by the congruence generated by the family ((u i = v i )) i∈I (see [START_REF] Eilenberg | Automata, Languages and Machines[END_REF] Ch 1 §5).

Example 1.1. Free group (as a presented monoid). -Let X be an alphabet (viewed as a set of generators), we can construct an implementation version of the free group Γ(X) as follows

• Create X, a disjoint copy of X, X = X + X with the involution x → x such that x = x

• Then Γ(X) = X * (xx = 1) x∈ X .

Example 1.2. Dihedral group D 5 (as a presented monoid). -

The Dihedral group D 5 = ⟨ {s 1 , s 2 } | s 2 1 = s 2 2 = (s 1 s 2 ) 5 = 1 ⟩ Mon is known as the symmetry group of the regular pentagon. iii) The Hilbert series of a monoid M is then the formal power series

Hilb(M, t) := m≥0 |M m | t m ∈ N[[t]]. (1.13)
To have such a Hilbert series it suffices that the monoid be a finitely graded set.

Assume that X is a finite alphabet, it is an exercise to check (by the definition of finitely additive proper length function) that a monoid M presented by

M = ⟨ X | ((u i = v i )) i∈I ⟩ Mon with |u i | = |v i |
for all i ∈ I is a finitely graded monoid.

9 l(xy) = l(x) + l(y) ∀x, y ∈ M . 10 l -1 (0) = {1 M }. 11 It is a morphism of monoids M → N, but, due to the lack of inverses, the condition l -1 (0) = {1 M } does not imply that it is injective, of course.

Remark 1.4. A remark for the exercise above is that the set X is a finite alphabet and relations r = {(u i = v i )} i∈I are homogeneous w.r.t. length meaning | u i |=| v i | for all i ∈ I. Moreover, let Bi = ⟨ {a, b} | ab = 1 ⟩ Mon be the bicyclic monoid [START_REF]Bicyclic semigroup[END_REF], then we can prove without difficulty that Bi admits no N-gradation because each additive proper length function l on Bi satisfies 0 = l(1) = l(ab) = l(a) + l(b) ≥ 2 which is a contradiction, this shows that we cannot relax the condition that the relator be homogeneous.

However, one can consider the Hilbert series of a set of distinguished representatives as the language L = {b s a r } r,s∈N is a section of the presented monoid Bi.

The condition homogeneity w.r.t. length whereas it is sufficient, is not necessary It also admits the square-free language as a section, but it is by no means the same monoid (the first one is a group whereas the second is not).

1.2. FREE OBJECTS.

In fact the correct setting for Hilb(V, t) is that of a free module or a vector space then and here as the monoids (1.14) and (1.17) are not graded, we give the Hilbert series of the "square free section" L where

• L 0 = 1 A * ;
• L 1 = {a, b};

• L 2 = {ab, ba};

• • • •
• For all m ≥ 1, one has L 2m = {(ab) m , (ba) m } and L 2m+1 = {a(ba) m , b(ab) m }.

However these monoids are not finitely graded for this decomposition because the grading is not compatible with the product (seen as a concatenation with rewriting) then

Hilb(M, t) = m≥0 |L m | t m (1.18) = 1 + m≥1 2t m = 1 + 2t 1 -t = 1 + t 1 -t . (1.19)
2. Consider another monoid with length presented as

M (A) := ⟨A | a 2 = b 2 ⟩ Mon = ⟨{a, b} | a 2 = b 2 ⟩ Mon (1.20)
One section of it is the language of words without any b 2 factor

L = A * \ A * b 2 A * . (1.21) In this case, M (A) is a finitely graded monoid M (A) = m∈N M (A) m where • M (A) 0 = 1 A * ; • M (A) 1 = W 1 = {a, b}; • M (A) 2 = W 2 = {aa, ab, ba}, | M (A) 2 |=| M (A) 1 | + | M (A) 0 |; • M (A) 3 = W 3 = {aaa, aab, aba, baa, bab} , | M (A) 3 |=| M (A) 2 | + | M (A) 1 |;
• M (A) 4 = W 4 = {aaaa, aaab, aaba, abaa, abab, baaa, baab, baba}, and then

| M (A) 4 |=| M (A) 3 | + | M (A) 2 |;
• M (A) 5 = W 5 = {aaaaa, aaaab, aaaba, aabaa, aabab, abaaa, abaab, ababa, baaaa, baaab, baaba, babaa, babab},

| M (A) 5 |=| M (A) 4 | + | M (A) 3 |; • • • • • For each m ≥ 2, M (
M (A) m |=| M (A) m-1 | + | M (A) m-2 |.
We obtain that | M (A) m |= F (m) is the Fibonacci number (with initial condition 13 ) and then the Hilbert series

F (0) = 1, F (1) = 2
Hilb(M (A), t) = m≥0 |M (A) m | t m = m≥0 F (m)t m = 1 + 2t + 3t 2 + 5t 3 + 8t 4 + 13t 5 + • • • = 1 + t 1 -t -t 2 .

Free associative algebras.

Here, we deal with free associative algebras with unit (i.e. C = k-AAU). See just below.

Free associative k-algebras and its universal property.

Let k be a commutative ring with unit. A noncommutative polynomial on X over k is a linear combination over k of words on X. We simply say polynomial when no confusion arises. If P is a polynomial, we write it as

P = w∈X * ⟨P | w⟩w.
The set of all polynomials is denoted by k⟨X⟩. It has a k-algebra structure, with a concatenation product

P Q = w∈X * ⟨P Q | w⟩w,
where

⟨P Q | w⟩ = w=uv ⟨P | u⟩⟨Q | v⟩.
The polynomial algebra k⟨X⟩ is the free associative k-algebra with unit (free k-AAU) generated by X. It means that, given the canonical embedding j X : X → k⟨X⟩, then 1.2. FREE OBJECTS.

for every associative k-algebra with unit A and set-theoretical map f : X → A, there exists a unique morphism of k-AAU f : k⟨X⟩ → A such that f • j X = f . We get an inclusion and one can prove that the pair (j X , k⟨X⟩) is a solution of the universal problem corresponding to the following diagram (see description in Subsection 1.2.1,

here for C = k-AAU) Set k-AAU X A k⟨X⟩. F f j X f (1.22)
Graded structures of the free associative k-algebra.

We shall explain graded structures of the free associative k-algebra with unit k⟨X⟩.

The reader can review in Bourbaki [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] or Reutenauer [START_REF] Reutenauer | Free Lie Algebras[END_REF] the concept of graded kalgebra (in the latter, homogeneous for the multidegree is called finely homogeneous).

For n ≥ 0, we denote by

k n ⟨X⟩ = span k {X n } (1.23)
the sub k-module generated by X n in k⟨X⟩ (the n-th tensor power of the free k-module with basis X). The grading by the total degree of the free associative k-algebra k⟨X⟩ can be described as a direct sum of k-modules

k⟨X⟩ = n∈N k n ⟨X⟩ (1.24)
with the concatenation multiplication

k n ⟨X⟩.k m ⟨X⟩ ⊆ k n+m ⟨X⟩. (1.25)
A member of one of the subspaces k n ⟨X⟩ (1.24) is called an homogeneous polynomial.

On the other hand, given an α ∈ N (X) there is a sub k-module of k⟨X⟩, namely

k α ⟨X⟩ = span k {X α }. (1.26) CHAPTER 1. PREAMBLE
The free associative k-algebra k⟨X⟩ can be also graded by multidegree

k⟨X⟩ = α∈N (X) k α ⟨X⟩ (1.27)
with the concatenation multiplication

k α ⟨X⟩.k β ⟨X⟩ ⊆ k α+β ⟨X⟩. (1.28)
We also call the grading (1.27) a grading by homogeneous submodules or finely homogeneous submodules. Let α = (α x ) x∈X ∈ N (X) and denote by k α ⟨X⟩ the space of finely homogeneous polynomials of partial degree α x in each letter x. Notice that the set X α is a basis of k α ⟨X⟩ (cf. Bourbaki [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch II §2.6 p.127 "multigraduation" or Reutenauer [START_REF] Reutenauer | Free Lie Algebras[END_REF] p.178 Ch 8 §1.6 where this module is noted E α ).

Free Lie algebras.

Here, we deal with free Lie algebras (i.e. C = k-Lie). See just below.

Lie polynomials.

On the polynomial algebra k⟨X⟩ with the concatenation product P Q = 1.2. FREE OBJECTS.

Moreover, L k (X) is homogeneous for the two gradings, where

L k (X) = n∈N L k (X) n ; L k (X) = α∈N (X) L k (X) α (1.31) with L k (X) n := L k (X) k n ⟨X⟩ (1.32)
(homogeneous Lie polynomials of total degree n) and, for a finitely supported α ∈ N (X)

L k (X) α := L k (X) k α ⟨X⟩ (1.33)
(homogeneous Lie polynomials of multidegree α).

Due to the fine grading of the free Lie algebra, for any (disjoint) partition X = B + Z, we can set

L k (X) B = α∈N (X) |α| Z =0 L k (X) α (1.34)
and

L k (X) BZ = α∈N (X) |α| Z >0 L k (X) α (1.35)
where, for any subset Y ⊂ X we set |α| Y := x∈Y α(x). It is straightforward to see that

L k (X) = L k (X) BZ ⊕ L k (X) B and that L k (X) BZ is a Lie ideal. Then we set 15 L k (X) = L k (X) BZ ⋊ L k (X) B
or, in the language of SES (short exact sequences), the following is split

0 L k (X) BZ L k (X) L k (X) B 0.
[(h 1 , b 1 ), (h 2 , b 2 )] = [h 1 , h 2 ] + α(b 1 )(h 2 ) -α(b 2 )(h 1 ), [b 1 , b 2 ] .
With our construction, we say that the Lie algebra g is a semi-direct product of b with h, denoted by

g := h ⋊ b. See also Definition 2.1.
CHAPTER 1. PREAMBLE Remark 1.5. As a remark, we add for the reader the following nice mnemonic about the orientation of the vertical bar in the ⋉ (or ⋊) notation: the factor "who acts" is the one with the screwdriver i.e. is on the bar's side, this action being by automorphisms for groups and by derivations for Lie algebras.

We will see later that this split SES can serve as a model for every semi-direct product. The classical Lazard elimination theorem (Theorem 2.3 below) will give a better understanding of this split SES ((1.36)), identifying L k (X) B and L k (X) BZ as (isomorphic images of) concretely defined free Lie algebras (together with their alphabets).

Presentation of a Lie algebra.

We here follow Bourbaki [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch 2 §2.3 16 . Let X be a set, g a Lie k-algebra and a = {a x } x∈X a subset of g. Let us consider a Lie morphism

f a : L k (X) → g, X ∋ x → a x .
(1.37)

The elements of the kernel of f a are called the relators of the set a. The set a is called generating (resp. free, basis ) if f a is surjective (resp. injective, bijective).

A presentation of g is an ordered pair (a, r) consisting of a generating set a = {a x } x∈X and a set r = {r j } j∈J of relators of a generating 17 J r := Ker(f a ) which is a Lie ideal of L k (X). We also say that g is presented by the set a related by the relators r j (j ∈ J)

and write

g = ⟨ a | r ⟩ k-Lie = L k (X) J r . (1.38)
Likewise, in other categories we could have defined a group by generators and relations (see [START_REF] Magnus | Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations[END_REF]) and written

G = ⟨ a | r ⟩ Grp .
(1.39) 16 Adapted to our situation which requires WLOG that relators be gathered within a set (because we will perform traces, see Subsection 2.2.2) rather than a family. 17 This time as a Lie ideal.

FREE OBJECTS.

Adjoint representation and derivations of Lie algebras.

For convenience, the adjoint representation within g will be extended to sequences of

elements (Q 1 , Q 2 , • • • , Q n ) ∈ Seq(g) as ad (Q 1 ,Q 2 ,••• ,Qn) := ad Q 1 • ad Q 2 • • • • • ad Qn ∈ End(g). (1.40)
We recall the classical notion of a Lie algebra derivation Definition 1.2. Let g be a Lie k-algebra, we denote End k (g) (or End(g)) the set of k-linear endomorphisms of g viewed as a k-module. Then D ∈ End k (g) is called a derivation of g if and only if for all u, v ∈ g we have

D([u, v]) = [D(u), v] + [u, D(v)]. (1.41)
The set of derivations of g, noted Der(g), is a Lie subalgebra of End(g) (for the usual bracket

[f, g] := f • g -g • f ).
Let g ∈ k-Lie, the adjoint representation is a Lie morphism ad g := ad : g → Der(g),

x → ad x .

(1.42)

We can now extend the adjoint representation ad g to U(g) by

ad U (g) (g 1 • • • g k ) := ad g g 1 • • • • • ad g g k . (1.43)
Moreover, our aim is to generalize this definition to the f -derivations, where f is a fixed Lie algebra morphism Definition 1.3. For any morphism of Lie algebras f : g 1 → g 2 , a k-linear map D :

g 1 → g 2 is called a f -derivation if D([u, v]) = [D(u), f (v)] + [f (u), D(v)] for any u, v ∈ g 1 .
Remark 1.6. i) We beware the reader that f -derivations do not form a Lie algebra but only form a sub k-module of Hom k (g 1 , g 2 ). We will denote this sub k-module by

Der (f ) (g 1 , g 2 ).
ii) It is easily checked that, for D ∈ Der (f ) (g

: L k (X) → L k (X) i.e. D ∈ Der(L k (X)).
Lemma 1.2. Let X be a set, g 2 be a Lie algebra and let f : L k (X) → g 2 be a morphism of Lie algebras. Every mapping of X into g 2 can be extended uniquely to a f -derivation

D : L k (X) → g 2 i.e. D ∈ Der (f ) (L k (X), g 2 ). We remark that Lemma 1.1 is as a consequence of Lemma 1.2 for f = Id L k (X) .
Proof. Let X be a set, let f be a morphism of Lie algebras L k (X) → g 2 , and let d be a mapping of X into g 2 . Notice that g 2 is a L k (X)-module determined by the formula:

u.a = [f (u), a], for any u ∈ L k (X) and a ∈ g 2 .
Prop 8, [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch II §2.8 say that there exists one and only one linear mapping D of L k (X) into g 2 extending d and satisfying the relation:

D([u, v]) = u.D(v) -v.D(u) for any u, v ∈ L k (X), it means that D([u, v]) = [f (u), D(v)] -[f (v), D(u)] = [D(u), f (v)] + [f (u), D(v)]
for any u, v ∈ L k (X). We obtain Lemma 1.2.

The following (easy) proposition gathers properties needed for the proof of our main result in Subsection 2.2.2.

Lemma 1.3. Let g be a Lie k-algebra, D ∈ End(g) (i.e. the set of linear endomorphism of g viewed as a k-module) and J a Lie ideal of g. Then one checks easily that 1. In order that exists a k-linear map [D] such that the following diagram commutes

g g g / J g / J D s J s J [D] (1.44)
it is necessary and sufficient that D(J ) ⊂ J , in this case the ideal J is usually called D-invariant.

1.3. AN EXAMPLE OF FREE OBJECT WITH RESPECT TO GRAPHS.

2. If, moreover, D is a derivation of g, then one has the induced map [D] ∈ Der( g / J ).

3. The inverse image D -1 (J ) (D being still a derivation) is a Lie subalgebra of g.

Remark 1.7. The reader is invited to remark that the proof of the above does not use the defining identities of a Lie algebra ([x, x] = 0 identically and Jacobi) and holds true mutatis mutandis to any k-algebra (non necessarily associative nor unital).

An example of free object with respect to graphs.

We now introduce a particular class of monoids. They are known under various names (monoid of commutations and rearrangements, monoid of traces, partially commutative monoid, Cartier-Foata monoid) and has many applications: in mathematics and combinatorics [START_REF] Cartier | Problèmes combinatoires de commutation et réarrangements[END_REF] as well as in computer sciences [START_REF] Diekert | Partial Commutation and Traces[END_REF]. It was introduced by Cartier and Foata in 1969 and defined by a presentation consisting in commutations between certain pairs of generators.

As relations of these presentations are only commutations, they can be read within the categories Mon, Grp, k-AAU and k-Lie (cf. Duchamp and Krob [START_REF] Duchamp | Free partially commutative structures[END_REF]). In fact, for (at least) these categories and M an object of one of them, the set of pairs (x, y) ∈ M 2 such that x and y commute is a reflexive and symmetric graph θ M ⊂ M 2 . We introduce the definition of an alphabet with commutations.

Definition 1.4. Let X ∈ Set be a set viewed as a alphabet. A commutation relation on X is a reflexive and symmetric graph θ ⊂ X 2 (i.e. θ = θ -1 and ∆

X := {(x, x)} x∈X , the diagonal of X, is a subset of θ).
The pairs (X, θ) where θ is a commutation relation on X form a category CommAlph, the arrows of which are f : A pair (X, θ X ) ∈ CommAlph being given, does there exist a pair (j (X,θ X ) , G(X, θ X ))

(X, θ X ) → (Y, θ Y ) such that • f : X → Y is a map • for all (x 1 , x 2 ) ∈ θ X we have (f (x 1 ), f (x 2 )) ∈ θ Y .
such that, for all A ∈ C and arrow f :

(X, θ X ) → F (A), we have a unique f ∈ Hom C (G(X, θ X ), A) such that f = F ( f ) • j X . Diagrammatically CommAlph C (X, θ X ) A G(X, θ X ). F f j (X,θ X ) f (1.45)
The theory of free partially commutative structures [START_REF] Duchamp | Free partially commutative structures[END_REF], which is the last one (in characteristic zero) for which Magnus theory 18 holds (see [START_REF] Duchamp | Factorisations dans le monoïde partiellement commutatif libre[END_REF] Thm 3) provides, as for the case of alphabets (sets), free objects. Let us give the two structures we will need as well as their constructions

• The free partially commutative monoid M (X, θ)

• The free partially commutative Lie algebra L k (X, θ).

As all categories in the list (1.1) have a mechanism of presentation 19 , then, unsurprisingly M (X, θ) and L k (X, θ) are quotients.

Firstly, M (X, θ) is the quotient of X * by the congruence 20 generated by the family (xy = yx) (x,y)∈θ . We will consider the canonical surjection s θ : X * → M (X, θ) as well as j θ : M (X, θ) → X * an arbitrary (but fixed) set-theoretical section of it. We will also need the notion of a Terminal Alphabet which is, in the model of Viennot, the set of 18 Magnus transformation, Lower central series of the Free group, Möbius counting of dimensions of the free Lie algebra, &c. 19 Which means definition of objects by generators and relations. 20 An equivalence relation compatible with right and left translations of the monoid X * (see [START_REF] Eilenberg | Automata, Languages and Machines[END_REF] Ch

TAlph(t) = {x ∈ X | t ∈ M (X, θ).x}.
Secondly, L k (X, θ) is the quotient of L k (X) by the ideal generated by the relator

r θ = {[x, y]} (x,y)∈θ .
The monoid M (X, θ) received a strikingly intuitive and powerful interpretation in terms of heaps of pieces by G. X. Viennot in [START_REF] Viennot | Heaps of Pieces : Basic definitions and combinatorial lemmas, in "Combinatoire enumerative[END_REF] (see also in Krattenthaler [START_REF] Krattenthaler | Commutation and Rearrangements by Pierre Cartier and Dominique Foata[END_REF]). Let B be a set (of pieces) that is identified with the set X, together with a symmetric and for all i ∈ {1, • • • , n}, x i ⪯ t x j if x i appears before x j in the word t (from right to left) and x i Rx j .

• ℓ t is a labelling of the elements of P t by elements of B.

Of course, an equivalence class of the word t in M (X, θ) corresponds to the same heap

H t in H(B, R).
Under this correspondence, the composition of equivalence classes of words induced by concatenation of words corresponds exactly to the composition of heaps. We thus obtain an isomorphism of monoids (1.47)

H C-F : M (X, θ) → H(B, R), [t] → H t . (1.46) Example 1.4. Let X = {b 1 , b 2 , • • • , b 7 } be a set. Given a commutation relation θ on X as in the list below (not mentioning {(b i , b i )} i=1,••• ,7 the diagonal of X; if a relation CHAPTER 1. PREAMBLE (b i , b j ) ∈ θ then also (b j , b i ) ∈ θ) (b 1 , b 2 ), (
For a word t = x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 = b 7 b 6 b 1 b 3 b 4 b 5 b 1 b 2 in M (X, θ)
, an equivalence class of the word t will be the following sequence of interchanges

b 7 b 6 b 1 b 3 b 4 b 5 b 1 b 2 ∼ b 7 b 6 b 1 b 4 b 3 b 5 b 1 b 2 ∼ b 7 b 6 b 1 b 4 b 5 b 3 b 1 b 2 ∼ b 7 b 6 b 1 b 5 b 4 b 3 b 1 b 2 ∼ b 7 b 6 b 5 b 1 b 4 b 3 b 1 b 2 ∼ b 7 b 5 b 6 b 1 b 4 b 3 b 1 b 2 ∼ b 5 b 7 b 6 b 1 b 4 b 3 b 1 b 2 ∼ b 5 b 7 b 1 b 6 b 4 b 3 b 1 b 2 ∼ b 5 b 7 b 1 b 4 b 6 b 3 b 1 b 2 ∼ b 5 b 7 b 1 b 4 b 6 b 3 b 2 b 1 ∼ b 5 b 7 b 1 b 4 b 6 b 2 b 3 b 1 ∼ b 5 b 7 b 1 b 4 b 2 b 6 b 3 b 1 ∼ b 5 b 7 b 1 b 2 b 4 b 6 b 3 b 1 ∼ b 5 b 7 b 2 b 1 b 4 b 6 b 3 b 1 ∼ • • •
Therefore, the corresponding heap H t = (P t , ⪯ t , ℓ t ) (where

P t = {x 1 , x 2 , • • • , x 8 } = {b 7 , b 6 , b 1 , b 3 , b 4 , b 5 , b 1 , b 2 } is the set of 8 pieces) under the isomorphism H C-F (1.46)
can be illustrated in Figure 1.2 as below. We notice that, contrariwise to [START_REF] Krattenthaler | Commutation and Rearrangements by Pierre Cartier and Dominique Foata[END_REF] we read our word t (from a heap of pieces H t in Figure 1.2) from top to bottom and left to right.

Remark 1.8.

Let t = x 1 • • • x n ∈ M (X, θ). One can show that, TAlph(t) = {x i | i ∈ [1, n] with (x j , x i ) ∈ θ for all j ∈ [i + 1, n]}.
Geometrically, under the isomorphism of monoids H C-F (1.46), when successive pieces are piled from bottom to top, the set Terminal Alphabet TAlph(t) can be described (as already said) as pieces (in the heap H t ) resting on the floor. Krattenthaler [START_REF] Krattenthaler | Commutation and Rearrangements by Pierre Cartier and Dominique Foata[END_REF], order of the letters has been reversed.

Remark 1.9. For (at least) the categories Mon, Grp, k-Lie, k-AAU, there is a forgetful functor to the category CommAlph of alphabets with commutations, that of pairs (X, θ) where ∆ X ⊂ θ ⊂ X × X and θ = θ -1 (in other words θ is a reflexive and

symmetric relation on X). An arrow f : (X, θ X ) → (Y, θ Y ) is a map f : X → Y such
that for all (x, y) ∈ θ X we have (f (x), f (y)) ∈ θ Y . All structures of (1.45) are free with respect to these functors.

Chapter 2

Lazard's elimination

(B + Z) = Free(C B [Z]) ⋊ Free(B) (2.2)
where X = B + Z is a set of generators (an alphabet) divided in two sectors (set partition) and C B [Z] is some code (i.e. a set built from the data of B and Z), Free is the free functor attached to the considered category and ⋊ is a sort of semi-direct product1 .

The study of quotients of Lazard's eliminations for k-Lie is particularly interesting.

It arises from dividing the classical elimination over a relator on the free Lie algebra The braid group B n on n strands is the group of isotopy classes of n-braids with concatenation given by glueing (see Figure 2.2).

L k (X)
• σ i σ i+1 σ i = σ i+1 σ i σ i+1 for i = 1, • • • , n -2. • σ i σ j = σ j σ i for | i -j |≥ 2.
strand i strand i+1

Figure 2.3: A standard generator σ i of the braid group B n .

To each braid in B n one can associate the permutation of the marked points, that is an element of the symmetric group S n . This leads to a natural homomorphism of group π : B n → S n characterized by

σ i → (i, i + 1).
The kernel of π is precisely the subgroup of B n formed by braids inducing the trivial permutation. It is called the pure braid group on n strands and denoted by PB n . Geometrically, the group PB n consists of those braids such that each strand starts and ends at the same point (see Figure 2.4). Moreover, we can verify that the pure braid group PB n coincides with the fundamental group of the complex configuration space Via Artin's presentation, PB n is the group generated by the twists

C n * = {z = (z 1 , . . . , z n ) ∈ C n |z i ̸ = z j if i ̸ = j} i.e. PB n = π 1 (C n * ).
x ij := (σ j-1 • • • σ i+1 )σ 2 i (σ -1 i+1 • • • σ -1 j-1 ) (for i < j, i, j = 1, • • • , n, see Figure 2.6) modulo the relations • (x ij , x kl ) = 1 for i < j < k < l,
• (x il , x jk ) = 1 for i < j < k < l,

• (x ij x ik , x jk ) = (x ij , x ik x jk ) = 1 for i < j < k, • (x kl x ik x -1 kl , x jl ) = 1 for i < j < k < l,
where, in a group (g, h) = g -1 h -1 gh is the commutator between g and h (see [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch II §4.4). j i Here, with i = j -3, we have x ij = σ j-1 σ j-2 σ2 j-3 σ -1 j-2 σ -1 j-1 (diagram: top to bottom and formula: left to right).

Given a pure braid β ∈ PB n , we can remove its last strand and obtain a pure braid ρ(β) ∈ PB n-1 . This yields a surjective homomorphism of groups ρ : PB n → PB n-1 .

Let F n-1 denote the free group on n -1 letters x 1 , • • • , x n-1 . It can be proved (cf.

Kassel and Turaev [START_REF] Kassel | Braid Groups[END_REF] Ch I §1.3) that we have a short exact sequence

0 F n-1 PB n PB n-1 0 ι ρ = forget last strand µ = insert straight last strand (2.3) where ι(x i ) := x in = (σ n-1 • • • σ i+1 )σ 2 i (σ -1 i+1 • • • σ -1 n-1 ) for i = 1, • • • , n -1.
Furthermore, one sees easily that this exact sequence is split (see the move µ in (2.3)), so that

PB n ∼ = F n-1 ⋊ PB n-1 .
For any group G, a graded Z-module is associated to G. It is defined (cf. Bourbaki [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch II §4.6) as

gr Z (G) := ∞ m=1 gr m (G) (2.4)
where the associated m-th quotient

gr m (G) := G m G m+1
is itself defined over {G m } m≥1 , the lower central series (cf. Bourbaki [START_REF] Bourbaki | Algebra[END_REF] Ch I §6.3 Def 5 and [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch II §4.6) of G is made up of normal subgroups 2

G 1 := G ⊇ G 2 := (G 1 , G) ⊇ • • • ⊇ G m+1 := (G m , G) ⊇ • • • .
It is remarkable that this module, gr Z (G), due to the identities of P. Hall and M.

Lazard (see [START_REF] Lazard | Sur les groupes nilpotents et les anneaux de Lie[END_REF]) is, in fact, a Lie Z-algebra (the bracket being the projection of the commutator in the group).

Then, calling t i,j , the projection of x ij into the graded Lie algebra gr Z (PB n ), it can be found in some paper by Ihara (cf. [START_REF] Ihara | Automorphisms of pure sphere braid groups and Galois representations[END_REF]

R[n] =          R 1 [n] [t i,j , t i,k + t j,k ] for 1 ≤ i < j < k ≤ n, R 2 [n] [t i,j + t i,k , t j,k ] for 1 ≤ i < j < k ≤ n, R 3 [n] [t i,j , t k,l ] for 1 ≤ i < j ≤ n, 1 ≤ k < l ≤ n,
DK k,n ∼ = DK Z,n ⊗ Z k. ( 2

.7)

3 There are three ways to reach these relations

• The lower central series of PB n and

• A theorem of Kohno on "small" representations of PB n ,

• Integrability condition for the KZ n differential equation.

Thus, an important consequence of our formulations is that we can construct a commutative diagram of k-modules with split short exact rows

0 gr k (F n-1 ) gr k (PB n ) gr k (PB n-1 ) 0 0 L k (x 1 , • • • , x n-1 ) DK k,n DK k,n-1 0 ∼ = ∼ = ∼ = (2.8)
where the graded Lie algebra gr k (G) = gr Z (G) ⊗ Z k for any group G. In particular, we obtain an isomorphism of k-modules

DK k,n ∼ = L k (x 1 , • • • , x n-1 ) ⊕ DK k,n-1 .
(2.9)

A natural question is how to construct a Lie isomorphism from the Drinfeld-Kohno Lie algebra to a semi-direct product of Lie algebras

DK k,n ∼ = -→ L k (x 1 , • • • , x n-1 ) ⋊ DK k,n-1 .
We call the phenomenon by the decomposition of Drinfeld-Kohno Lie algebra and it will be completely achieved in Corollary 2.17. Furthermore, we realize that this decomposition can be read directly in terms of alphabets partitioned in two blocks T n = T n-1 +T n , this is the "raison d'être" of our main result here (the reader can refer to Subsection 2.2.2). We introduce the notion of quotients of Lazard's eliminations which generalizes the classical elimination on the free Lie algebra L k (X) to a more general scheme. More precisely, given X = B + Z a set partitioned in two blocks and a relator r ⊂ L k (X) which is compatible with the alphabet partition, our main results are to give a Lie isomorphism [see Theorem 2.6 point (iii)]

L k (X) / J ∼ = -→ L k (X) BZ J Z BZ ⋊ L k (X) B J B
and to determine a necessary and sufficient condition (see Proposition 2.9) so that

L k (Z) ∼ = L k (X) BZ J Z BZ .
The case of the infinitesimal pure braid relator is an example of a good relator satisfying all hypotheses in our main theorem and Proposition 2.9, thus it gives rise to the existence of the decomposition of Drinfeld-Kohno Lie algebra i.e there is DK k,n ∼ = We shall now study a Theorem of N. Bourbaki [START_REF] Bourbaki | General Topology[END_REF] which allows to consider semidirect products as categorical colimits. In the language of MO [108]

L k (x 1 , • • • , x n-1 ) ⋊ DK k,n-1
f (ϕ h (n)) = g(h)f (n)g(h -1 ) (2.10)
for all n ∈ N , h ∈ H. Then there is a unique homomorphism k : N ⋊ H → G extending f and g in the usual sense (i.e. k(n, h) = f (n).g(h) the product in G between f (n) and g(h)), where the group operation in N ⋊ H is given by

(n 1 , h 1 ).(n 2 , h 2 ) = (n 1 ϕ h 1 (n 2 ), h 1 h 2 ).
This is for groups, but we will tailor a similar property for Lie algebras remarking that (2.1) can be, in bivariate notations, reformutated as .11) this is equivalent to saying that the following diagram commutes

f (ϕ(h, n)) = Ad G (g(h), f (n)), ( 2
H × N G × G N G. g×f ϕ Ad G f
As we want to highlight the similarity between semi-direct products of groups and Lie algebras, we recall below the definition of it.

[(h 1 , b 1 ), (h 2 , b 2 )] = [h 1 , h 2 ] + α(b 1 )(h 2 ) -α(b 2 )(h 1 ), [b 1 , b 2 ] .
We denote this Lie algebra by g := h ⋊ b and call it "the semi-direct product of b with h". themselves by derivations (resp. inner adjoint representation ad). We will have the choice between the indexed notation Ad h (g) := hgh -1 (resp. ad h (g) = [h, g]) and a bivariate one Ad(h, g) = hgh -1 (resp. ad(h, g) = [h, g]). In order to ease the writing of diagrams and equivariance, we will adopt below the bivariate notation.

Adaptation of Bourbaki's Proposition (i.e. Theorem 2.1) to our situation is then the following:

Proposition 2.2. Let k be a commutative ring with unit and g i , i = 1, 2 be two Lie k-algebras. We suppose given also a Lie k-algebra morphism α : g 2 → Der(g 1 ). Then let f 1 : g 1 → g, f 2 : g 2 → g be two homomorphisms into a Lie k-algebra g, such that

f 1 (α(b, a)) = ad g (f 2 (b), f 1 (a)) (2.12)
for all b ∈ g 2 , a ∈ g 1 (α and ad g are here written in bivariate notation in the obvious way). Then there is a unique homomorphism f : g 1 ⋊ g 2 → g extending f 1 and f 2 in the usual sense.

Again, this is equivalent to saying that the diagram (2.13) below commutes.

We are led to the following definition of SDT ("Semi-Direct Twist") for the categories of groups and Lie algebras Definition 2.2. i) A semi-direct twist in the category of groups is a triplet (G 2 , G 1 , α)

where α : G 2 × G 1 → G 1 is such that b → α(b, -) is a group morphism G 2 → Aut(G 1 ).
ii) A semi-direct twist in the category of Lie k-algebras is a triplet (g 2 , g 1 , α) where

α : g 2 × g 1 → g 1 is such that b → α(b, -) is a Lie algebra morphism g 2 → Der(g 1 ).
In Grp (resp. k-Lie), a morphism between (G 2 , G

G 2 × G 1 H 2 × H 1 G 1 H 1 f 2 ×f 1 α β f and g 2 × g 1 h 2 × h 1 g 1 h 1 f 2 ×f 1 α β f commute.
We are now led to define SDT Grp the category of semi-direct twists in Grp and SDT k-Lie the category of semi-direct twists in k-Lie.

We have the following useful comments

Commentary 2. i) The functor F from Grp to SDT Grp defined by G → G, G, Ad G is right-adjoint to the functor SDT Grp → Grp defined by (G 2 , G 1 , α) → G 1 ⋊ G 2 and
the same holds for the analogous functors in Lie algebras.

ii) More precisely, one has the following universal diagrams

SDT Grp Grp (G 2 , G 1 , α) G G 1 ⋊ G 2 F f 2 ×f 1 j f and SDT k-Lie k-Lie (g 2 , g 1 , α) g g 1 ⋊ g 2 . F f 2 ×f 1 j f
Remark 2.3. We would like here to formulate two remarks about equation (2.12)

Firstly, it can be proved only on generators.

More precisely, let G 2 = {b j } j∈J and G 1 = {a i } i∈I be set of generators of the Lie algebras g 2 and g 1 respectively and assume that the diagram (2.13) commutes for

(b, a) ∈ G 2 × G 1 .
Now, given a fixed generator b j and using Jacobi identity, the two k-linear maps D 1 :

g 1 → g, a → f 1 • α ⊗ (b j ⊗ a) and D 2 : g 1 → g, a → ad g ⊗ • (f 2 ⊗ f 1 )(b j ⊗ a) are f 1 - derivations i.e. D 1 , D 2 ∈ Der (f 1 ) (g 1 , g).
By assumption, we have that D 1 -D 2 is zero on generators {a i } i∈I of g 1 thus, by Remark 1.6 (ii), D 1 -D 2 = 0. This means that 2.2. A GENERALIZATION OF LAZARD'S ELIMINATION THEOREM. for all a ∈ g 1 one has

f 1 • α ⊗ (b j ⊗ a) = ad g ⊗ • (f 2 ⊗ f 1 )(b j ⊗ a)
and this is true for all fixed b j ∈ G 2 . To summarize, the above equation is true for

(a, b) ∈ g 1 × G 2 .
We then define the following submodule

m = {b ∈ g 2 | (∀a ∈ g 1 ) f 1 • α ⊗ (b ⊗ a) = ad g ⊗ •(f 2 ⊗ f 1 )(b ⊗ a)}.
and leave to the reader to check (by Jacobi identity) that m is a Lie subalgebra of g 2 .

Then, as inclusion G 2 ⊆ m follows from the summary, one derives m = g 2 and then

Eq. (2.12) is established.

Secondly Eq. (2.12) is equivalent to saying that the following commutes

g 2 ⊗ g 1 g ⊗ g g 1 g f 2 ⊗f 1 α ⊗ ad g ⊗ f 1 (2.13)
(here α ⊗ and ad g ⊗ are linear maps respectively induced from the bilinear maps α : g 2 × g 1 → g 1 and ad g : g × g → g).

A generalization of Lazard's elimination theorem.

We here introduce the classical Lazard's elimination theorem and extend it to a more general scheme, namely theory of quotients of Lazard's eliminations. The main applications are to derive answers to Pr. Schützenberger's questions about the Partially Commutative Free Lie algebra (cf. Duchamp and Krob [START_REF] Duchamp | Free partially commutative structures[END_REF]) and the decomposition of Drinfeld-Kohno Lie algebra described in Subsection 2.2.3.

Classical Lazard's elimination.

Let us recall briefly Lazard's elimination theorem in our setting. CHAPTER 2. LAZARD'S ELIMINATION Theorem 2.3 (Lazard's elimination theorem, see also in [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch II §2.9 Props 9 and 10). Let X = B + Z be a set partitioned in two blocks. We have an isomorphism of split short exact sequences

0 L k (B * Z) L k (X) L k (B) 0 0 L k (X) BZ L k (X) L k (X) B 0 j B|Z (=rn) rn Id p B|Z j B j p (2.14)
where B * Z is the set of words B * Z = {uz} u∈B * ,z∈Z (all letters are in B save the last one which is in Z) and where the maps rn and j B are as follows

• the mapping rn is, by universal property, the unique Lie morphism

L k (B * Z) → L k (X) BZ such that, for u = b 1 • • • b k ∈ B * and z ∈ Z we get rn(uz) = ad L k (X) b 1 • • • • • ad L k (X) b k (z) =: ad L k (X) (u) (z) (2.15)
bracketing, see [START_REF] Reutenauer | Free Lie Algebras[END_REF] Ch 1 §3 p.20 (indeed, we here use the same symbol rn for the Lie morphism L k (B * Z) → L k (X) and for the restriction to its image as in the diagram (2.14)).

• for convenience to describe a fixed double symbol formed (j, p) of any short exact sequence, the mapping j B|Z in the first arrow of the diagram (2.14) is in fact the morphism rn : L k (B * Z) → L k (X) and p B|Z is the Lie algebra homomorphism

L k (X) → L k (B) that sends each b ∈ B to b and sends each z ∈ Z to 0. • if j B : L k (B) → L k (X)
is the subalphabet embedding, (so that the restriction to its image is the Lie isomorphism

j B : L k (B) → L k (X) B ) then j B • p B|Z is the projector on L k (X) B = α∈N (X) |α| Z =0 L k (X) α . The kernel of p B|Z is L k (X) BZ = α∈N (X) |α| Z >0 L k (X) α .
• diagram (2.14) is a split SES, one of its section is given by j B .

A GENERALIZATION OF LAZARD'S ELIMINATION THEOREM.

Commentary 3. i) The map rn acts as a substitution, for example

rn ([b 1 b 2 z 1 , b 3 z 2 ]) = [[b 1 , [b 2 , z 1 ]] , [b 3 , z 2 ]]
for b i ∈ B and z i ∈ Z.

ii) What says Theorem 2.3 above is that

rn : L k (B * Z) → L k (X) BZ
is an isomorphism i.e. given Q ∈ L k (X) BZ (each Lie monomial has at least one z ∈ Z in its multidegree), there exists a) words (u

1 z 1 , • • • , u n z n ) b) a Lie polynomial P = P (u 1 z 1 , • • • , u n z n ) ∈ L k (B * Z)
such that rn(P ) = P (rn(u 1 z 1 ), • • • , rn(u n z n )) = Q and that P is unique. For example, we have here,

t (1) b ([b 1 b 2 z 1 , b 3 z 2 ]) = [bb 1 b 2 z 1 , b 3 z 2 ] + [b 1 b 2 z 1 , bb 3 z 2 ] .
• By universal property, the map b → t

(1) b : B → Der(L k (B * Z)) is extended as a morphism of Lie algebras Q → t (2) Q : L k (B) → Der(L k (B * Z)) rewritten at once bivariately as t (2) Q (P ) =: α(Q, P ) ∈ L k (B * Z) (i.e. we define a map α : L k (B) × L k (B * Z) → L k (B * Z) by α (Q, P ) = t (2) Q (P )).
• We get a map α ⊗ as in (2.13) (with g 2 = L k (B) and

g 1 = L k (B * Z)). This action α (or α ⊗ ) allows us to construct L k (B * Z) ⋊ L k (B) (supported by L k (B * Z) ⊕ L k (B)
) according to Definition 2.1 (see also [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch 1 §1.8 and our Proposition 2.2).

CHAPTER 2. LAZARD'S ELIMINATION

• The pair of Lie homomorphisms f 1 : L k (B * Z) → L k (X) and f 2 : L k (B) → L k (X) defined by f 1 (uz) := rn(uz) (where rn is the right-normed bracketing, see above) and f 2 (b) := b satisfy the equivariance condition6 of (2.12) w.r.t. α, hence, by Prop. Proposition 2.2, there exists a unique Lie k-algebra morphism

f : L k (B * Z) ⋊ L k (B) → L k (X) extending f i , i = 1, 2. • In the inverse direction, a Lie algebra morphism L k (X) → L k (B * Z) ⋊ L k (B)
can be constructed by the universal property of L k (X), sending each generator b ∈ B to (0, b) and each generator z ∈ Z to (z, 0).

• The two obtained arrows are proven mutually inverse by a direct computation on the generators.

• So far, we have proved that L k (X) ∼ = L k (B * Z)⋊L k (B). In order to complete the proof of Theorem 2.

A (T , (x 1 , • • • , x n ))
(where T is a binary tree with n leaves (noted

|T | l = n) and (x 1 , • • • , x n ) ∈ A n ) by the recursion ev A (T , (x 1 , • • • , x n )) =          x 1 if |T | l = 1, ev A (T 1 , (x 1 , • • • , x p )) * ev A (T 2 , (x p+1 , • • • , x n )) if T = (T 1 , T 2 ) and |T 1 | l = p. (2.16)
As a particular case, we can define an operator ev g (T ,

(x 1 , • • • , x n )) on a Lie k-algebra 2.2. A GENERALIZATION OF LAZARD'S ELIMINATION THEOREM.
g by replacing the algebra operation * by the Lie bracket [

•, •] i.e. for (x 1 , • • • , x n ) ∈ g n ev g (T , (x 1 , • • • , x n )) =          x 1 if |T | l = 1, [ev g (T 1 , (x 1 , • • • , x p )), ev g (T 2 , (x p+1 , • • • , x n ))] if T = (T 1 , T 2 ) and |T 1 | l = p.
(2.17) Now, we define a special sequence of trees (right-normed tree or Dynkin combs) with n leaves, noted D n , defined recursively by

D 1 = • ; D n+1 = (•, D n ).
(2.18)

Then, for a Lie k-algebra g and any sequence (x 1 , • • • , x n ) ∈ g n (or written as a word

x 1 • • • x n ∈ g * when there is no ambiguity), we can set rn((x 1 , • • • , x n )) = rn(x 1 • • • x n ) := ev g (D n , (x 1 , • • • , x n )). (2.19)
The action of rn as a Lie morphism is therefore that of a substitution: for example, its action on Lie monomials of L k (B * Z) is as follows. For every tree with n leaves T and list of words u i z i ∈ B * Z (could be called blocks, because B * Z is a code in the sense of [START_REF] Lothaire | Combinatorics on words[END_REF], Prop 1.2.1)

rn ev L k (B * Z) (T , (u 1 z 1 , • • • , u n z n )) = ev L k (X) BZ T , (rn(u 1 z 1 ), • • • , rn(u n z n )) .
(2.20)

For example, let X = B + Z be a set partitioned in two blocks B = {b 1 , b 2 } and

Z = {z 1 , z 2 , z 3 }. Then the image of b 1 z 3 , b 1 b 2 z 1 , b 3 b 3 z 2 under the Lie algebra morphism rn is rn b 1 z 3 , b 1 b 2 z 1 , b 3 b 3 z 2 = rn(b 1 z 3 ), rn(b 1 b 2 z 1 ) , rn(b 3 b 3 z 2 ) = [b 1 , z 3 ], [b 1 , [b 2 , z 1 ]] , [b 3 , [b 3 , z 2 ]] .
ii) We have already called "monomial" (see Proposition Let X = B ⊔ Z be a set partitioned in two blocks and φ : X → g a (set-theoretical) map such that8

1. φ(B) is a generating set of b as a Lie algebra.

2. φ(Z) is a generating set of h as a Lie ideal of g. Then 1. φ(X) is a generating set of g as a Lie algebra.

2. One has a commutative diagram of Lie algebras with split short exact rows (and commuting sections)

0 L k (B * Z) L k (X) L k (B) 0 0 h g b 0. j B|Z (=rn) φ 1 p B|Z φ 3 φ 2 j p (2.21)
The arrows j B|Z and p B|Z here are the ones constructed as in Theorem 2.3, j, p being the canonical maps for the (internal) decomposition g = h ⊕ b and

• φ 3 , φ 2 are extensions of φ by universal property of Diagram (1.30)

• for u = b 1 • • • b k ∈ B * , z ∈ Z, we have φ 1 (uz) := ad g φ(b 1 ) • • • • • ad g φ(b k ) (φ(z)). (2.22)
In order to make a smooth transition with the subsequent point (i.e. 2. Let X = B + Z be a set partitioned in two blocks and φ : X → g 3 a (set-theoretical) map such that 1. φ(B) ⊂ {0} × g 2 and pr 2 • φ(B) is a generating set of g 2 as a Lie algebra.

2. φ(Z) ⊂ g 1 × {0} and pr 1 • φ(Z) is a generating set of g 1 as a Lie algebras with operators (here, operators are provided by α : g 2 → Der(g 1 )).

Then 1. φ(X) is a generating set of g 3 as a Lie algebra.

2. One has a commutative diagram of Lie algebras with split short exact rows

0 L k (B * Z) L k (X) L k (B) 0 0 g 1 g 3 g 2 0. j B|Z (=rn) φ 1 p B|Z φ 3 φ 2 j 1 pr 2 (2.23)
The arrows j B|Z and p B|Z here are the ones constructed as in Theorem 2.3, j 1 , pr 2 being the canonical maps for the (external) decomposition g 3 = g 1 ⊕ g 2 and

• φ 3 (resp. φ 2 ) is the extension of φ (resp. pr 2 •φ) by universal property of Diagram (1.30) • for u = b 1 • • • b k ∈ B * , z ∈ Z, we have φ 1 (uz) := α[φ(b 1 )] • • • • • α[φ(b k )](φ(z)). (2.24)

Quotients of Lazard's eliminations.

In this subsection, we deal with a special kind of relators i.e. relators being compatible with an elimination scheme. This situation encompasses presented Lie algebras like Drinfeld-Kohno or partially commutative ones (cf. Duchamp and Krob [START_REF] Duchamp | Free partially commutative structures[END_REF]). The situation will be described in the subsequent section.

Definition of the compatibility and an example.

Let k be a ring. Let X = B + Z be a set partitioned in two blocks. We suppose given a relator r = {r j } j∈J ⊂ L k (X) (cf. [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch II §2.3 9 ) which is compatible with the alphabet partition i.e. there exists a partition of the set of indices

J = J Z ⊔ J B such that • r B = {r j } j∈J B = r ∩ L k (X) B and r Z = {r j } j∈J Z = r ∩ L k (X) BZ .
The notations being as above, we construct the ideals

• J B is the Lie ideal of L k (X) B generated by {r j } j∈J B
• J , J Z and J BZ are the Lie ideals of L k (X) generated respectively by r, r Z and

r BZ := {ad Q z} Q∈J B ,z∈Z . Example 2.1. i) Let us recall that T n+1 = {t ij } 1≤i<j≤n+1
is the set of variables and the infinitesimal pure braid relator R[n + 1] (2.6) in the free Lie algebra L k (T n+1 ).

A typical example is for the graded set

T n+1 = T n ⊔ T n+1 (i.e. T n+1 = {t i,n+1 } 1≤i≤n )
and the infinitesimal pure braid relator r

:= R[n + 1] ⊂ L k (T n+1 ).
In this case, we observe that the relator r Tn is equal to

R[n + 1] ∩ L k (T n+1 ) Tn = R[n]
and the relator (2.25)

r T n+1 = R[n + 1] ∩ L k (T n+1 ) TnT n+1 is equal to the following formulas          R † 1 [n + 1] [t i,j , t i,n+1 + t j,n+1 ] for 1 ≤ i < j ≤ n, R † 2 [n + 1] [t i,j + t i,n+1 , t j,n+1 ] for 1 ≤ i < j ≤ n, R † 3 [n + 1] ±[t i,j , t k,n+1 ] for 1 ≤ i < j ≤ n, 1 ≤ k ≤ n,
Then we can construct the following Lie ideals 9 With I = X.

A GENERALIZATION OF LAZARD'S ELIMINATION THEOREM.

•

J Tn = J R[n] is the Lie ideal of L k (T n ) generated by the infinitesimal pure braid relator r Tn = R[n]. • J T n+1 (resp. J TnT n+1 ) is the Lie ideal of L k (T n+1 ) generated by the relator r T n+1 (resp. r TnT n+1 = {ad Q z} Q∈J R[n] ,z∈T n+1 ). • J = J R[n+1] is the Lie ideal of L k (T n+1 ) generated by the infinitesimal pure braid relator R[n + 1]. ii) Of course, for instance, if we considered the relator R[4] ∪ {[t 1,2 , t 2,3 ] + [t 1,4 , t 3,4 ]},
then compatibility would no longer be fulfilled, for the added relator would neither belong to L k (X) B nor to L k (X) BZ , and thus J would not be J Z ⊔ J B any more.

Main Result: Quotients of Lazard's eliminations.

When we have such a type of relator, we can state the following theorem.

Theorem 2.6. (Main Result) Let k be a ring and X = B + Z be a set partitioned in two blocks. We suppose given a relator r = {r j } j∈J ⊂ L k (X) (cf. [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch II §2.310 ) which is compatible with the alphabet partition i.e. there exists a partition of the set of

indices J = J Z ⊔ J B such that • r B = {r j } j∈J B = r ∩ L k (X) B and r Z = {r j } j∈J Z = r ∩ L k (X) BZ .
With these data, we construct the ideals

• J B is the Lie ideal of L k (X) B generated by {r j } j∈J B
• J , J Z and J BZ are the Lie ideals of L k (X) generated respectively by r, r Z and

r BZ := {ad Q z} Q∈J B ,z∈Z = {[Q, z]} Q∈J B ,z∈Z .
With these notations, we get the following properties:

i) we have J Z BZ := J Z +J BZ ⊂ L k (X) BZ . Moreover, J Z BZ is a Lie ideal of L k (X) BZ (and even, by definition, of L k (X))
ii) the action of L k (X) B on Der(L k (X) BZ ) (by internal ad) passes to quotients as an action

α : L k (X) B → Der L k (X) BZ J Z BZ (2.26)
such that r B ⊂ Ker(α) and then, we get an action

[α] : L k (X) B J B → Der L k (X) BZ J Z BZ (2.27)
iii) we can construct an isomorphism (and its inverse) from presented Lie algebra L k (X) / J by the set r = {r j } j∈J of relators onto the semi-direct product of Lie algebras L k (X) BZ J Z BZ ⋊ L k (X) B J B which will be denoted by

β 25 : L k (X) / J ∼ = -→ L k (X) BZ J Z BZ ⋊ L k (X) B J B (2.28)
iv) in fact, one has a commutative diagram of Lie algebras with split short exact rows

0 L k (X) BZ L k (X) L k (X) B 0 0 L k (X) BZ J Z BZ L k (X) / J L k (X) B J B 0 j s J Z BZ p s J s J B [j] [p] (2.29)
where, for any ideal I, s I stands for the natural quotient map.

Proof. i) The formula (1.35) implies that L k (X) BZ = α∈N (X) |α| Z >0 L k (X) α is a Lie ideal of L k (X) which contains r Z = r ∩ L k (X) BZ . Hence, J Z (the Lie ideal of L k (X) generated by r Z ) is a subset of L k (X) BZ .
Similarly for the case

r BZ = {ad Q z} Q∈J B ,z∈Z ⊂ L k (X) BZ , we easily show that J BZ ⊂ L k (X) BZ . Therefore J Z BZ = J Z + J BZ ⊂ L k (X) BZ . Then by taking the intersection, it is seen that J Z BZ ∩ J B ⊂ L k (X) BZ ∩ L k (X) B = {0} (see (2.14)) and then J Z BZ ∩ J B = {0}.
The second assertion of (i) can be obtained from the fact that

J Z BZ is a Lie ideal of L k (X), thus J Z BZ = J Z BZ ∩L k (X) BZ is a Lie ideal of L k (X) BZ .
ii) Let us recall briefly the adjoint representation ad :

L k (X) B → Der(L k (X) BZ ) (2.30)
which is defined by ad

Q (P ) = [Q, P ] for any Q ∈ L k (X) B and P ∈ L k (X) BZ . It is well-known that ad is a Lie algebra morphism. Let Q ∈ L k (X) B then, due to 2.2. A GENERALIZATION OF LAZARD'S ELIMINATION THEOREM. the fact that J Z BZ = J Z + J BZ is ad Q -invariant and by Lemma 1.3, we have a commutative diagram L k (X) BZ L k (X) BZ L k (X) BZ J Z BZ L k (X) BZ J Z BZ . ad Q s J Z BZ s J Z BZ α(Q) (2.31)
This construction is sufficient in order to get a well-defined morphism of Lie

algebras α : L k (X) B → Der L k (X) BZ J Z BZ (2.32)
that is induced from the adjoint representation ad. Let us show that (b) There is a Lie algebra morphism

r B ⊂ Ker(α). ( 2
[α] : L k (X) B J B → Der( L k (X) BZ J Z BZ ) (2.34) which factorizes α as α = [α] • s J B .
Proof. We recall that the adjoint representation within L k (X) can be extended to

sequences of Lie polynomials (Q 1 , Q 2 , • • • , Q n ) ∈ Seq(L k (X)) by equation (1.40) ad (Q 1 ,Q 2 ,••• ,Qn) = ad Q 1 • ad Q 2 • • • • • ad Qn ∈ End(L k (X)).
Subsequences of letters will be noted as words as follows

(b 1 , • • • , b p u , Q, b p+1 , • • • , b p+q v ) = (u, Q, v). (2.35)
With these notations, we have the following

CHAPTER 2. LAZARD'S ELIMINATION Lemma 2.8. Let Q ∈ J B and u = b 1 • • • b k ∈ B * , then there exists two finite sequences (u 1 , • • • , u N ) ∈ (B * ) N and (Q 1 , • • • , Q N ) ∈ J N B such that ad (Q,u) = ad (Q,b 1 ,••• ,b k ) = N i=1 ad (u i ,Q i ) .
(2.36)

Proof. We prove the fact by induction on k.

If k = 0, we are tautologically done.

If k > 0, we write u = b 1 v with b 1 ∈ B, then ad (Q,u) = ad (Q,b 1 v) = ad Q • ad b 1 • ad (v) = [ad Q , ad b 1 ] • ad (v) + ad b 1 • ad Q • ad (v) = ad [Q,b 1 ] • ad (v) + ad b 1 • ad Q • ad (v) = ad [Q,b 1 ] • ad (v) + ad b 1 • ad (Q,v) . Now we observe that [Q, b 1 ] ∈ J B and |v| = |u| -1, then applying the induction hypothesis for ([Q, b 1 ], v) ∈ J B × B k-1 and (Q, v) ∈ J B × B k-1 , we get the result (of Lemma 2.8).
End of the proof of Lemma 2.7.

-

Let Q ∈ L k (X) B
, due to the fact that L k (X) BZ and J Z BZ are ad Q -invariant (they are ideals) 11 , we have a commutative diagram (where, for any ideal J , s J stands for the natural quotient map)

L k (X) BZ L k (X) BZ L k (X) BZ J Z BZ L k (X) BZ J Z BZ . ad Q s J Z BZ s J Z BZ α(Q) (2.37) Now, for Q ∈ J B and (u, z) ∈ B * × Z, let us show that ad Q (ad (u) (z)) ∈ J BZ .
From equation (2.36) of Lemma 2.8, we can write

ad Q (ad (u) z) = ad (Q,u) (z) = i∈F ad (u i ,Q i ) (z) = i∈F ad (u i ) • ad Q i (z) (2.38)
with Q i ∈ J B and u i ∈ B * , then clearly the sum belongs to J BZ by the definition of r BZ = {ad Q z} Q∈J B ,z∈Z . In other words, for all (u, z)

∈ B * × Z, ad (u) (z) ∈ ad -1 Q (J BZ ). But from the point 3 of Lemma 1.3 we know that ad -1 Q (J BZ ) is a Lie 2.2. A GENERALIZATION OF LAZARD'S ELIMINATION THEOREM. subalgebra of L k (X) BZ but we also know that {ad (u) (z)} (u,z)∈B * ×Z is a generating set of L k (X) BZ , it follows that ad -1 Q (J BZ ) = L k (X) BZ i.e. ad Q (L k (X) BZ ) = J BZ and then α(Q) • s J Z BZ = s J Z BZ • ad Q = 0. (2.39)
From the fact that s J Z BZ is onto, we see that

α(Q) = 0. As a conclusion (Q ∈ J B ) implies (α(Q) = 0) which is the claim.
As a consequence of the construction of the Lie algebra morphism [α] the kmodule

g X := L k (X) BZ J Z BZ ⊕ L k (X) B J B
is endowed with the structure of a Lie algebra given by the semi-direct product

g X := L k (X) BZ J Z BZ ⋊ L k (X) B J B .
Here, the Lie bracket is given by the following formula

[(P 0 , Q 0 ), (P 1 , Q 1 )] = ([P 0 , P 1 ] + [α](Q 0 )(P 1 ) -[α](Q 1 )(P 0 ), [Q 0 , Q 1 ]). (2.40)
With our construction, we thus have the split short exact sequence of Lie algebras

0 → L k (X) BZ J Z BZ → g X → L k (X) B J B → 0. (2.41) iii) In order to prove that L k (X) / J ∼ = L k (X) BZ J Z BZ ⋊ L k (X) B J B in k-Lie,
we follow the steps below: a. A mapping β 21 : X → g X is given by the formula

β 21 (x) :=    (0, [x]) if x ∈ B ([x], 0) if x ∈ Z, (2.42) here [x] = x + J B if x ∈ B and [x] = x + J Z BZ if x ∈ Z.
The universal property of the free Lie algebra L k (X) shows that there is a Lie algebra morphism

β 23 : L k (X) → g X such that the following triangle X L k (X) g X β 22 β 21 β 23 (2.43) CHAPTER 2. LAZARD'S ELIMINATION
is commutative, where β 22 is the embedding map. We claim that

β 23 (ad (u) (z)) = (ad (u) (z) + J Z BZ , 0), (2.44)
for any u ∈ B * and z ∈ Z. Indeed, this follows by induction on the length of word u,

• If |u| = 0: for any z ∈ Z, by equation (2.42) and diagram (2.43) we have (2.42) and diagram (2.43) we also have

β 23 (z) = β 21 (z) = (z + J Z BZ , 0). • If |u| = 1: for any u = b ∈ B and z ∈ Z,
β 23 (ad (u) (z)) = β 23 ([b, z]) = [β 23 (b), β 23 (z)] = [(0, [b]), ([z], 0)] = ([α]([b])([z]), 0) = ([b, z] + J Z BZ , 0) = (ad (u) (z) + J Z BZ , 0). • Assume that β 23 (ad (u) (z)) = (ad (u) (z) + J Z BZ , 0) for any u ∈ B * and |u| = k -1. For this assumption, for any u = b 1 b 2 • • • b k ∈ B * and z ∈ Z, by formulas (2.40),
β 23 (ad (u) (z)) = β 23 [b 1 , ad (b 2 •••b k ) (z)] = β 23 (b 1 ), β 23 (ad (b 2 •••b k ) (z)) = (0, [b 1 ]), ([ad (b 2 •••b k ) (z)], 0) (by ind. on k) = [α]([b 1 ])([ad (b 2 •••b k ) (z)]), 0 = [b 1 , ad (b 2 •••b k ) (z)] + J Z BZ , 0 = ad (b 1 •••b k ) (z) + J Z BZ , 0 = ad (u) (z) + J Z BZ , 0 . b. If j ∈ J = J Z ⊔ J B , then β 23 (r j
) is, according to the properties of (2.42) and

(2.43), equal to

β 23 (r j ) =    (0, [r j ]) = (0, 0) if j ∈ J B ([r j ], 0) = (0, 0) if j ∈ J Z .
(2.45) beacause, from (2.42), β 23 (r j ) arrives in the sectors of

g X := L k (X) BZ J Z BZ ⊕ L k (X) B J B
as precised in 2.45 and we know that, each time, the result is zero.

Thus, considering r = {r j } j∈J as a set, we have β 23 (r) = 0 and then β 23 (J ) = 0.

A GENERALIZATION OF LAZARD'S ELIMINATION THEOREM.

This shows that β 23 induces a morphism of Lie algebras

β 25 := [β 23 ] : L k (X) / J → g X . (2.46)
Given the natural surjection here named (2.47) c. In this part, using the equivariant property of Lie algebras of Section 2.1, we describe the inverse isomorphism (called here β 33 ) as follows

β 24 : L k (X) ↠ L k (X) / J , we can construct a commutative diagram, namely X g X L k (X) L k (X) / J .
β 33 : g X = L k (X) BZ J Z BZ ⋊ L k (X) B J B → L k (X) / J
In fact,

• There is the obvious embedding map j : L k (X) BZ → L k (X). We can easily verify that j(J Z BZ ) = J Z BZ ⊆ J , then j gives rise to a Lie morphism

g 1 := [j] : L k (X) BZ J Z BZ → L k (X) / J .
By (2.44) and (2.46), one obtains

β 25 • g 1 (ad (u) (z) + J Z BZ ) = [β 23 ](ad (u) (z) + J ) = β 23 (ad (u) (z)) = (ad (u) (z) + J Z BZ , 0)
, for any u ∈ B * and z ∈ Z. From the above calculation over generators {ad (u) (z)} (u,z)∈B * ×Z + J Z BZ , it follows that

β 25 • g 1 (P 0 ) = (P 0 , 0), ∀P 0 ∈ L k (X) BZ J Z BZ .
(2.48)

• Observe that there is the embedding map s :

L k (X) B → L k (X) which is a section of p : L k (X) → L k (X) B (as in 2.14
). Since J B ⊆ J and then s(J B ) ⊆ J , so there is a Lie algebra morphism

g 2 := [s] : L k (X) B J B → L k (X) / J .
By (2.42), (2.43) and (2.46), one observes that

β 25 • g 2 (b + J B ) = [β 23 ](b + J ) = β 21 (b) = (0, b + J B ) , CHAPTER 2. LAZARD'S ELIMINATION
for any b ∈ B. We calculated this over its generators, one derives the relation

β 25 • g 2 (Q 0 ) = (0, Q 0 ), ∀Q 0 ∈ L k (X) B J B .
(2.49)

• Moreover, we show directly that morphisms

g 1 : L k (X) BZ J Z BZ → L k (X) / J and g 2 : L k (X) B J B → L k (X) / J in
the category k-Lie satisfy the equivariant property (2.12) (or diagrammatically (2.13)) i.e. for any

Q 0 = Q + J B ∈ L k (X) B J B and P 0 = P + J Z BZ ∈ L k (X) BZ J Z BZ one has g 1 [α](Q 0 )(P 0 ) = [g 2 (Q 0 ), g 1 (P 0 )]. (2.50)
Indeed, we give the following our proof without difficulty

g 1 [α](Q 0 )(P 0 ) = g 1 [α](Q + J B )(P + J Z BZ ) = g 1 [Q, P ] + J Z BZ = [Q, P ] + J = [Q + J , P + J ] = [g 2 (Q 0 ), g 1 (P 0 )].
d. In terms of equivariant property (2.50) and by Proposition 2.2, they are sufficient to conclude that there is a unique morphism of Lie algebras

β 33 : g X = L k (X) BZ J Z BZ ⋊ L k (X) B J B → L k (X) / J (2.51)
which extends g 1 and g 2 in the usual sense i.e. β 33 (P 0 , 0) = g 1 (P 0 ) and

β 33 (0, Q 0 ) = g 2 (Q 0 ), for any P 0 ∈ L k (X) BZ J Z BZ and Q 0 ∈ L k (X) B J B . We obtain two Lie algebra morphisms β 33 • β 25 : L k (X) / J → L k (X) / J and β 25 • β 33 : g X → g X .
We now show that β 25 is the inverse of β 33

• By formula (2.42) and morphisms (2.43), (2.46) and (2.51), it give us to compute in detail the following behavior: for any x ∈ X, then

β 33 •β 25 ([x]) = β 33 • [β 23 ]([x]) = β 33 (β 21 (x)) =    β 33 ((0, [x])) = g 2 ([x]) = [x] if x ∈ B β 33 (([x], 0)) = g 1 ([x]) = [x] if x ∈ Z. Hence, β 33 • β 25 ([x]) = [x] = Id L k (X) / J ([x]
). As a consequence we clearly derive that β 33 • β 25 = Id L k (X) / J .

A GENERALIZATION OF LAZARD'S ELIMINATION THEOREM.

• Moreover, by formulas (2.48), (2.49) and the morphism (2.51), for any

(P 0 , Q 0 ) ∈ L k (X) BZ J Z BZ ⊕ L k (X) B J B , one has β 25 • β 33 (P 0 , Q 0 ) = β 25 • β 33 ((P 0 , 0) + (0, Q 0 )) = β 25 • (β 33 (P 0 , 0) + β 33 (0, Q 0 )) = β 25 (g 1 (P 0 ) + g 2 (Q 0 )) = β 25 (g 1 (P 0 )) + β 25 (g 2 (Q 0 )) = (P 0 , 0) + (0, Q 0 ) = (P 0 , Q 0 ).
The above calculation amounts to assert that

β 25 • β 33 = Id g X .
e. As a consequence, β 25 is an isomorphism of Lie algebras. Hence, in the category k-Lie, we constructed explicitly the isomorphism and its inverse

L k (X) / J L k (X) BZ J Z BZ ⋊ L k (X) B J B .
L k (X) BZ L k (X) L k (X) B 0 0 L k (X) BZ J Z BZ L k (X) / J L k (X) B J B 0. j s J Z BZ p s J s J B [j] [p]
(2.53)

QED

Notation.

-An alternative proof of the Theorem above 2.6, using B-gradings can be found in Commentary 5.

Applications.

Elimination of the subalphabet Z.

In certain cases (which is that of the Lie algebras DK k,n ), it can happen that the left factor of the semi-direct product (2.28) be isomorphic to L k (Z). We start from the CHAPTER 2. LAZARD'S ELIMINATION commutative diagram (2.29) with an additional arrow

L k (Z) 0 L k (X) BZ L k (X) L k (X) B 0 0 L k (X) BZ J Z BZ L k (X) / J L k (X) B J B 0 j Z j s J Z BZ p s J s J B [j] [p]
(2.54)

where j Z is the subalphabet embedding such that

Im(j Z ) = L k (X) Z = α∈N (X) |α| B =0 L k (X) α . (2.55)
We are now in the position to state the following Proposition 2.9. With the notations as in Theorem 2.6, let us consider the composite

map β = s J Z BZ • j Z , then a.
In order that β be injective, it is necessary and sufficient that

J Z BZ ∩ L k (X) Z = {0}.
b. In order that β be surjective, it is necessary and sufficient that, for all (b, z) ∈ B × Z, we had

s J Z BZ ([b, z]) ∈ s J Z BZ (L k (X) Z ). (2.56) Proof. a. Firstly, it is clear that the composite s J Z BZ •j Z is injective ⇐⇒ Ker(s J Z BZ • j Z ) = {0} ⇐⇒ J Z BZ ∩ L k (X) Z = Ker(s J Z BZ • j Z ) = {0}.
b. Secondly, we now have to prove that the composite map s J Z BZ • j Z is surjective if and only if for all (b, z) ∈ B × Z, we have

s J Z BZ ([b, z]) ∈ s J Z BZ (L k (X) Z ). (2.57)
Let us call β the composite map s J Z BZ • j Z . The proof goes as follows " ⇒ ": Assume that β is surjective. This assumption shows that for any (b, z) ∈ If n = 1 this is the hypothesis.

B × Z, there exits Q ∈ L k (Z) such that s J Z BZ ([b, z]) = β(Q) = s J Z BZ (j Z (Q)) and the fact that j Z (Q) ∈ L k (X) Z proves the claim. " ⇐ ": We first prove that, for all (b, Q) ∈ B × L k (Z) we have s J Z BZ ([b, j Z (Q)]) ∈ s J Z BZ (L k (X) Z ). ( 2
If n ≥ 2 (and Q homogeneous), then Q is a finite sum Q = i∈F [Q i1 , Q i2 ] with Q ij ∈ L k (Z) n ij and n = n i1 + n i2 . Now, we have [b, Q] = i∈F [[b, Q i1 ], Q i2 ] + [Q i1 , [b, Q i2 ]] (2.59)
and then the claim is a consequence of the induction hypothesis 12 and the fact

that s J Z BZ (L k (X) Z ) is a Lie subalgebra. Now we prove, by induction on |u| (u ∈ B * ) that, for all Q ∈ L k (Z) s J Z BZ ad (u) (j Z (Q)) ∈ s J Z BZ (L k (X) Z ). (2.60) If |u| = 0 this is trivial, otherwise u = bv for (b, v) ∈ B × B * .
From the induction hypothesis (understated)

s J Z BZ ad (v) (j Z (Q)) = s J Z BZ (R)
for some R ∈ L k (X) Z . But we have

s J Z BZ ad (u) (j Z (Q)) = s J Z BZ ad (bv) (j Z (Q)) = s J Z BZ [b, ad (v) (j Z (Q))] = s J Z BZ [b, ad (v) (j Z (Q)) -R] + s J Z BZ ([b, R]) = s J Z BZ ([b, R]) ∈ s J Z BZ (L k (X) Z ) by (2.58).
We just proved (2.60).

Thus, it permits us to verify that formula {s

J Z BZ (ad (u) (z)) | (u, z) ∈ B * × Z} ⊂ Im(s J Z BZ • j Z ) which is obviously the Lie subalgebra of L k (X) BZ J Z BZ . We remind that {s J Z BZ (ad (u) (z)) | (u, z) ∈ B * × Z} is a generating set of the Lie algebra L k (X) BZ J Z BZ , this yields Im(s J Z BZ • j Z ) = L k (X) BZ J Z BZ .
(2.61)

The above formula i.e. (2.61) gives a consequence that s J Z BZ • j Z is surjective. About the Free (partially commutative) Lie algebras, Pr. Schützenberger asked the following questions [START_REF] Schützenberger | [END_REF] 1. Is the free partially commutative Lie algebra torsion free (over Z)?

2. If yes (in which case it is linearly free over Z), is it possible to construct combinatorial bases of it?

3. To which extent can it be considered as "free"? (more than "as a module").

Question 3 has been answered in Section 1.3. The two remaining ones can be answered by the following adaptation of §2.2.3.

Theorem 2.10. Let (X, θ) be an alphabet with commutations and M (X, θ) be the free partially commutative monoid. We consider a partition of X, X = B + Z such that Z is totally non-commutative i.e. no two letters of Z commute between themselves (θ∩Z 2 = ∆ Z ). As defined in Section 1.3, s θ is the canonical surjection X * → M (X, θ).

We also consider j θ : M (X, θ) → X * , an arbitrary set-theoretical section of s θ . For t ∈ M (X, θ), we define the terminal alphabet of it

TAlph(t) = {x ∈ X | t ∈ M (X, θ).x} (2.62)
as the set of last letters of preimages of t w.r.t. s θ and the code

C B (Z) = {s θ (uz)|u ∈ B * , z ∈ Z, TAlph(s θ (uz)) = {z}} ⊂ M (X, θ) (2.63) Let C = j θ (C B (Z)) ⊂ B * Z and j C be the composite map L k (C) → L k (B * Z) rn -→ L k (X) BZ (
L k (C) 0 L k (X) BZ L k (X) L k (X) B 0 0 L k (X) BZ J Z BZ L k (X) / J L k (X) B J B 0. j C j s J Z BZ p s J s J B [j] [p]
(2.64)

A GENERALIZATION OF LAZARD'S ELIMINATION THEOREM.

Then, with the above hypotheses (Z totally non-commutative and C = j θ (C B (Z))), s J Z BZ • j C is an isomorphism. In particular, the left factor of the semi-direct product (2.28), here L k (X) BZ J Z BZ is a free Lie algebra.

Proof. For the proof that s J Z BZ • j C is one-to-one, we will need the three following lemmas.

Lemma 2.11. Let g be a Lie k-algebra, G a set of generators of g as a Lie k-algebra and R ⊂ g. Then, the ideal J (R) generated as an ideal by R is linearly (i.e. as a module) generated by the elements

{ad t (h)} t∈Seq(G) h∈R .
(2.65)

Proof of Lemma 2.11 : For M a k-submodule of g, the set

S(M ) = {g ∈ g | ad g (M ) ⊂ M } (2.66)
is a Lie subalgebra of g from the Jacobi relation. Then, let N be the submodule of g generated by the elements of (2.65). By the fact that {ad t (h)} t∈Seq(G) h∈R ⊂ J (R), one has N ⊂ J (R). Moreover, we have, by construction, G ⊂ S(N ) hence g ⊂ S(N ) which proves that N is an ideal containing R and then J (R). We obtain the claim. □ Consequence 2.12. Corresponding to r θ , and applying Lemma 2.11 to g = L k (B), G = B, we see that the ideal J B is generated (as a submodule) by the elements

{ad (u) ([b 1 , b 2 ])} u∈B * (b 1 ,b 2 )∈θ B (2.67)
then applying Lemma 2.11 to g = L k (X), G = X, we see that the ideal J Z is generated (as a submodule) by the elements

{ad (u) ([b, z])} u∈X * (b,z)∈θ∩(B×Z)
.

(2.68) Lemma 2.13. Let g be a Lie algebra and (p 1 , • • • , p n , q) ∈ Seq(g), then there exists

t = σ c(σ) σ ∈ Z[S n ] such that [ad (p 1 ,••• ,p n-1 ) (p n ), q] = σ∈Sn c(σ) ad (p σ(1) ,••• ,p σ(n) ) (q). (2.69) CHAPTER 2. LAZARD'S ELIMINATION
Proof of Lemma 2.13 : Let Y = {y 1 , • • • , y n , z} be an auxiliary alphabet of n + 1 letters and φ be the morphism L k (Y ) → g defined by φ(y i ) = p i and φ(z) = q. Then, using the partition

Y = {y 1 , • • • , y n } + {z} =: B + Z, we see that Q = [ad (y 1 ,••• ,y n-1 ) (y n ), z] ∈ L k (Y ) BZ . Then [ad (y 1 ,••• ,y n-1 ) (y n ), z] = u∈{y 1 ,••• ,yn} * c(u) ad (u) (z). (2.70) But Q is of multidegree (1, • • • , 1, 1) ∈ N (Y )
and then, each u in the support of the decomposition (2.70) can be written

u = y σ(1) • • • y σ(n) for some permutation σ ∈ S n . Hence [ad (y 1 ,••• ,y n-1 ) (y n ), z] = σ∈Sn c(σ) ad (y σ(1) ,••• ,y σ(n) ) (z). (2.71)
Now, with the notations of Lemma 2.13,

[ad (p 1 ,••• ,p n-1 ) (p n ), q] = φ([ad (y 1 ,••• ,y n-1 ) (y n ), z]) = σ∈Sn c(σ) φ(ad (y σ(1) ,••• ,y σ(n) ) (z)) = σ∈Sn c(σ) ad (p σ(1) ,••• ,p σ(n) ) (q). ( 2
(u) ([b 1 , b 2 ])} u∈B * (b 1 ,b 2 )∈θ B
. Then, here J BZ is generated by the elements

{[ad (u) ([b 1 , b 2 ]), z]} u∈B * (b 1 ,b 2 )∈θ B , z∈Z (2.73)
and then, from Lemma 2.13 and tracking the position of

[b 1 , b 2 ], J BZ is generated by elements {ad (u),[b 1 ,b 2 ],(v) (z)} u,v∈B * (b 1 ,b 2 )∈θ B , z∈Z (2.74)
(where, recalling that in the part "End of the proof of Lemma 2.7", for any Q ∈ J B and

(v, z) ∈ B * × Z, from equation (2.36) of Lemma 2.8, we have shown that ad (Q),(v) (z) ∈ J BZ . Thus, for Q = [b 1 , b 2 ], we can write {ad (u),[b 1 ,b 2 ],(v) (z)} u,v∈B * (b 1 ,b 2 )∈θ B , z∈Z ⊂ J BZ ).

A GENERALIZATION OF LAZARD'S ELIMINATION THEOREM.

The last lemma characterizes the kernel of a morphism which performs identifications and annihilation of letters Lemma 2.15. Let Y 1 , Y 2 be alphabets and φ 0 :

Y 1 → Y 2 ∪ {0} ⊂ L k (Y 2 ) with Y 2 ⊂ φ 0 (Y 1 ). Let φ 1 , be the morphism L k (Y 1 ) → L k (Y 2
) constructed from φ 0 by the mechanism (1.30). Then the kernel of φ 1 is the ideal generated by the elements

r eq = {x -x ′ } φ 0 (x)=φ 0 (x ′ )∈Y 2 ; r nil = {x} φ 0 (x)=0 .
Proof of Lemma 2.15 : Still with r eq and r nil defined as above, call R the relator 13i.e. R = r eq ∪ r nil . It is easily checked that R ⊂ Ker(φ 1 ) and then J (R) ⊂ Ker(φ 1 ) (J (R) being the ideal generaled by R). We then have a morphism

φ 3 : L k (Y 1 ) J (R) → L k (Y 2 ). (2.75)
It is surjective due to the condition Y 2 ⊂ φ 0 (Y 1 ). Now let us consider the following candidate to be an inverse of φ 3 . Remarking that, for y 2 ∈ Y 2 and y 1 a preimage of y 2 (φ 1 (y 1 ) = y 2 ), the class y 1 + J (R) is independent from the choice of y 1 . We define

φ 4 : L k (Y 2 ) → L k (Y 1 ) J (R)
L k (C) L k (X) BZ L k (X) BZ J Z BZ j C s J Z BZ (2.76)
and, as

α 31 := rn : L k (B * Z) → L k (X) (see (2.14)) is injective with image L k (X) BZ ,
we also note

α 31 = rn : L k (B * Z) → L k (X)
BZ the corresponding isomorphism and

β 13 : L k (X) BZ → L k (B * Z) its inverse.
We define φ : Let us prove that Ker(φ • β 13 ) = α 31 (Ker(φ)) ⊃ J Z + J BZ = J Z BZ . In fact, from (2.74), we know that J BZ is generated by elements

L k (B * Z) → L k (C)
{ad (u),[b 1 ,b 2 ],(v) (z)} u,v∈B * (b 1 ,b 2 )∈θ B , z∈Z . Set Q = ad (u),[b 1 ,b 2 ],(v) (z)
and remark that from the Jacobi identity one has

Q = ad (u),b 1 ,b 2 ,(v) (z) -ad (u),b 2 ,b 1 ,(v) (z) = α 31 (ub 1 b 2 vz) -α 31 (ub 2 b 1 vz) = α 31 (ub 1 b 2 vz -ub 2 b 1 vz).
(2.78)

But ub 1 b 2 vz -ub 2 b 1 vz ∈ Ker(φ) because b 1 and b 2 commute in M (X, θ)
, hence one has α 31 (Ker(φ)) ⊃ J BZ . Now, from (2.68), we know that J Z is generated by the elements

{ad (u) ([b, z])} u∈X * (b,z)∈θ∩(B×Z)
.

Set Q = ad (u) ([b, z]) and remark that Q = ad (u) ([b, z]) = ad (ub) (z) -ad (uz) (b) = α 31 (ubz -uzb) (2.79)
but, again, φ(ubz) = φ(uzb) because b and z commute in M (X, θ). Then we arrive at

α 31 (Ker(φ))) ⊃ J Z .
And now, we can construct the factorization to quotient of φ • β 13 as follows Let us show that s J Z BZ • j C and α are mutually inverse and, firstly remark that, for uz ∈ B * Z, j θ s θ (uz) is the unique representative of uz within C. Then, for uz ∈ C, we have β 13 j C (uz) = uz. This shows that, for uz ∈ C, we have

L k (C) L k (X) BZ J Z BZ L k (X) BZ L k (B * Z) L k (X) BZ L k (X) BZ J Z BZ . j C α β 13 s J Z BZ α 31 φ β 13 s J Z BZ
φβ 13 j C (uz) = φ(uz) = j θ s θ (uz) = uz and then φ • β 13 • j C = Id L k (C) . From this, we get α • s J Z BZ • j C = φ • β 13 • j C = Id L k (C) .
(2.80)

A GENERALIZATION OF LAZARD'S ELIMINATION THEOREM.

In particular α is onto and

s J Z BZ • j C is into. Let us prove (which is sufficient) that s J Z BZ • j C is onto. Our strategy is to show that Im(s J Z BZ • j C ) contains s J Z BZ α 31 (B * Z).
Let us show that, if uz ∈ B * Z and TAlph(s θ (uz)) ̸ = {z}, we have s J Z BZ α 31 (uz) = 0. Indeed, we claim that, for any uz

= b 1 • • • b k z ∈ B * Z such that (b 1 , z) ∈ θ and (b 1 , b j ) ∈ θ for all j ∈ [2, k], then α 31 (uz) ∈ J Z BZ . Indeed, if |u| = 1 then α 31 (uz) = [b 1 , z] ∈ J Z ⊆ J Z BZ since (b 1 , z) ∈ θ ∩(B ×Z). If |u| = 2 i.e. u = b 1 b 2 ∈ B * , (b 1 , z) ∈ θ and (b 1 , b 2 ) ∈ θ, then α 31 (uz) = [b 1 , [b 2 , z]] = [b 2 , [b 1 , z]] + [z, [b 2 , b 1 ]] = ad b 2 ([b 1 , z]) -ad [b 2 ,b 1 ] (z) ∈ J Z BZ since ad b 2 ([b 1 , z]) ∈ J Z and ad [b 2 ,b 1 ] (z) ∈ J BZ . If |u| = k ≥ 3 i.e. u = b 1 b 2 • • • b k ∈ B * ,
it is obtained from the induction hypothesis and by the formula

α 31 (uz) = ad (b 1 b 2 •••b k ) (z) = [b 1 , [b 2 , ad (b 3 •••b k ) (z)]] = ad b 2 ([b 1 , ad (b 3 •••b k ) (z)]) -ad [b 2 ,b 1 ] (ad (b 3 •••b k ) (z)) = ad b 2 (α 31 (b 1 b 3 • • • b k z)) -ad [b 2 ,b 1 ] (ad (b 3 •••b k ) (z)). Now, if uz = b 1 • • • b k z ∈ B * Z and TAlph(s θ (uz)) ̸ = {z}, by Remark 1.8 there exists i ∈ [1, k] so that (b i , z) ∈ θ and (b i , b j ) ∈ θ for all j ∈ [i + 1, k]. Thus, s J Z BZ α 31 (uz) = 0 because α 31 (uz) = ad (b 1 •••b i-1 ) (α 31 (b i • • • b k z)) ∈ J Z BZ (here α 31 (b i • • • b k z) ∈ J Z BZ by the claim above).
Otherwise, if TAlph(s θ (uz)) = {z}, we have j θ s θ (uz) ∈ C and α 31 (uz -j θ s θ (uz)) ∈ J BZ because J BZ contains all the commutations of θ B , then

s J Z BZ α 31 (uz) = s J Z BZ α 31 (uz -j θ s θ (uz) + j θ s θ (uz)) = s J Z BZ α 31 (j θ s θ (uz)) ∈ s J Z BZ α 31 (C) = s J Z BZ j C (C). (2.81) All in all s J Z BZ α 31 (B * Z) ⊂ s J Z BZ • j C (L k (C)
). Thus, s J Z BZ • j C and α are mutually inverse. We proved our theorem.

QED CHAPTER 2. LAZARD'S ELIMINATION

It would be interesting to have alternative proofs for answers to Schützenberger's questions about the Partially Commutative Free Lie algebra (cf. Duchamp and Krob [START_REF] Duchamp | Free partially commutative structures[END_REF] Thm III.3) as a consequence of our main theorem (Theorem 2.6).

Corollary 2.16. (Lazard's Partially Commutative Elimination) Let X be a set equipped with a commutation relation θ and B be a subset of X such that Z = X -B is totally non-commutative. Then there is an isomorphism from the free partially commutative Lie algebra L k (X, θ) to the semi-direct of product of Lie algebras, namely

L k (X, θ) ∼ = L k (C) ⋊ L k (B, θ B ) in k-Lie.
(2.82)

Proof. A partial result of our main theorem [Theorem 2.6 (iii)] deals the following decomposition

L k (X, θ) ∼ = k-Lie L k (X) BZ J Z BZ ⋊ L k (B, θ B ).
Therefore, in the category k-Lie, the formula

L k (C) ∼ = L k (X) BZ J Z
BZ is immediately seen from the fact that the composite s J Z BZ • j C is isomorphism of Lie algebras by our result in Theorem 2.10. We thus obtain the corollary.

Knizhnik-Zamolodchikov equation, Drinfeld-Kohno Lie algebra and its decomposition.

In this part, we give a sample of the application of our main theorem (Theorem 2.6) and by Proposition 2.9 to the decomposition of Drinfeld-Kohno Lie algebras. We remind the reader that the decomposition has deep relations with the study of special solutions of Knizhnik-Zamolodchikov equations (KZ n , for n = 3, 4) by polylogarithms and hyperlogarithms (cf. Drinfeld [START_REF] Drinfeld | Quantum group[END_REF][START_REF] Drinfeld | On quasitriangular quasi-hopf algebra and a group closely connected with Gal(Q/Q)[END_REF], Cartier [START_REF] Cartier | Fonctions polylogarithmes, nombres polyzetas et groupes pro-unipotents[END_REF], Brown [START_REF] Brown | Iterated integrals in quantum field theory[END_REF], Oi and Ueno [START_REF] Oi | KZ equation on the moduli space M 0,5 and harmonic product of multiple polylogarithms[END_REF]).

For n ≥ 2, assume that T n = {t i,j } 1≤i<j≤n is a set of n 2 endomorphisms t i,j of W, where W is a finite-dimensional vector space over C. We consider the Knizhnik-Zamolodchikov (KZ) equation (cf. Knizhnik and Zamolodchikov [START_REF] Knizhnik | Current algebra and Wess-Zumino model in two dimensions[END_REF], Drinfeld [START_REF] Drinfeld | Quantum group[END_REF][START_REF] Drinfeld | On quasitriangular quasi-hopf algebra and a group closely connected with Gal(Q/Q)[END_REF], Cartier [START_REF] Cartier | Développements récents sur les groupes de tresses. Applications à la topologie et à l'algèbre[END_REF],

Kassel [START_REF] Kassel | Quantum groups[END_REF])

(KZ n ) dF(z) = Ω n (z)F(z) (2.83)
defined over the complex configuration space

C n * = {z = (z 1 , • • • , z n ) ∈ C n |z i ̸ = z j for i ̸ = j},
where the system (so-called the KZ connection of 1-forms)

Ω n (z) = 1≤i<j≤n t i,j 2iπ d log(z i -z j ) (2.84)
and F = F(z) is a function defined on an open subset of C n * with values in the complex space End C (W ).

As a consequence of a classical integrability criterium (cf. Drinfeld [START_REF] Drinfeld | Quantum group[END_REF], Cartier [START_REF] Cartier | Développements récents sur les groupes de tresses. Applications à la topologie et à l'algèbre[END_REF], Kohno [START_REF] Kohno | Monodromy representations of braid groups and Yang-Baxter equations, Annales de l'institut Fourier[END_REF]), the system (2.84) is completely integrable if and only if dΩ n -Ω n ∧ Ω n = 0, and this imposes relations between the endomorphisms t i,j . These relations are precisely the infinitesimal pure braid relations. Considering now t i,j as abstract variables (or generators) and no longer endomorphisms, we repeat these relations here i.e. T n = {t i,j } 1≤i<j≤n satisfy the following infinitesimal pure braid relations

R[n] =          R 1 [n] [t i,j , t i,k + t j,k ] for 1 ≤ i < j < k ≤ n, R 2 [n] [t i,j + t i,k , t j,k ] for 1 ≤ i < j < k ≤ n, R 3 [n] [t i,j , t k,l ] for 1 ≤ i < j ≤ n, 1 ≤ k < l ≤ n,
F(z) = (z 3 -z 1 ) 1 2iπ (t 1,2 +t 1,3 +t 2,3 ) S( z 2 -z 1 z 3 -z 1 ) (2.86)
where S(z) satisfies the first order differential equation with three regular singular points at 0, 1 and

∞ d dz S(z) = 1 2iπ t 1,2 z + t 2,3 z -1 S(z).
By setting x 0 := t 1,2 2iπ and x 1 := -t 2,3 2iπ , we then transform X := {x 0 , x 1 } as an object in Set, this arises to consider the first order noncommutative differential equation (see Assume that k is a commutative ring with unit. We also transform the set of endomorphisms T n as an object in Set. The Drinfeld-Kohno Lie algebra14 DK k,n is then presented as

(3.38))    d(S) = (ω 0 (z)x 0 + ω 1 (z)x 1 )S, (N CDE) lim z∈Ω,z→0 S(z)e -x 0 log(z) = 1 H(Ω)⟨⟨X⟩⟩ , asymptotic initial condition,
DK k,n = ⟨ T n | R[n] ⟩ k-Lie = L k (T n ) J R[n] (2.87) where J R[n] is the Lie ideal of the free Lie algebra L k (T n ) generated by R[n] (2.6).
By using the Knizhnik-Zamolodchikov equations, Kohno proved in [START_REF] Kohno | Série de Poincaré-Koszul associée aux groupes de tresses pures[END_REF] that DK Z,n can be identified with gr Z (PB n ) the graded Lie algebra of the pure braid group PB n .

The Drinfeld-Kohno Lie algebra DK k,n ∼ = DK Z,n ⊗ Z k is also called the Lie algebra of infinitesimal braids.

Corollary 2.17. (Decomposition of Drinfeld-Kohno Lie algebra, cf. Etingof et al. [START_REF] Etingof | The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points[END_REF]) Given k a commutative ring with unit and n ≥ 0, there is an isomorphism of Lie algebras from Drinfeld-Kohno Lie algebra to the semi-direct product of Lie algebras

DK k,n+1 ∼ = L k (X n ) ⋊ DK k,n , (2.88)
where X n is any alphabet of cardinality n.

Proof. Recall in Example 2.1, we decompose T n+1 = T n + T n+1 and consider the in-

finitesimal pure braid relator r = R[n + 1] ⊂ L k (T n+1
). We then can easily check through direct calculation that r is a good relator satisfying all hypotheses in Theorem 2.6 and Proposition 2.9. This provides us with effective tools to derive the existence of the decomposition of Drinfeld-Kohno Lie algebra, namely, there is a Lie isomorphism

DK k,n+1 ∼ = L k (X n ) ⋊ DK k,n
, where we identified T n+1 with the alphabet X n .

Remark 2.6. The above method gives, as a consequence, that the Drinfeld-Kohno Lie algebra is an iterated semi-direct product of free Lie algebras, more precisely

DK k,n+1 ∼ = L k (X n ) ⋊ (L k (X n-1 ) ⋊ (• • • ⋊ L k (X 1 )) • • • )
in the category k-Lie (see Etingof et al. [START_REF] Etingof | The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points[END_REF] §3.10). We also remark that, in [START_REF] Chen | Gröbner-Shirshov bases for Lie algebras[END_REF] Cor 4.4, the authors used Gröbner-Shirshov bases for the Drinfeld-Kohno Lie algebra to also show that DK k,n+1 is an iterated semi-direct product of free Lie algebras (the reader can study an exposition 15 of the theory of Gröbner-Shirshov bases for associative algebras, Lie algebras, groups, semigroups, Ω-algebras, operads, etc. in the survey of L. A. Bokut and Y. Chen [START_REF] Bokut | Gröbner-Shirshov bases and their calculation[END_REF]).

We will return to this point with strange and generalized gradings in Subsection 2.3.3.

Lazard elimination as a free object.

In this section, we investigate more categorical frameworks for Lazard's elimination in k-Lie. In the first subsection, we introduce the category of Short Exact Sequences with Section (SESS) in k-Lie. In the last subsection, we will study the category of B-graded Lie algebras, where B = ({0, 1}, ∨) is the Boolean semigroup, which can be proved to be equivalent to the previous category in Proposition 2.18. Finally, we will perform Lazard's elimination as a free functor from the category of double sets to the category of B-graded Lie algebras.

Category of SESS in Lie algebras.

Assume that we have a SES of Lie algebras

0 -→ g l j -→ g p -→ g r -→ 0, (2.89)
(we also say that g is an extension of g r by g l In this vein, we have the following Remark 2.7. i) In general, any SES (2.89) is not necessarily split. For example, the Heisenberg Lie algebra (see Blasiak et al. [START_REF] Blasiak | Combinatorial algebra for second-quantized Quantum Theory[END_REF]), denoted by L H , is presented as

L H = ⟨ a † , a, e | [a, a † ] = e, [a † , e] = [a, e] = 0 ⟩ k-Lie . (2.90)
If we consider the one-dimensional Lie algebra

L 1 = ⟨ z | ∅ ⟩ k-Lie
and the two-dimensional abelian Lie algebra

L 2 = ⟨ x, y | [x, y] = 0 ⟩ k-Lie ,
the reader can verify that 0

-→ L 1 j -→ L H p -→ L 2 -→ 0 is a SES of Lie algebras,
but not split, where j : L 1 → L H , z → e is an injective Lie homomorphism and

p : L H → L 2 , a † → x, a → y, e → 0 is
-→ h -→ g = h ⊕ b -→ b -→ 0 (2.91)
but it is better than a simple extension because this SES is split and the Lie algebra g is a semi-direct product of b with h, denoted by g := h ⋊ b.

LAZARD ELIMINATION AS A FREE OBJECT.

As a consequence of the remark above, if SES (2.89) is split then g = h ⋊ b and g ≃ g r ⋊ g l in k-Lie.

Let us define the category of SESS in k-Lie, denoted by k-SSLie, as follows

• Objects: an object is a SESS of Lie algebras 0 g l g g r 0

j p σ (2.92)
• Morphisms: a morphism between two objects

0 g l 1 g 1 g r 1 0 j 1 p 1 σ 1 and 0 g l 2 g 2 g r 2 0 j 2 p 2 σ 2 is a commutative diagram in k-Lie with SESS arrows 0 g l 1 g 1 g r 1 0 0 g l 2 g 2 g r 2 0. j 1 φ l p 1 σ 1 φ φ r j 2 p 2 σ 2
(2.93)

An equivalence of categories and a Lazard elimination functor.

Given a set S, an object X in Set is said to be S-graded if it can be written as

X = s∈S X s a disjoint
M = s∈S M s subsets M s .M t ⊆ M s+t k-Lie Semigroup g = s∈S g s submodules [g s , g t ] ⊆ g s+t k-AA Semigroup A = s∈S A s submodules A s .A t ⊆ A s+t k-AAU Monoid A = s∈S A s submodules A s .A t ⊆ A s+t
= {1 M } then M = m∈N M m a N-graded
monoid is called succinctly "graded monoid" in Definition 1.1. We recall the Hilbert series of a finitely graded moinoid M is the formal power series

Hilb(M, t) = m≥0 | M m | t m ∈ N[[t]] (⊂ Q[[t]]).
More generally, this definition is still valid when internal structures of the components are subsets (i.e. for Set, Mon or Grp).

2. From now on and until the end of this remark k is assumed to be a field. Then, in case of k-AAU, an N-graded (or graded for short) associative algebra with unit

A = m∈N A m is said finitely graded if each k-module A m is finite dimensional.
The Hilbert series of a finitely graded associative algebra A is the formal power series

Hilb(A, t) := m≥0 dim k A m .t m ∈ Q[[t]]. (2.94)
More generally, this definition is still valid when internal structures of the components are free submodules (i.e. for k-Lie or k-AAU).

LAZARD ELIMINATION AS

Hilb(U(DK k,n+1 ), t) = n+1 i=2 1 1 -(i -1)t ∈ Q[[t]].
Remark 2.9. Bourbaki's book [START_REF] Bourbaki | Algebra[END_REF], Algebra Ch II §11.1 deals with graded k-algebras over a commutative monoid. However, for other structures (like Lie algebras), semigroups can replace monoids with the same crucial properties (homogeneous components, generators &c.). This will be sufficient for our purposes (indeed for the infinite Drinfeld-Kohno Lie algebra DK k,∞ , we will need the additive commutative semigroup (N ≥2 , ∨) × (N ≥1 , +), see in Subsection 2.3.3).

Here, we will use semigroups for Lie algebras as in the above table.

The category of B-graded Lie algebras.

For a Boolean semigroup B = ({0, 1}, ∨) 16 , a B-graded Lie algebra g can be described as follows: g = g 1 ⊕ g 0 is the direct sum of two submodules g 1 and g 0 such that

[g 0 , g 0 ] ⊆ g 0∨0 = g 0 , [g 0 , g 1 ] ⊆ g 0∨1 = g 1 , [g 1 , g 0 ] ⊆ g 1∨0 = g 1 , [g 1 , g 1 ] ⊆ g 1∨1 = g 1 . (2.95)
Thus, the B-graded Lie algebra g can be written as g = h ⊕ b, where h := g 1 being a

Lie ideal and b := g 0 a Lie subalgebra (hence we have the semi-direct product g = h⋊b).

CHAPTER 2. LAZARD'S ELIMINATION

We now define the category of B-graded Lie algebras over k, denoted by B-GrLie, as follows

• Objects: an object is a Lie k-algebra g = h ⋊ b, h is a Lie ideal and b a Lie subalgebra of g.

• Morphisms: a morphism of two objects

g 1 = h 1 ⋊ b 1 and g 2 = h 2 ⋊ b 2 is a Lie algebra homomorphism φ : g 1 → g 2 such that φ(h 1 ) ⊆ h 2 and φ(b 1 ) ⊆ b 2 .
We then can define a functor F : k-SSLie → B-GrLie from the category of SESS in k-Lie to the category of B-graded Lie algebras by the following

• F ((2.92)) def = (g = h ⋊ b)
, where h = Ker(p) the Lie ideal and b = Im(σ) the Lie subalgebra of g, that is an object in B-GrLie;

• F ((2.93)) def = (φ : g 1 = h 1 ⋊ b 1 → g 2 = h 2 ⋊ b 2 ), here F ((2.93)) is a morphism in B-GrLie because φ(h 1 ) = φ(Ker(p 1 )) = φ(Im(j 1 )) = φ • j 1 (g l 1 ) = j 2 • φ l (g l 1 ) ⊆ j 2 (g l 2 ) = Im(j 2 ) = Ker(p 2 ) = h 2 and φ(b 1 ) = φ(Im(σ 1 )) = φ • σ 1 (g r 1 ) = σ 2 • φ r (g r 1 ) ⊆ σ 2 (g r 2 ) = Im(σ 2 ) = b 2 .
Moreover, one has the following Proof. We first prove that F is essentially surjective. Indeed, for every object g = h ⋊ b ∈ B-GrLie, we take the following natural SESS in k-Lie

0 h g = h ⊕ b b 0 j p σ (2.96)
which is an object in k-SSLie. We then easily see that F ((2.96)) = g in B-GrLie

which corresponds to Id g : g → g because F ((2.96)) = (g = Ker(p) ⋊ Im(σ) = h ⋊ b).
We verified that F is an essentially surjective functor. Further, for each pair of objects

0 g l 1 g 1 g r 1 0 j 1 p 1 σ 1 (2.
F : Hom k-SSLie (x 1 , x 2 ) → Hom B-GrLie (F (x 1 ), F (x 2 )) (2.93) → φ : g 1 = h 1 ⋊ b 1 → g 2 = h 2 ⋊ b 2
is a bijection in Set by the following properties

• Surjectivity: in fact, for any morphism φ :

g 1 = h 1 ⋊ b 1 → g 2 = h 2 ⋊ b 2 in Hom B-GrLie (F (x 1 ), F (x 2 )), we construct two Lie homomorphisms φ l := r 2 • φ • j 1 : g l 1 → g l 2 and φ r := p 2 •φ•σ 1 : g r 1 → g r 2 ,
where r 2 : g 2 → g l 2 is a retract of j 2 i.e. r 2 is a Lie algebra homomorphism such that r 2 • j 2 = Id g l 2 . One notes that since φ(h 1 ) ⊆ h 2 by assumption, we get that φ

• j 1 (g l 1 ) = φ(h 1 ) ⊆ h 2 = j 2 (g l 2 )
and then for any

a l 1 ∈ g l 1 there exists a l 2 ∈ g l 2 such that φ•j 1 (a l 1 ) = j 2 (a l 2 ) (a l 2 is unique since j 2 is injective). We can now obtain that φ l (a l 1 ) = r 2 • φ • j 1 (a l 1 ) = r 2 • j 2 (a l
2 ) = a l 2 , and then j 2 •φ l (a l 1 ) = j 2 (a l 2 ) = φ•j 1 (a l 1 ). Thus, j 2 •φ l = φ•j 1 . Moreover, we also remark that φ(b 1 ) ⊆ b 2 by assumption, then φ

• σ 1 (g r 1 ) = φ(b 1 ) ⊆ b 2 = σ 2 (g r 2 )
, thus for any a r 1 ∈ g r 1 there exists a r 2 ∈ g r

2 such that φ•σ 1 (a r 1 ) = σ 2 (a r 2 ) (a r 2 is unique since σ 2 is injective). It is sufficient to obtain that φ r (a r 1 ) = p 2 • φ • σ 1 (a r 2 ) = p 2 • σ 2 (a r 2 ) = a r 2 , we thus have σ 2 • φ r (a r 1 ) = σ 2 (a r 2 ) = φ • σ 1 (a r 1 )
. We arrive at σ 2 • φ r = φ • σ 1 . As a consequence, we can obtain the following commutative diagram in k-Lie with SESS arrows

0 g l 1 g 1 g r 1 0 0 g l 2 g 2 g r 2 0. j 1 φ l p 1 σ 1 φ φ r j 2 p 2 σ 2 (2.99)
We therefore conclude that the function F is a surjection because F ((2.99)) = (φ :

g 1 = h 1 ⋊b 1 → g 2 = h 2 ⋊b 2 )
, where (2.99) is a morphism in Hom k-SSLie (x 1 , x 2 ).

• Injectivity: we start with a morphism φ :

g 1 = h 1 ⋊ b 1 → g 2 = h 2 ⋊ b 2 in Hom B-GrLie (F (x 1 ), F (x 2 )
) as above, we assume that there is an another mor-

CHAPTER 2. LAZARD'S ELIMINATION phism 0 g l 1 g 1 g r 1 0 0 g l 2 g 2 g r 2 0 j 1 ψ l p 1 σ 1 φ ψ r j 2 p 2 σ 2 (2.100)
in Hom k-SSLie (x 1 , x 2 ) such that F ((2.100)) = (φ :

g 1 = h 1 ⋊ b 1 → g 2 = h 2 ⋊ b 2 ).
By (2.99) and (2.100), observe that for any

a l 1 ∈ g l 1 then j 2 • ψ l (a l 1 ) = φ • j 1 (a l 1 ) = j 2 • φ l (a l 1 )
, one has ψ l (a l 1 ) = φ l (a l 1 ) because j 2 is injective. We obtain that ψ l = φ l . Similarly, it is not hard to show that ψ r = φ r . Consequently two morphisms (2.99) and (2.100) are equal in Hom k-SSLie (x 1 , x 2 ). We proved the injectivity of the function F . Thus, F is a fully faithful functor. We proved our proposition.

It follows from the above proposition that F is an equivalence of these categories.

Thus, two categories k-SSLie and B-GrLie are equivalent 17 . In a more explicit way, we can construct an inverse functor G : B-GrLie → k-SSLie as follows

• For an object g = h ⋊ b ∈ B-GrLie, by (2.96), we set

G(g = h ⋊ b) def = (0 h g = h ⊕ b b 0) j p σ
which is obviously an object in k-SSLie;

• For a morphism φ :

g 1 = h 1 ⋊ b 1 → g 2 = h 2 ⋊ b 2 in B-GrLie, we set G(φ) def = 0 h 1 g 1 b 1 0 0 h 2 g 2 b 2 0 j 1 φ |h 1 p 1 σ 1 φ φ |b 1 j 2 p 2 σ 2
which is a commutative diagram in k-Lie with SESS arrows i.e. a morphism in k-SSLie.

C → G • F, β : 1 D → F • G.
Moreover, Proposition 2.18 is a criterion for testing whether any functor F : C → D is a part of an equivalence of categories in this sense, see [START_REF] Kashiwara | Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften[END_REF] Prop 1.3.13.

Commentary 5. i) Let us provide an alternative proof of Theorem 2.6 using the tools above.

1. The fact that the alphabet X is bisected as X = B + Z induces a B-grading in L k (X) with B → 0, Z → 1 (in fact, by Lemma 5.3, a regrading of the fine grading of L k (X)).

2. Likewise the partitioning r = r B ⊔ r Z means that the relators are homogeneous w.r.t. this B-grading and, hence the Lie ideal J generated by r is itself B-graded, let us denote J = J 1 ⊕ J 0 be its homogeneous decomposition.

3. The only thing we have to prove is that J 0 = J B and J 1 = J Z + J BZ which can be done as follows (a) The inclusion J B + J Z + J BZ ⊂ J is straightforward considering the definitions of the summands.

(b) We observe that (c) Then, as r ⊂ J B + J Z ⊂ J B + J Z + J BZ , we get the reverse inclusion

J B + J Z + J BZ is a Lie ideal of L k (X).
J B + J Z + J BZ ⊃ J . (d) Finally we have J 0 = J ∩L k (X) B = J B and J 1 = J ∩L k (X) BZ = J Z +J BZ
by counting the degrees.

Then, the conclusions of Theorem 2.6 can be revisited as follows i)

J Z BZ (resp. J B ) is the 1-component (resp. 0-component) of J i.e. J 1 = J Z + J BZ = J Z BZ (resp. J 0 = J B ).
ii-iv) The remainder of the theorem is a consequence of the following general fact which says that a Lie ideal of g which is B-graded as a submodule is a Bgraded Lie ideal (i.e. the kernel of a B-morphism, see Proposition 5.2 in Appendix 5.4.1).

Lemma 2.19. Let g be a B-graded Lie algebra and h be a Lie ideal of g which is {0, 1}-graded as a submodule. Then

CHAPTER 2. LAZARD'S ELIMINATION

i. h 0 is a Lie ideal of g 0 , h 1 is a Lie ideal of g (and then of g 1 ).

ii. We have the following commutative diagram

0 g 1 g g 0 0 0 g 1 h 1 g h g 0 h 0 0. j 1 s 1 p j 0 s s 0 [j 1 ] [p] [j 0 ]
Proof. This is a particular case of Proposition 5.2 with S = B.

ii) For implementation purposes, one can remark that the Lie ideal J B is the set of Lie polynomials on {ad (u) (r j )} u∈B * ,j∈J B .

A free functor on the category of double sets.

Our next aim is to investigate that Lazard's elimination defines a free functor from the category of double sets to the category of B-graded Lie algebras with respect to the forgetful functor F : B-GrLie → Set 2 defined in the following way

• Given an object g = h ⋊ b ∈ B-GrLie, then F (g = h ⋊ b) def = (h, b)
, where h and b are only underlying sets (forgetting their Lie structures), so is an object in Set 2 (the product of the category Set with itself);

• Given a B-GrLie-morphism g 1 = h 1 ⋊ b 1 φ → g 2 = h 2 ⋊ b 2 , we define F (φ) as F (φ) def = (h 1 , b 1 ) F (φ) -→ (h 2 , b 2 )
the underlying (double) set-theoretical map corresponding to

F (φ)((h 1 , b 1 )) = (φ(h 1 ), φ(b 1 )).
We first start with a pair of sets (Z, B) ∈ Set 2 , and then under the Lie homomorphism

α : L k (B) → Der(L k (B * Z)) (appearing in the proof of Theorem 2.
3) we can construct the below classical elimination which is a SESS of Lie algebras

0 L k (B * Z) L k (B * Z) ⋊ L k (B) L k (B) 0. j B|Z p B|Z s B|Z (2.101)
In this situation, we now set g(Z, B)

:= L k (B * Z) ⋊ L k (B) = g 1 (Z, B) ⊕ g 0 (Z, B)
with the grading g 1 (Z, B) := j B|Z (L k (B * Z)) the Lie ideal of the Lie algebra g(Z, B)

Z → L k (B * Z) → j B|Z (L k (B * Z)) and B → L k (B) → s B|Z (L k (B)), respectively.
Amazingly, we now remark that the pair (j (Z,B) , g(Z, B)) satisfies the following universal problem: for any object (Z, B) in Set 2 and L = L 1 ⋊ L 0 in B-GrLie, for each

morphism f = (f Z , f B ) : (Z, B) → F (L) = (L 1 , L 0 ) in Set 2 (this simply means that f Z : Z → L 1 and f B : B → L 0 are morphisms in Set), there exists a unique f ∈ Hom B-GrLie (g(Z, B), L) such that f = F ( f ) • j (Z,B) , it means that the below diagram commutes Set 2 B-GrLie (Z, B) L = L 1 ⋊ L 0 g(Z, B) = g 1 (Z, B) ⋊ g 0 (Z, B). F f =(f Z ,f B ) j (Z,B) f (2.102)
In fact, if we take

uz = b 1 • • • b k z ∈ B * Z then [f B (b 1 ), • • • , [f B (b k ), f Z (z)] • • • ] ∈ L 1 since
L 1 is a Lie ideal of L. We thus construct a Lie morphism L k (B * Z)

f 1 -→ L 1 that is the unique extension of a map f BZ : B * Z → L 1 , b 1 • • • b k z → [f B (b 1 ), • • • , [f B (b k ), f Z (z)] • • • ]
L k (B) ⊗ L k (B * Z) L ⊗ L L k (B * Z) L. f 0 ⊗ f 1 α ⊗ ad L ⊗ f 1 (2.103)
The equivariant diagram (2.103) say that there is a unique Lie morphism f :

g(Z, B) = L k (B * Z) ⋊ L k (B) → L extending f 0 and f 1 in the usual sense i.e. f • s B|Z (L k (B)) = CHAPTER 2. LAZARD'S ELIMINATION f 0 (L k (B)) and f • j B|Z (L k (B * Z)) = f 1 (L k (B * Z)).
In particular, we observe that

f (g 1 (Z, B)) = f • j B|Z (L k (B * Z)) = f 1 (L k (B * Z)) ⊆ L 1 and f (g 0 (Z, B)) = f • s B|Z (L k (B)) = f 0 (L k (B)) ⊆ L 0 , proving that f : g(Z, B) = g 1 (Z, B) ⋊ g 0 (Z, B) → L = L 1 ⋊ L 0 is a morphism in B-GrLie.
Moreover, the following lemma is sufficient

for our investigation Lemma 2.20. One has f ∈ Hom B-GrLie (g(Z, B), L) is the unique morphism in B-GrLie such that f = F ( f ) • j (Z,B) .
Proof. It is immediate to verify by calculation that for any (z, b) ∈ (Z, B) then ,B) . On the other hand, if there is a morphism g : g(Z, B) → L in B-GrLie which is an another solution of the diagram (2.102), we then see that for each b ∈ B, one notes that g

F ( f ) • j (Z,B) [(z, b)] = F ( f )(j Z (z), j B (b)) = f (j Z (z)), f (j B (b)) = ( f 1 (z), f 0 (b)) = f BZ (z), f B (b) = (f Z (z), f B (b)) = f ((z, b)), thus clearly f = F ( f )•j (Z
• s B|Z (b) = g(j B (b)) = f B (b) = f 0 (b), thus g • s B|Z (L k (B)) = f 0 (L k (B)). Moreover, for each uz = b 1 • • • b k z ∈ B * Z, then g • j B|Z (z) = g(j Z (z)) = f Z (z) = f BZ (z) = f 1 (z) and hence g • j B|Z (uz) = g((uz, 0)) = g([(0, b 1 ), • • • , [(0, b k ), (z, 0)] • • • ]) = [g • s B|Z (b 1 ), • • • , [g • s B|Z (b k ), g • j B|Z (z)] • • • ] = [f B (b 1 ), • • • , [f B (b k ), f Z (z)] • • • ] = f BZ (uz) = f 1 (uz), we arrive at g • j B|Z (L k (B * Z)) = f 1 (L k (B * Z)).
As a consequence, the morphism g : g(Z, B) → L extends f 0 and f 1 in the usual sense. Thus, g = f by the uniqueness of the equivariant extension. We verified our lemma.

As a result, the solution of the universal problem (2.102) provides a free functor L : Set 2 → B-GrLie, (Z, B) → g(Z, B), so-called the Lazard elimination functor, which is left-adjoint to the forgetful functor F : B-GrLie → Set 2 .

Remark 2.10. To prove directly that these functors L and F determine an adjunction

Set 2 B-GrLie, L F ⊣ we can construct a pair (1 Set 2 η ⇒ F • L, L • F ε ⇒ 1 B-GrLie )
of natural transformations (called the unit and counit of the adjunction) satisfying the triangle identities

L Lη / / 1 L # # L • F • L εL F ηF / / 1 F # # F • L • F F ε L F 2.
] := {[t i,j ] = t i,j + J R[n+1] | i ∈ [1, j -1]},
that is in fact a free Lie algebra over k because DK

(j) k,n+1 ∼ = L k (T j ) ∼ = L k (X j-1 ) as Lie
algebras, where X j-1 is a set of j - It is not hard to show that for all i, j ∈ [2, n + 1], one has

1 elements {x 1 , • • • , x j-1 }. T n+1 = T 2 T 3 T 4 . . . . . . T n+1 t 1,
[DK (i) k,n+1 , DK (j) k,n+1 ] ⊆ DK (i∨j) k,n+1
and moreover we get that the Drinfeld-Kohno Lie algebra DK k,n+1 is an iterated semidirect product of free Lie algebras (see Remark 2.6)

DK k,n+1 = DK (n+1) k,n+1 ⋊ DK (n) k,n+1 ⋊ (• • • ⋊ DK (2) k,n+1 ) • • • CHAPTER 2. LAZARD'S ELIMINATION ∼ = L k (X n ) ⋊ (L k (X n-1 ) ⋊ (• • • ⋊ L k (X 1 )) • • • ). (2.104) In particular, DK k,n+1 = 2≤j≤n+1 DK (j) k,n+1 is a ([2, n + 1], ∨)-graded Lie algebra.
Strange gradings allow not only to manage semi-direct products but, more complex elimination schemes like iterated decompositions. Indeed, suppose we had an elimination scheme (0.1)

ST RU CT ⟨x 1 , x 2 , . . . , x n ⟩ ∼ = N ICE⟨x 1 , x 2 , . . . , x n ⟩ ⋄ ST RU CT 1 ⟨x 1 , . . . , x n-1 ⟩
where N ICE et ST RU CT 1 stand for algebraic structures generated (sometimes freely) by generators x i . Iterating it, we get

ST RU CT ⟨x 1 , x 2 , . . . , x n ⟩ ∼ = N ICE⟨x 1 , x 2 , . . . , x n ⟩ ⋄ (N ICE⟨x 1 , x 2 , . . . , x n-1 ⟩ ⋄ (• • • ⋄ N ICE⟨x 1 ⟩) • • • ).
In the next part, we can even manage infinite decompositions with (N ≥2 , ∨) or nonlinear eliminations with other semigroups.

Remark 2.11. Iterated semi-direct decompositions (i.e. formulas like Equation (2.104))

provide a natural grading by the semigroup (I, ∨) where

I = {i 1 < i 2 < • • • < i n } for arbitrary Lie algebras such that g = g in ⋊ g i n-1 ⋊ (• • • ⋊ g i 1 ) • • • . (2.105)
The direct limit of Drinfeld-Kohno Lie algebras.

If we consider the direct system that is a chain of embeddings

DK k,2 f 2 -→ DK k,3 -→ • • • -→ DK k,n fn -→ DK k,n+1 f n+1 -→ • • • (2.106)
where the structure Lie monomorphisms f n : DK k,n → DK k,n+1 are defined in the following way Proof. We construct the structure Lie homomorphisms ϕ n : DK k,n → DK k,∞ that are determined by setting

f n (P + J R[n] ) = P + J R[n+1] , where P ∈ L k (T n ) ⊂ L k (T n+1
R[∞] =          R 1 [∞] [t i,j , t i,k + t j,k ] for 1 ≤ i < j < k < +∞, R 2 [∞] [t i,j + t i,k , t j,k ] for 1 ≤ i < j < k < +∞, R 3 [∞] [t i,j , t k,l ] for 1 ≤ i < j < +∞, 1 ≤ k < l < +∞,
ϕ n (P + J R[n] ) = P + J R[∞] , where P ∈ L k (T n ) ⊂ L k (T ∞ ),
ϕ n = ϕ m • f nm whenever n ≤ m, where f nn = Id DK k,n and f nm = f m-1 • • • • • f n .
Suppose we are given a target (g, ψ n ) with ψ n : DK k,n → g in k-Lie, then we construct a Lie homomorphism u 0 : L k (T ∞ ) → g that is a unique extension of the mapping from

T ∞ to g corresponding to t i,j → ψ j ([t i,j ]) = ψ j (t i,j + J R[j] ) (where 1 ≤ i < j < +∞) by
universal property of Diagram (1.30). Further, we now show that for any polynomial

P of type R[∞]
we have u 0 (P ) = 0 because we have the following properties

• for each 1 ≤ i < j < k < +∞ then u 0 ([t i,j , t i,k + t j,k ]) = [u 0 (t i,j ), u 0 (t i,k ) + u 0 (t j,k )] = ψ j ([t i,j ]), ψ k ([t i,k ])+ψ k ([t j,k ]) = ψ k •f jk ([t i,j ]), ψ k ([t i,k ])+ψ k ([t j,k ]) = ψ k ([t i,j ]), ψ k ([t i,k ])+ψ k ([t j,k ]) = ψ k [t i,j , t i,k +t j,k ]+J R[k] = ψ k (0) = 0, and then similarly one has u 0 ([t i,j + t i,k , +t j,k ]) = ψ k [t i,j + t i,k , t j,k ] + J R[k] = ψ k (0) = 0;
• for each 1 ≤ i < j < +∞, 1 ≤ k < l < +∞ such that |{i, j, k, l}| = 4, assume that l = max{i, j, k, l} then we can obtain u 0 ([t i,j , t k,l ]) = [u 0 (t i,j ), u 0 (t k,l )] =

ψ j ([t i,j ]), ψ l ([t k,l ]) = ψ l •f jl ([t i,j ]), ψ l ([t k,l ]) = ψ l ([t i,j ]), ψ l ([t k,l ]) = ψ l [t i,j , t k,l ]+ J R[l] = ψ l (0) = 0.
We verified that u 0 (P ) = 0, thus clearly J R[∞] is in the kernel of u 0 . We arrive at a conclusion that u 0 induces a Lie homomorphism u : DK k,∞ → g, and moreover u • ϕ n = ψ n for each n ≥ 2 because they equal on its generators: for each 1

≤ i < j ≤ n then u • ϕ n (t i,j + J R[n] ) = u(t i,j + J R[∞] ) = ψ j (t i,j + J R[j] ) = ψ n • f jn (t i,j + J R[j] ) = ψ n (t i,j + J R[n]
). As a result, we can take a commutative diagram of Lie algebras Under the universal enveloping functor

DK k,n DK k,m DK k,∞ g.
U : k-Lie → k-AAU, g -→ U(g) (2.108)
which is a left adjoint to the Liezation functor F : k-AAU → k-Lie, we have the following Corollary 2.22. Consider the direct system that is a chain of embeddings

U(DK k,2 ) U (f 2 ) -→ U(DK k,3 ) -→ • • • -→ U(DK k,n ) U (fn) -→ U(DK k,n+1 ) U (f n+1 ) -→ • • • ,(2.109)
then the universal enveloping algebra of infinite Drinfeld-Kohno Lie algebra is indeed the direct limit of such structure morphisms (2.109) in k-AAU

U(DK k,∞ ) = lim -→ U(DK k,n ).
Proof. As a left-adjoint, the universal enveloping functor preserves colimits, so in particular it sends direct limits in k-Lie to direct limits in k-AAU. We therefore obtain that

U(DK k,∞ ) = U(lim -→ DK k,n ) = lim -→ U(DK k,n ).
(m) k,∞ the Lie subalgebra of DK k,∞ generated by [T m ] := {[t i,m ] = t i,m + J R[∞] | i ∈ [1, m -1]}, then one verifies without difficulty that DK (m) k,∞ ∼ = L k (T m ) ∼ = L k (X m-1
) is indeed a free Lie algebra over k.

T ∞ = T 2 T 3 T 4 . . . . . . T m . . . t 1,
(m) k,∞ , DK (n) k,∞ ] ⊆ DK (m∨n) k,∞ (for all m, n ∈ N ≥2 )
and moreover we have an infinite iterated semi-direct product of free Lie algebras

DK k,∞ = • • • ⋊ DK (m) k,∞ ⋊ (DK (m-1) k,∞ ⋊ (• • • ⋊ DK (2) k,∞ ) • • • ) . (2.110)
As a consequence, we write the infinite Drinfeld-Kohno algebra as the direct sum

DK k,∞ = m≥2 DK (m) k,∞
which is a (N ≥2 , ∨)-graded Lie algebra over k. Furthermore, due to the injections DK k,n → DK k,∞ , the set T ∞ = {t i,j } 1≤i<j can be considered as a subset of DK k,∞ and CHAPTER 2. LAZARD'S ELIMINATION due to the semi-direct decompositions DK k,∞ is a free module (see the formula (2.110))

and embeds within its enveloping algebra U(DK k,∞ ). We have the following Proposition 2.23. Under the natural projection s ∞ : k⟨T ∞ ⟩ → U(DK k,∞ ), the image of the set of words of the form

t i 1 ,j 1 t i 2 ,j 2 . . . t in,jn ∈ T * ∞ (where n ≥ 0 and 2 ≤ j 1 ≤ j 2 ≤ . . . ≤ j n ) is a k-linear basis of the algebra U(DK k,∞ ).
Proof. See Proposition 5.7 in Appendix 5.4.2.

S-graded Lie algebras and properties of them and their enveloping algebras with respect to Hilbert series.

Definition 2.3. Assume that (S, +) is a commutative semigroup satisfying "Condition (D)" 18 . Let A = s∈S A s be a S-graded algebra in finite dimensions, the Hilbert series of A is given by

Hilb(A) = s∈S dim k (A s ).s ∈ Q[[S]],
where the total semigroup algebra Q ] is a Q-module of all infinite sum s∈S α s s and the convolution product Proof. As a S-graded algebra A ⊕ B = s∈S A s ⊕ B s = s∈S (A ⊕ B) s , we get that

s 1 ∈S α s 1 s 1 s 2 ∈S β s 2 s 2 = s∈S    s 1 ,s 2 ∈S s 1 +s 2 =s α s 1 β s 2    s.
Hilb(A ⊕ B) = s∈S dim k (A s ⊕ B s ).s = s∈S dim k (A s ).s + s∈S dim k (B s ).s = Hilb(A) + Hilb(B).
Moreover, by using "Condition (D)", we arrive at a finitely S-graded structure

A ⊗ B = s 1 ∈S A s 1 ⊗ s 2 ∈S B s 2 = s∈S    s 1 ,s 2 ∈S s 1 +s 2 =s A s 1 ⊗ B s 2    = s∈S (A ⊗ B) s ,
then the Hilbert series is

Hilb(A ⊗ B) = s∈S dim k    s 1 ,s 2 ∈S s 1 +s 2 =s A s 1 ⊗ B s 2    .s = s∈S    s 1 ,s 2 ∈S s 1 +s 2 =s dim k (A s 1 ) dim k (B s 2 )    .s = s 1 ,s 2 ∈S dim k (A s 1 ) dim k (B s 2 ).(s 1 + s 2 ) = s 1 ∈S dim k (A s 1 ).s 1 s 2 ∈S dim k (B s 2 ).s 2 = Hilb(A).Hilb(B).
Furthermore, for the grading of enveloping algebras of S-graded Lie algebras see Appendix 5.4.2.

Remark 2.12. Assume that (S, +) is locally finite 19 commutative semigroup and g = s∈S g s is a S-graded Lie algebra. If g is a finitely S-graded i.e. each k-module g s is free of finite rank, so is the S ⊔ {0}-graded enveloping algebra U(g) (5.23).

CHAPTER 2. LAZARD'S ELIMINATION

Now we will deal with a more general graded type of the infinite Drinfeld-Kohno Lie algebra and then its enveloping algebra as follows Example 2.4. Let us recall that the infinite Drinfeld-Kohno Lie algebra

DK k,∞ = m≥2 DK (m) k,∞
is a (N ≥2 , ∨)-graded Lie algebra and moreover each component

DK (m) k,∞ ∼ = L k (X m-1 ) = n≥1 L k (X m-1 ) n
in the category k-Lie (see Subsection 1.2.5 for more details). Let us denote by DK

(m,n) k,∞
the set of all homogeneous Lie polynomials of total degree n of each free Lie algebra DK (m) k,∞ , that is Lie algebra isomorphic to L k (X m-1 ) n . Furthermore, it is obviously a free k-module with a Lyndon basis P l , where l ∈ LynX m-1 and the length | l |= n, here we used the indexed set (LynX m-1 , <) which is the totally ordered set of all Lyndon words over X m-1 . As a consequence, under the one to one correspondence between l and P l , one then has dim k (DK

(m,n) k,∞ ) = dim k (L k (X m-1 ) n ) =
#{Lyndon words of length n on X m-1 }, denoted Lyn(m -1, n). We hence deduce that the infinite Drinfeld-Kohno Lie algebra

DK k,∞ = m≥2 DK (m) k,∞ = m≥2 n≥1 DK (m,n) k,∞
is in fact equipped with a (N ≥2 , ∨) × (N ≥1 , +)-graded Lie algebra structure. Therefore, the enveloping algebra U(DK k,∞ ) inherits a [(N ≥2 , ∨)×(N ≥1 , +)]⊔{0}-graded structure (5.23)

U(DK k,∞ ) = s∈S N ∪{0} U s (DK k,∞ ),
where

S N := (N ≥2 , ∨) × (N ≥1 , +) and U s (DK k,∞ ) = T s (DK k,∞ ) J s , here we used T s (DK k,∞ ) = w∈[S N ] * µ(w)=s
T w (DK k,∞ ) as in the formula (5.22). Note that the Hilbert series ) by the following

Hilb(U(DK k,∞ )) = s∈S N ∪{0} dim k (U s (DK k,∞ )).s ∈ Q[[S N ∪ {0}]], where for each s ∈ S N ∪ {0}, dim k (U s (DK k,∞ )) = (m -1) n if s = (m, n) ∈ S N and dim k (U 0 (DK k,∞ )) = 1
Hilb(U(DK k,∞ ), t 1 , t 2 ) = m≥2 1 + t m 1 (m -1)t 2 1 -(m -1)t 2 ∈ Q[[t 1 , t 2 ]]
where, for each m ∈ N ≥2 ,

1 + t m 1 (m -1)t 2 1 -(m -1)t 2 = 1 + n≥1 (m -1) n t m 1 t n 2 .
Let us give a calculation example for this multiplication,

(1 + t 2 1 t 2 1 -t 2 )(1 + t 3 1 2t 2 1 -2t 2 ) = 1 + t 2 1 t 2 1 -t 2 + t 3 1 2t 2 1 -2t 2 + t 2∨3 1 t 2 .2t 2 (1 -t 2 )(1 -2t 2 ) = 1 + t 2 1 t 2 1 -t 2 + 2t 3 1 t 2 1 -2t 2 + 2t 3 1 t 2 2 (1 -t 2 )(1 -2t 2 ) .
U(DK k,n+1 ) = s∈T N ∪{0} U s (DK k,n+1 ),
where T N := ([2, n + 1], ∨) × (N ≥1 , +). Therefore, the Hilbert series

Hilb(U(DK k,n+1 ), t 1 , t 2 ) = 2≤i≤n+1 1 + t i 1 (i -1)t 2 1 -(i -1)t 2 ∈ Q[[t 1 , t 2 ]].
When t 1 = 1, we can recover Kohno's formula appearing in Example 2.3. 20 In other words, two series m1,n1≥0 α m1,n1 t m1 1 t n1 2 and m2,n2≥0 β m2,n2 t m2 1 t n2 2 being given in

Q[[t 1 , t 2 ]], then their convolution product is mi,ni≥0 α m1,n1 β m2,n2 t m1∨m2 1 t n1+n2 2 = m,n≥0      mi,ni≥0 m1∨m2=m n1+n2=n α m1,n1 β m2,n2      t m 1 t n 2
the internal sum being finite because the monoid of monomials has "D" property.

2.4 Smash product algebra and Lazard's elimination.

In this section, we investigate other aspects of Lazard's elimination within k-AAU the category of unital associative k-algebras. We first introduce crossed and smash products of algebras and discuss some relevance with semi-direct products of Lie algebras and the universal enveloping functor by Example 2.7 and Proposition 2.27. In the last two examples of this section, we will study a practical application to achieve Lazard's elimination and the quotient of Lazard's elimination in k-AAU.

Let k be a commutative ring with unit. We now study the crossed product of algebras, especially a smash product of a bialgebra and an associative algebra with unit. The reader who is only interested in studying these products may turn to R.K.

Molnar [START_REF] Molnar | Semi-direct products of Hopf algebras[END_REF], S. Montgomery [START_REF] Montgomery | Hopf Algebras and Their Actions on Rings[END_REF] and A. Borowiec, W. Marcinek [START_REF] Borowiec | On crossed product of algebras[END_REF], which can be read independently to what is presented as follows.

Definition 2.4. Let A, B, H be three objects in k-AAU. Assume that there are monomorphisms i A : A → H and i B : B → H in k-AAU. We say that H is a crossed (twisted) product of A and B if the canonical mapping Φ :

A ⊗ B → H, a ⊗ b → (i A ⊗ i B )(a ⊗ b) is an isomorphism in k-Mod.
As an immediate consequence of the above definition, a crossed product of two objects A and B is unique up to an isomorphism in k-AAU.

Example 2.6. The standard tensor product of algebras A ⊗ B is a crossed product of two objects A and B in k-AAU (here Φ = Id), where i A :

A → A ⊗ B, a → a ⊗ 1 B and i B : B → A ⊗ B, b → 1 A ⊗ b are the natural monomorphisms in k-AAU 21 .
The following definition is necessary for describing the main theorem of crossed product of algebras.

2.4. SMASH PRODUCT ALGEBRA AND LAZARD'S ELIMINATION.

Definition 2.5. Suppose given two objects A and B in k-AAU. A morphism τ :

B ⊗ A → A ⊗ B in k-Mod is called an algebra cross if it satisfies the following conditions c1) τ (1 B ⊗ a) = a ⊗ 1 B , c2) τ • (m B ⊗ Id A ) = (Id A ⊗m B ) • (τ ⊗ Id B ) • (Id B ⊗τ ), d1) τ (b ⊗ 1 A ) = 1 A ⊗ b, d2) τ • (Id B ⊗m A ) = (m A ⊗ Id B ) • (Id A ⊗τ ) • (τ ⊗ Id A ).
m τ = (m A ⊗ m B ) • (Id A ⊗τ ⊗ Id B ) is an object in k-AAU, denoted by A ⋊ τ B := (A ⊗ B, m τ , 1 A⊗B ). Moreover, A ⋊ τ B is a crossed product of A and B if k is a field.
Proof. We want to prove that A ⋊ τ B ∈ k-AAU.

Associativity can be proved by direct computation (as below for the unit (2.111)) or found in the literature (see also Proposition 2.2 in [START_REF] Van Daele | The Yang-Baxter and pentagon equation[END_REF] or Proposition 2.3 and Remark 2.4 (1) in [START_REF] Čap | On twisted tensor products of algebras[END_REF]) or even diagrammatically (i.e. using Penrose-like calculus). In fact, the diagram

(A ⊗ B) ⊗ (A ⊗ B) ⊗ (A ⊗ B) (A ⊗ B) ⊗ (A ⊗ B) (A ⊗ B) ⊗ (A ⊗ B) A ⊗ B
Id A⊗B ⊗mτ mτ ⊗Id A⊗B mτ mτ commutes by using the above relations (c1),(c2), (d1),(d2) and the associative laws

m A • (Id A ⊗m A ) = m A • (m A ⊗ Id A ), m B • (Id B ⊗m B ) = m B • (m B ⊗ Id B )
. The only thing left to prove is that 1 A⊗B = 1 A ⊗ 1 B is a unit for the multiplication m τ . Then, for the fact that it is a unit on the right we just need the condition (c1) as τ (1

B ⊗a) = a⊗1 A , then m τ [(1 A ⊗ 1 B ) ⊗ (y 1 ⊗ y 2 )] = (m A ⊗ m B ) • (Id A ⊗τ ⊗ Id B )[(1 A ⊗ 1 B ) ⊗ (y 1 ⊗ y 2 )] = (m A ⊗ m B )[Id A (1 A ) ⊗ τ (1 B ⊗ y 1 ) ⊗ Id B (y 2 )] = (m A ⊗ m B )[1 A ⊗ y 1 ⊗ 1 B ⊗ y 2 ] = y 1 ⊗ y 2 .
(2.111)

For the unit on the right, we must use the condition (d1) as

m τ [(x 1 ⊗ x 2 ) ⊗ (1 A ⊗ 1 B )] = (m A ⊗ m B ) • (Id A ⊗τ ⊗ Id B )[(x 1 ⊗ x 2 ) ⊗ (1 A ⊗ 1 B )] = (m A ⊗ m B )[x 1 ⊗ 1 A ⊗ x 2 ⊗ 1 B ] = x 1 ⊗ x 2 .
We also observe that if k is a field then i A :

A → A ⋊ τ B, a → a ⊗ 1 B and i B : B → A ⋊ τ B, b → 1 A ⊗ b are the natural monomorphisms in k-AAU.
Further, the mapping

Φ : A ⊗ B → A ⋊ τ B, a ⊗ b → m τ • (i A ⊗ i B )(a ⊗ b) = (m A ⊗ m B ) • (Id A ⊗τ ⊗ Id B ) • (i A ⊗ i B )(a ⊗ b) = a ⊗ b is a canonical isomorphism in k-Mod. Thus, A ⋊ τ B is a crossed product of A and B.
Remark 2.13. i) In general, when k is a unital commutative ring and τ : B ⊗ A → A ⊗ B is an algebra cross, we will also say that A ⋊ τ B is a crossed (twist) product of A and B if no confusion arises.

ii) The standard twist map τ : B ⊗ A → A ⊗ B defined as τ (b ⊗ a) = a ⊗ b satisfies all conditions for the algebra cross in Definition 2.5 and then A ⋊ τ B is the standard tensor product of algebras A and B.

iii) (See A. Borowiec and W. Marcinek [START_REF] Borowiec | On crossed product of algebras[END_REF]) Let f 1 :

A 1 → B 1 , f 2 : A 2 → B 2 be
two morphisms in k-AAU. Given two algebra crosses τ :

A 2 ⊗ A 1 → A 1 ⊗ A 2 and σ : B 2 ⊗ B 1 → B 1 ⊗ B 2 , then f : A 1 ⋊ τ A 2 → B 1 ⋊ σ B 2 , a ⊗ b → f 1 (a) ⊗ f 2 (b) is a crossed
product algebra homomorphism if and only if the following diagram commutes

A 2 ⊗ A 1 B 2 ⊗ B 1 A 1 ⊗ A 2 B 1 ⊗ B 2 . f 2 ⊗f 1 τ σ f 1 ⊗f 2 iv) It can be shown that if τ : B ⊗ A → A ⊗ B is an algebra cross, then τ 0 := τ 12 τ τ 12
(where τ 12 is the standard tensor flip x ⊗ y → y ⊗ x) is an algebra cross

τ 0 : A 0 ⊗ B 0 → B 0 ⊗ A 0
where the (-) 0 operator means passing to the opposite algebra.

We now give the general theory of the crossed product of algebras to a particular case that is efficient in practice. Let A be an associative algebra with unit 1 A i.e.

if it satisfies i) b ▷ (a 1 a 2 ) = (1)(2) (b (1) ▷ a 1 )(b (2) ▷ a 2 )
, where we have used Sweedler's notation 22

∆ B (b) = (1)(2) b (1) ⊗ k b (2) , ii) and b ▷ 1 A = ϵ B (b)1 A .
It is easy to verify that the following corollary is true.

Corollary 2.26. (Smash Product A♯B) With the preceding conditions and the following

multiplication m ♯ : (A ⊗ B) ⊗ (A ⊗ B) → A ⊗ B, m ♯ [(x 1 ⊗ x 2 ) ⊗ (y 1 ⊗ y 2 )] = (1)(2)
x 1 (x

(1) 2 ▷ y 1 ) ⊗ x (2) 2 y 2 (2.112) A♯B = (A ⊗ B, m ♯ , 1 A⊗B ) is an object in k-AAU.
This algebra is called the smash product algebra between the bialgebra B and the left B-module algebra A.

Proof. If the mapping ▷ : B ⊗ A → A is a left B-module algebra action then one can easily check through direct calculation that the mapping τ : 2) is an algebra cross in k-Mod. According to Theorem 2.25, we deduce that A♯B ≡ A ⋊ τ B is an object in k-AAU, where the multiplication

B ⊗ A → A ⊗ B, b ⊗ a → (1)(2) b (1) ▷ a ⊗ b (
m ♯ = (m A ⊗ m B ) • (Id A ⊗τ ⊗ Id B ) = m τ .
Remark 2.14. Assume that the bialgebra B and the left B-module algebra A are endowed with Hopf structures satisfying suitable circumstances introduced in Molnar Here is an important and natural example of smash product algebras.

Example 2.7. Let g i , i = 1, 2 be two objects in k-Lie and α : g 2 → Der(g 1 ) be a morphism in k-Lie. We first extend α from g 2 to Der(U(g 1 )) ⊂ End(U(g 1 )) as 22 See Montgomery [START_REF] Montgomery | Hopf Algebras and Their Actions on Rings[END_REF] Ch 1 §1.4.2.

CHAPTER 2. LAZARD'S ELIMINATION

in Bourbaki [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch I §2.8 Prop 7. Moreover, we can also extend α as a morphism α U : U(g 2 ) → End(U(g 1 )) in k-AAU by the universal property (5.37). Together with a bialgebra structure (U(g 2 ), µ U , 1 k , ∆ U , ϵ U ), we then obtain a left U(g 2 )-module algebra

action ▷ : U(g 2 ) ⊗ U(g 1 ) → U(g 1 ), b ⊗ a → b ▷ a = α U (b)(a)
. As we already constructed above, by Corollary 2.26, the k-module U(g 1 ) ⊗ U(g 2 ) can be endowed with a smash product structure U(g 1 )♯U(g 2 ) = (U(g 1 ) ⊗ U(g 2 ), 1 k ⊗ 1 k ), where the multiplication is

m ♯ [(u 1 ⊗ u 2 ) ⊗ (v 1 ⊗ v 2 )] = (1)(2) u 1 α U (u (1)
2

)(v 1 ) ⊗ u (2) 2 v 2 . (2.113)
This brings us to the following proposition Proposition 2.27. Let g 1 and g 2 be two objects in k-Lie. We suppose given also a Lie k-algebra morphism α : g 2 → Der(g 1 ) and f i : g i → g, two Lie homomorphisms into a Lie k-algebra g satisfying the equivariance (2.13)

g 2 ⊗ g 1 g ⊗ g g 1 g, f 2 ⊗f 1 α ⊗ ad g ⊗ f 1
or equivalently (2.12) i.e.

f 1 (α(b, a)) = ad g (f 2 (b), f 1 (a))
for all b ∈ g 2 , a ∈ g 1 . Then one has 1. There is a unique morphism f : g 1 ⋊ g 2 → g in k-Lie extending f 1 and f 2 in the usual sense.

2. There is a unique morphism f * : U(g 1 )♯U(g 2 ) → U(g) in k-AAU extending f * i := U(f i ) : U(g i ) → U(g) (for i = 1, 2) in the usual sense, where U : k-Lie → k-AAU, g → U(g) is the universal enveloping functor.

If

f : g 1 ⋊ g 2 → g is an isomorphism in k-Lie then f * : U(g 1 )♯U(g 2 ) → U(g) is an isomorphism in k-AAU.

Proof.

1. It is indeed a consequence of Proposition 2.2.

2.4. SMASH PRODUCT ALGEBRA AND LAZARD'S ELIMINATION.

Let us define a k-module morphism f

* : U(g 1 )♯U(g 2 ) → U(g) sending u 1 ⊗ u 2 to f * 1 (u 1 )f * 2 (u 2 )
. We now prove that f * is a morphism in k-AAU. In fact, one has

f * • m ♯ [(u 1 ⊗ u 2 ) ⊗ (v 1 ⊗ v 2 )] = f * [ (1)(2) u 1 α U (u (1)
2

)(v 1 ) ⊗ u (2) 2 v 2 ] = (1)(2) f * 1 (u 1 )f * 1 (α U (u (1)
2

)(v 1 ))f * 2 (u (2) 2 )f * 2 (v 2 ) = f * 1 (u 1 ) (1)(2) f * 1 (α U (u (1)
2

)(v 1 ))f * 2 (u (2) 2 ) f * 2 (v 2 )
and then

f * • m ♯ [(u 1 ⊗ u 2 ) ⊗ (v 1 ⊗ v 2 )] = f * (u 1 ⊗ u 2 )f * (v 1 ⊗ v 2 ) if we have an equa- tion (1)(2) f * 1 (α U (u (1)
2

)(v 1 ))f * 2 (u (2)
2

) = f * 2 (u 2 )f * 1 (v 1
) in U(g). This formula can be obtained from inductive processes and by the following extension of the equivariance (2.13) (where the below right-hand side is the right-normed bracketing in

U(g)) f * 1 (α U (v)(u)) = ad U (g) f 2 (b 1 ) • • • • • ad U (g) f 2 (b k ) [f * 1 (u)] (2.114) for all v = b 1 • • • b k ∈ U(g 2 ) and u ∈ U(g 1 ), where b i in g 2 and f * 2 (v) = f 2 (v 1 ) • • • f 2 (v k ). More precise, if v 1 ∈ g 1 and u 2 ∈ g 2 then ∆ U (u 2 ) = u 2 ⊗ 1 k + 1 k ⊗ u 2 , thus one has (1)(2) f * 1 (α U (u (1)
2

)(v 1 ))f * 2 (u (2)
2

) = f 1 (α U (u 2 )(v 1 )) + f 1 (v 1 )f 2 (u 2 ) = ad U (g) f 2 (u 2 ) [f 1 (v 1 )] + f 1 (v 1 )f 2 (u 2 ) = f 2 (u 2 )f 1 (v 1 ) = f * 2 (u 2 )f * 1 (v 1 ); in general case if v 1 = av ′ 1 ∈ U(g 1 ) and u 2 = bu ′ 2 ∈ U(g 2 ), where (a, b) ∈ g 1 × g 2 and (v ′ 1 , u ′ 2 ) ∈ U(g 1 ) × U(g 2 ),
we leave it as a small exercise to interested readers (hints: we notice that ∆

U (u 2 ) = ∆ U (b)∆ U (u ′ 2 ) = (b ⊗ 1 k + 1 k ⊗ b)( (1)(2) u ′ (1) 2 ⊗ u ′ (2) 2 ) = (1)(2) bu ′ (1) 2 ⊗ u ′ (2) 2 + u ′ (1) 2 ⊗ bu ′ (2)
2

). Further, the uniqueness of the algebra homomorphism f * comes from the definition of the mapping because

f * (u 1 ⊗ u 2 ) = f * • m ♯ [(u 1 ⊗ 1 k ) ⊗ (1 k ⊗ u 2 )] = f * (u 1 ⊗ 1 k )f * (1 k ⊗ u 2 ) = f * 1 (u 1 )f * 2 (u 2 ).
3. If g : g → g 1 ⋊ g 2 is an inverse Lie homomorphim of f . Then, the k-algebra morphism f * can be reversed by constructing an algebra homomorphism g * : CHAPTER 2. LAZARD'S ELIMINATION U(g) → U(g 1 )♯U(g 2 ) as an extension of g : g → g 1 ⋊ g 2 → U(g 1 )♯U(g 2 ) by the universal property (5.37). It immediately implies that f * • g * = Id U (g) and

g * • f * = Id U (g 1 )♯U (g 2 ) .
It turns out that this result is very useful, in examples below a lot more algebra isomorphism structures are presented, which are just a little bit harder to state immediately to the reader. More precisely, we now treat the first application of the scheme to the free associative algebra (k⟨X⟩, conc, 1 X * ). ii) the algebra homomorphism f * between the smash product

L k (X) b 1 • • • • • ad L k (X) b k (z) = ad L k (X) (u) (z) for each u = b 1 • • • b k ∈ B *
U(L k (B * Z))♯U(L k (B))
and the enveloping algebra U(L k (X)) which sends

u 1 ⊗ u 2 → f * 1 (u 1 )f * 2 (u 2 ) ex- tending f * 1 : U(L k (B * Z)) → U(L k (X)) and f * 2 : U(L k (B)) → U(L k (X)
) is an isomorphism in k-AAU by obtaining from Proposition 2.27 points (2),(3) and the fact that f is a Lie isomorphism as a consequence of Theorem 2.3 which constructed Lazard's elimination in k-Lie.

iii) then, recall that the free associative algebra k⟨X⟩ is identified with U(L k (X)) 2.4. SMASH PRODUCT ALGEBRA AND LAZARD'S ELIMINATION. the universal enveloping algebra of the free Lie algebra L k (X) 23 , thus one has k⟨B * Z⟩ = U(L k (B * Z)) and k⟨B⟩ = U(L k (B)). We call these phenomenons

k⟨B * Z⟩♯k⟨B⟩ ∼ = -→ k⟨X⟩ in k-AAU by Lazard's elimination in k-AAU.
This example deals with a more general type of the above example for quotients of Lazard's eliminations in k-Lie.

Example 2.9. As in Subsection 2.2.2, we suppose given a set X = B + Z (partitioned in two blocks) and a relator r = {r j } j∈J ⊂ L k (X) which is compatible with the alphabet partition i.e. there exists a partition of the set of indices

J = J Z ⊔ J B such that r B = {r j } j∈J B = r ∩ L k (X) B and r Z = {r j } j∈J Z = r ∩ L k (X) BZ . As
we have seen in Subsection 2.2.2, we considered that J B is the Lie ideal of L k (X) B generated by {r j } j∈J B , J , J Z and J BZ are the Lie ideals of L k (X) generated by r, r Z and r BZ = {ad Q z} Q∈J B ,z∈Z respectively. With these constructions above, by Theorem 2.6, we have the following iii) a morphism of Lie algebras (2.28)

i) a morphism [α] : L k (X) B J B → Der( L k (X) BZ J Z BZ ) of Lie algebras, where J Z BZ = J Z + J BZ is the Lie ideal of L k (X) BZ . ii) g 1 : L k (X) BZ J Z BZ → L k (X) / J and g 2 : L k (X) B J B → L k (X) / J
β 33 : L k (X) BZ J Z BZ ⋊ L k (X) B J B → L k (X) / J
extending g 1 and g 2 in the usual sense is an isomorphism in k-Lie (meaning the quotient of Lazard's elimination in k-Lie).

We are now ready to give an another important application of Proposition 2.27 point

(2) and point (3). More precisely, the algebra homomorphism (β 33 ) * from the smash

product algebra U( L k (X) BZ J Z BZ )♯U( L k (X) B J B ) to the universal enveloping al- gebra U( L k (X) / J ) extending g * 1 : U( L k (X) BZ J Z BZ ) → U( L k (X) / J ) and g * 2 : U( L k (X) B J B ) → U( L k (X) / J ) in the usual sense i.e. (β 33 ) * (u 1 ⊗u 2 ) = g * 1 (u 1 )g * 2 (u 2
) is an isomorphism in k-AAU. We call these phenomenons by the quotient of Lazard CHAPTER 2. LAZARD'S ELIMINATION elimination in k-AAU.

We would now like to treat this consequence to special cases:

• Firstly, applying the quotient of Lazard's elimination (in k-AAU) to Lazard's Partially Commutative Elimination i.e. Corollary 2.16, if X is a set equipped with a commutation relation θ and B is a subset of X such that Z = X -B is totally non-commutative, then there is an isomorphism in k-AAU

U(L k (X, θ)) ∼ = U(L k (C))♯U(L k (B, θ B )).
(2.115)

• Furthermore, if we consider the decomposition of Drinfeld-Kohno Lie algebra in Corollary 2.17, one has an isomorphism

U(DK k,n+1 ) ∼ = U(L k (X n ))♯U(DK k,n ) (2.116)
in k-AAU, where X n is any alphabet of cardinality n.

TABLE OF LAZARD ELIMINATION FORMULAS.

Table of Lazard elimination formulas.

We summarize Lazard's elimination principle (2.2)

Free(B + Z) = Free(C B [Z]) ⋊ Free(B)
in each category formed (2.1)

Mon, Grp, k-Lie and k-AAU by the following table

Category

Abbreviation Elimination formula (free) This chapter is about some characters, their convolutions and their extensions is at hand (then B is a Hopf algebra), this monoid is a group (inversion being provided by precomposition with the antipode, see [START_REF] Duchamp | Three variations on the linear independence of grouplikes in a coalgebra[END_REF]).

Monoids Mon X * = (B * Z) * B * Groups Grp Γ(X) = Γ(C B (Z)) ⋊ Γ(B) Lie k-algebras k-Lie L k (X) ∼ = L k (B * Z) ⋊ L k (B) Unital associative k-algebras k-AAU k⟨X⟩ ∼ = k⟨B * Z⟩♯k⟨B⟩
Here will be dealt mainly with two types of characters

• Shuffle characters which will be provided by the algebra of polylogarithms (at first indexed with noncommutative polynomials C⟨x 0 , x 1 ⟩ and extended to series of Dom(Li) ⊊ C⟨⟨x 0 , x 1 ⟩⟩ (as in Figure 3.2).

• Stuffle characters that are provided by Harmonic sums defined on a word w =

y s 1 • • • y sr by H w = H s 1 ,••• ,sr = N ≥n 1 >•••>nr>0 1 n s 1 1 • • • n sr r .
• The link between Polylogarithms, Harmonic sums and then to MZV which are (some of) their limits being provided (classically) by Taylor expansion around zero (see Proposition 3.13) of some Polylogarithms, the transfer of the extended domain Dom(Li) to Harmonic sums is warranted by normal families of functions (a.k.a. such in the work [START_REF] Montel | Leçons sur les familles normales de fonctions analytiques et leurs applications[END_REF]).

Shuffle and Stuffle characters are particular cases of convolution characters on enveloping algebras, the link with Lazard elimination and, more generally, with semidirect products is highlighted by formula (3.7).

On the side of indexation, one remarks that the Stuffle product is a perturbation3 of the Shuffle product. Many such perturbations can be found in the literature (see a non exhaustive table of them in [START_REF] Duchamp | The mechanics of shuffle products and their siblings[END_REF]) as well as their shifted variants ( [START_REF] Bouillot | A two-parameter deformation of the quasishuffle and new bases of quasi-symmetric functions[END_REF][START_REF] Duchamp | Deformations of Algebras: Twisting and Perturbations[END_REF][START_REF] Grinberg | The enriched q-monomial basis of the quasisymmetric functions[END_REF][START_REF] Hirose | Blockshuffle identities for multiple zeta values[END_REF]), some of them directly linked to conjectures about MZV ( [START_REF] Hirose | Blockshuffle identities for multiple zeta values[END_REF]).

The structure of the chapter is the following:

In the next part (i.e. Section 3.2), we define what is the convolution product in general (i.e. between linear maps, not only characters and infinitesimal ones). This allows to express the generating series of the identity (i.e. (Id End ) gen ) as an infinite product of exponentials of rank one infinitesimal characters (this formula is a resolution of the identity). To this end, we first provide two examples (in Example 3.1), one with the shuffle and one with the stuffle product. We end this paragraph with the general formula (which holds for enveloping algebras of linearly k-free Lie algebras, k being itself a Q-algebra).

We saw that a crucial step is this expression of (Id End ) gen is to express it as a summable series of tensor products obtained through dual bases. A remarkable combinatorial realization of such bases in duality is provided within the next section (see Section 3.3) and is performed through half-shuffle and Zinbiel algebras.

The end of the chapter is devoted to extensions of some characters relating to special functions: Polylogarithms for the shuffle product (a character with values in holomorphic functions) and Harmonic functions for the Stuffle. The transfer of the properties of the extended domain from Li to Harmonic sums is performed by row inequalities and columns limits (see Lemma 3.17).

Convolution algebra and factorization.

Throughout this section k is assumed to be a Q-algebra.

Let us first give a famous example of the aimed factorization (see [START_REF] Racinet | Series generatrices non-commutatives de polyzetas et associateurs de Drinfel'd[END_REF][START_REF] Reutenauer | Free Lie Algebras[END_REF]) for the special case when the Lie algebra g is the free Lie algebra (i.e. g = L k (X)).

We consider the usual Hopf enveloping algebra U(L k (X)), this algebra inherits the following Hopf structure H conc (X) = (k⟨X⟩, conc, 1 X * , ∆ ¡ , ϵ) 4 . As we will see below, formulas (3.1) (with their proper indexation) hold true for multihomogeneous bases, in particular bases like Hall, Lyndon, Viennot, Schützenberger which are monomial i.e.

obtained by bracketting. For simplicity, we return here to the Lyndon basis 5 (P l ) l∈LynX of the free Lie algebra L k (X) and (S w ) w∈X * , computed from (P w ) w∈X * by duality (where (P w ) w∈X * is the Poincaré-Birkhoff-Witt -PBW for short -basis associated to (P l ) l∈LynX for the standard lexicographic order) is such that, by restriction, (S l ) l∈LynX is 6 a tran-4 See also Remark 5.6, Appendix 5.4.4.

5 For this basis, we have P x = x if x ∈ X and P l = [P l1 , P l2 ] if st(l) = (l 1 , l 2 ) (a pair of Lyndon words (l 1 , l 2 ) is called the standard factorization of a Lyndon word l / ∈ X if l = l 1 l 2 and l 2 is the longest Lyndon proper right factor of l). 6 Here,

S x = x if x ∈ X and S l = xS u if l = xu ∈ LynX \ X.

CONVOLUTION ALGEBRA AND FACTORIZATION.

scendence basis of the unital shuffle algebra (k⟨X⟩, ¡,1 X * ) 7 , indexed by (LynX, <) the totally ordered set of all Lyndon words over X. The PBW basis (P w ) w∈X * of the free associative algebra k⟨X⟩ and its dual family 8 (S w ) w∈X * can be expressed as follows:

each word w ∈ X * can be written uniquely as a decreasing product of Lyndon words where the first equality follows Equation (5.13). This decomposition is called MRS 9 factorization.

w = l i 1 1 • • • l i k k (where l i ∈ LynX, l 1 > • • • > l k and i 1 , • • • , i k ∈ N), then P w = P i 1 l 1 • • • P i k l k and S w = S ¡i 1 l 1 ¡ • • • ¡ S ¡i k l k i 1 ! • • • i k ! . ( 3
Furthermore, note that the Lyndon basis (P l ) l∈LynX (as any Hall or monomial basis) is multi-homogeneous with respect to the N (X) -grading, where by Subsection 1.2.5, a

Lie polynomial P ∈ L k (X) = α∈N (X) L k (X) α is called multi-homogeneous if there exists α = (α x ) x∈X ∈ N (X) so that P ∈ L k (X) α . Now, let g = g 1 ⊕g 2 be a Lie k-algebra decomposed (internal) into two Lie subalgebras. Assume that g is free k-module then we say that a basis B of g is compatible with this decomposition if

B = B 1 + B 2 ,
where

B 1 = B ∩ g 1 , B 2 = B ∩ g 2 .
Proposition 3.1. Every basis of the free Lie algebra L k (X) which is multi-homogeneous

10 is compatible with the classical Lazard elimination L k (X) = L k (X) BZ ⋊ L k (X) B .
Proof. Assume that B is a multi-homogeneous basis of the free Lie algebra L k (X). We

7 See Reutenauer's [94] §5.2, Thm 5.3, Cors 5.5 and 5.6 for more details. 8 In fact, the coordinate family, in this case a basis of the multihomogneous graded dual, a subspace of the dual space k⟨⟨X⟩⟩. 9 After Mélançon, Reutenauer and Schützenberger (see [START_REF] Reutenauer | Free Lie Algebras[END_REF]). 10 e.g. monomial bases, Hall, Lyndon, Viennot, Schützenberger [START_REF] Lothaire | Combinatorics on words[END_REF][START_REF] Reutenauer | Free Lie Algebras[END_REF][START_REF] Schüzenberger | Sur une propriété combinatoire des algèbres de Lie libres pouvant être utilisée dans un problème de mathématiques appliquées, Séminaire Dubreil-Jacotin Pisot (Algèbre et théorie des nombres)[END_REF][START_REF] Viennot | Algèbres de Lie libres et monoıdes libres[END_REF].

CHAPTER 3. CHARACTERS now put

B 1 := B ∩ L k (X) BZ = B ∩     α∈N (X) |α| Z >0 L k (X) α    
and

B 2 := B ∩ L k (X) B = B ∩     α∈N (X) |α| Z =0 L k (X) α     . Now, for each P ∈ B then P ∈ L k (X) α , where α = (α x ) x∈X ∈ N (X) . Clearly, if |α| Z > 0 then P ∈ B ∩ L k (X) BZ = B 1 , otherwise if |α| Z = 0 one has P ∈ B ∩ L k (X) B = B 2 .
We then obtain that B = B 1 + B 2 . We verified our result.

General case. -Given a Lie algebra g over k a commutative ring with unit and its universal enveloping algebra U(g), let us consider the usual Hopf algebra

(U(g), µ U , 1 k , ∆ U , ϵ U ) as mentioned in Appendix 5.4.4. On the k-module End(U(g)),
there is a structure of unital associative algebra, called the convolution algebra, in which the product of two k-linear maps f, g ∈ End(U(g)) is defined by the formula

f ⋆ g = µ U • (f ⊗ g) • ∆ U ∈ End(U(g)), whose unit element is 1 End = 1 k • ϵ U ∈ End(U(g)).
The inverse element of the identity map Id End with respect to the convolution product is given by the antipode S U , the antiautomorphism of U(g) characterized by S U (x) = -x for any x ∈ g.

At (U(g), ∆ U , ϵ U ), the (full) dual space U * (g) = Hom(U(g), k) is a unital associative algebra induced from the transposes of ∆ U and ϵ U , respectively. More precisely, the We suppose now that g is free as a k-module. Then, due to the fact that U(g) is also k-linearly free, the canonical map Φ : U * ⊗ U → End(U), is into. One can show that the image of Φ is dense in End(U) when endowed with the topology of pointwise convergence (see Example 5.3).

convolution product m U * : U * (g) ⊗ U * (g) → U * (g) (resp. a unit λ U * : k → U * (g)) is the composite map U * (g) ⊗ U * (g) → (U(g) ⊗ U(g)) * ∆ * U --→ U * (g) (resp. k → k * ϵ * U -→ U * (g)) (cf.

CONVOLUTION ALGEBRA AND FACTORIZATION.

Suppose further that k is a Q-algebra and fix a totally ordered basis B = (b i ) i∈I of the Lie algebra g (which is therefore supposed to be is a free k-module). This datum leads to an associated PBW basis of U(g), denoted by (B α ) α∈N (I) , which is constructed by the following multi-index notation: for any α = i∈I α i e i ∈ N (I) , where the elementary multi-indices e i ∈ N (I) is defined for all i ∈ I by e i (j) = δ j i (Kronecker delta) and supp(α

) ⊆ {i 1 < • • • < i k }, we set B α = b α i 1 i 1 • • • b α i k i k = (B e i 1 ) α i 1 • • • (B e i k ) α i k ∈ U(g). (3.3) Notice that B α+β ̸ = B α B β if supp(α) ∩ supp(β) ̸ = ∅
and that the standard order considered here is, in the free case, the opposite lexicographic order. At the present stage, it is straightforward to transform the above PBW basis of U(g) to its dual (so-called) basis 11 (B α ) α∈N (I) of U * (g) in terms of the duality given by

⟨B α | B β ⟩ = δ β α (the Kronecker delta), for all α, β ∈ N (I) ,
where the pairing

⟨• | •⟩ : U * (g) ⊗ U(g) → k, φ ⊗ u → φ(u)
is the usual one. More precisely, for these constructions, we have Theorem 3.2. One has the following

B α ⋆ B β = (α + β)! α!β! B α+β , B α = (b ⋆ i 1 ) ⋆α i 1 ⋆ • • • ⋆ (b ⋆ i k ) ⋆α i k α i 1 ! • • • α i k ! = (B e i 1 ) ⋆α i 1 ⋆ • • • ⋆ (B e i k ) ⋆α i k α i 1 ! • • • α i k ! , (3.4)
where any α ∈ N (I) is expressed in the above notation and with the formula α! := i∈I α i !.

Proof. We can easily check that the coproduct ∆ U of any PBW basis is given by

∆ U (B γ ) = α,β∈N (I) α+β=γ γ! α!β! B α ⊗ B β , for any γ ∈ N (I) (3.5)
because ∆ U is a morphism of algebras and the elements of g are primitive.

Now, for each α, β ∈ N (I) , it follows that 11 In fact, the family of coordinate forms.

⟨B α ⋆ B β | B γ ⟩ = ⟨m U * (B α ⊗ B β ) | B γ ⟩
CHAPTER 3. CHARACTERS = (m U * (B α ⊗ B β ))(B γ ) = (B α ⊗ B β )(∆ U (B γ )) = (B α ⊗ B β ) α 1 ,β 1 ∈N (I) α 1 +β 1 =γ γ! α 1 !β 1 ! B α 1 ⊗ B β 1 = α 1 ,β 1 ∈N (I) α 1 +β 1 =γ γ! α 1 !β 1 ! B α (B α 1 )B β (B β 1 ) = α 1 ,β 1 ∈N (I) α 1 +β 1 =γ γ! α 1 !β 1 ! δ α 1 α δ β 1 β = δ γ α+β γ! α!β! .
This leads to the equation

B α ⋆ B β = (α+β)! α!β! B α+β . By telescoping the products, for each α 1 , • • • , α k ∈ N (I) we obtain B α 1 ⋆ • • • ⋆ B α k = (α 1 + α 2 )! α 1 !α 2 ! (α 1 + α 2 + α 3 )! (α 1 + α 2 )!α 3 ! • • • (α 1 + • • • + α k )! (α 1 + • • • + α k-1 )!α k ! B α 1 +•••+α k = (α 1 + • • • + α k )! α 1 ! • • • α k ! B α 1 +•••+α k .
In particular, considering α i e i = e i + • • • + e i α i times ∈ N (I) , where i ∈ I, α i ∈ N and the elementary multi-index e i ∈ N (I) , the dual basis element B α i e i can then be expressed as

B α i e i = (e i !) α i (α i e i )! B e i ⋆ • • • ⋆ B e i α i times = (B e i ) ⋆α i α i ! .
Thus, for any multi-index α

= i∈I α i e i = α i 1 e i 1 +• • •+α i k e i k ∈ N (I) , where supp(α) ⊆ {i 1 < • • • < i k }, one has B α = (α i 1 e i 1 )! • • • (α i k e i k )! (α i 1 e i 1 + • • • + α i k e i k )! B α i 1 e i 1 ⋆ • • • ⋆ B α i k e i k = α i 1 ! • • • α i k ! α! B α i 1 e i 1 ⋆ • • • ⋆ B α i k e i k = B α i 1 e i 1 ⋆ • • • ⋆ B α i k e i k (since α! = α i 1 ! • • • α i k !) = (B e i 1 ) ⋆α i 1 ⋆ • • • ⋆ (B e i k ) ⋆α i k α i 1 ! • • • α i k ! .
We just proved our theorem.

We [START_REF] Duchamp | Kleene Stars in Shuffle algebras, CAP Days[END_REF]).

Correspondingly, the image of the identity map Id End under the ring isomorphism Φ -1 is denoted by (Id End ) gen . We can carry out a systematic formulation of the generalized identity as follows Theorem 3.4 ( [START_REF] Duchamp | Kleene Stars in Shuffle algebras, CAP Days[END_REF]). Let k be a Q-algebra and g be a Lie k-algebra endowed with a totally ordered basis B = (b i ) i∈I of the Lie algebra g (hence free as a k-module) then i) The following infinite product identity holds in the complete topological associative algebra U * (g) ⊗U(g)

(Id End ) gen = α∈N (I) B α ⊗ B α = ↗ i∈I exp(B e i ⊗ B e i ). (3.6) ii) Moreover, if the ordered basis B = (b i ) i∈I is split in two successive parts B 1 = (b i ) i∈I 1
and B 2 = (b i ) i∈I 2 where I = I 1 → + I 2 is an ordinal sum, we have

(Id End ) gen = ↗ i∈I exp(B e i ⊗ B e i ) = ↗ i∈I 1 exp(B e i ⊗ B e i ) × ↗ i∈I 2 exp(B e i ⊗ B e i ).
(3.7)

Proof. i) First equality follows from (5.13). The last right-hand side of (3.6) is

↗ i∈I exp(B e i ⊗ B e i ) = ↗ i∈I n≥0 1 n! (B e i ) ⋆n ⊗ (B e i ) n
that is equal to

i 1 <••••••<i k 0≤α i 1 ,••• ,α i k 1 α i 1 ! • • • α i k ! (B e i 1 ) ⋆α i 1 ⋆ • • • ⋆ (B e i k ) ⋆α i k ⊗ (B e i 1 ) α i 1 • • • (B e i k ) α i k
by the usual expression. Let us however say a word about convergence. We consider the nets of finite partial sums and products

S F = α∈N (I) supp(α)⊂F B α ⊗ B α P F = ↗ i∈F exp(B e i ⊗ B e i )
for all F ⊂ f inite I. Convergence of these nets to (Id End ) gen goes as follows.

The nets Φ(S F ) and Φ(P F ) converge pointwise to Id End due to the following CHAPTER 3. CHARACTERS Lemma 3.5. Let M be a free module with basis B = (β i ) i∈I , endowed with the discrete topology, then, for a net N = (φ α ) α∈A within End(M ), TFAE 1. N converges pointwise to φ lim .

2. For all i ∈ I, φ α (β i ) converges to φ lim (β i ).

Proof. A net N = (φ α ) α∈A within End(M ) converges pointwise to ℓ ∈ End(M ) if and only if

(∀F ⊂ f inite M )(∃B ∈ A)(∀α ≥ B)(φ α | F = ℓ| F ) (3.8)
Now, due to the equation (3.3) and Theorem 3.2, we deduce our result.

ii) First proof. -Second equality is a particular case of ordinal partition of indices for infinite ordered products. Now, for the reader who wants to feel what is happening inside (3.7), we give a second -combinatorial -proof.

Second proof. -By using I = I 1 → + I 2 , the last right-hand side of (3.7) is

↗ i∈I 1 exp(B e i ⊗ B e i ) × ↗ i∈I 2 exp(B e i ⊗ B e i ) = ↗ i∈I 1 n≥0 1 n! (B e i ) ⋆n ⊗ (B e i ) n × ↗ i∈I 2 n≥0 1 n! (B e i ) ⋆n ⊗ (B e i ) n =     i 1 <••••••<i k ∈I 1 0≤α i 1 ,••• ,α i k 1 α i 1 ! • • • α i k ! (B e i 1 ) ⋆α i 1 ⋆ • • • ⋆ (B e i k ) ⋆α i k ⊗ (B e i 1 ) α i 1 • • • (B e i k ) α i k     ×     i 1 <••••••<i k ∈I 2 0≤α i 1 ,••• ,α i k 1 α i 1 ! • • • α i k ! (B e i 1 ) ⋆α i 1 ⋆ • • • ⋆ (B e i k ) ⋆α i k ⊗ (B e i 1 ) α i 1 • • • (B e i k ) α i k     = i 1 <••••••<i k ∈I 0≤α i 1 ,••• ,α i k 1 α i 1 ! • • • α i k ! (B e i 1 ) ⋆α i 1 ⋆ • • • ⋆ (B e i k ) ⋆α i k ⊗ (B e i 1 ) α i 1 • • • (B e i k ) α i k = ↗ i∈I n≥0 1 n! (B e i ) ⋆n ⊗ (B e i ) n = ↗ i∈I exp(B e i ⊗ B e i ) = (Id End ) gen .
We have then completed the second proof. i) Formula (3.7) generalizes without difficulty to the case when the totally ordered set

I splits as I = I 1 → + • • • → + I m (m-factor ordinal sum) as (Id End ) gen = ↗ i∈I 1 exp(B e i ⊗ B e i ) × • • • × ↗ i∈Im exp(B e i ⊗ B e i ). (3.9)
Recall that the Drinfeld-Kohno Lie algebra DK k,n+1 = 2≤j≤n+1 DK

(j) k,n+1 is a ([2, n + 1], ∨)-graded Lie algebra in which each factor DK (j)
k,n+1 is a free Lie algebra L k (T j ) (see Equation (2.104)). Thus, under the n-factor ordinal sum

T n+1 = T 2 → + • • • → + T n+1 ,
the Lie algebra DK k,n+1 has a k-linear basis B = l∈I DK P l , where we used the totally ordered set

I DK := Lyn(T 2 ) → + • • • → + Lyn(T n+1
). As a consequence, we have the following infinite product identity in the complete topological associative algebra

U * (DK k,n+1 ) ⊗U(DK k,n+1 ) (over k a Q-algebra) (Id End ) gen = ↘ l∈I DK exp(S l ⊗ P l ) = ↘ l∈Lyn(T n+1 ) exp(S l ⊗ P l ) × • • • × ↘ l∈Lyn(T 2 )
exp(S l ⊗ P l ).

ii) We now define a deformation of the shuffle product (in fact an interpolation between shuffle and stuffle products 12 ).

Let Y be the infinite alphabet {y k | k ≥ 1} and take a parameter q in k. If the ground ring was R and q is seen as formal, take k = R[q], later, but not now, we will require that k be a Q-algebra. The q-deformed stuffle product, noted q , is defined by the following recursion

u q 1 Y * = 1 Y * q u = u, (3.10)
y k 1 u q y k 2 v = y k 1 (u q y k 2 v) + y k 2 (y k 1 u q v) + qy k 1 +k 2 (u q v) (3.11) for all y k 1 , y k 2 ∈ Y and u, v ∈ Y * .
CHAPTER 3. CHARACTERS is connected, cocommutative and N-graded.

Then, due to the fact that (-1) n-1 n .(I + ) ⋆ n is summable (see [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch II §1.6 or Cartier [START_REF] Cartier | A primer of Hopf algebras[END_REF] Thm 3.6.1), H q (Y ) = U(Prim(H)) and using the Eulerian projector

π 1 = log(I) = n≥1 (-1) n-1 n (I + ) ⋆ n
(i.e. the Eulerian projector as in [START_REF] Reutenauer | Free Lie Algebras[END_REF]), one constructs two bases (of k⟨Y ⟩) in duality Π w and Σ w . Then

(Id) gen = w∈Y * w ⊗ w = w∈Y * Σ w ⊗ Π w = ↘ l∈LynY exp(Σ l ⊗ Π l ),
for details, see [START_REF] Bui | Schüzenberger's factorization on the (completed) Hopf algebra of q-stuffle product[END_REF].

Example 3.3. A motivation. -

In this example, we show an application of Lazard's elimination to the combinatorics of polylogarithms.

Among multihomogeneous bases (Hall, Schützenberger-Viennot-Hall, Shirshov, etc.), one is particularly interesting, the Lyndon basis and its dual family (which is this case admits a nice combinatorial resursion). With the same notations as in Section 3.1 and as in the beginning of Section 3.2, we first construct {P w } w∈X * from the Lyndon basis (see [START_REF] Lothaire | Combinatorics on words[END_REF] and [START_REF] Reutenauer | Free Lie Algebras[END_REF]) and its dual family {S w } w∈X * which satisfies the following recursion ([94] Thm 5.3)

S l = xS u , for l = xu ∈ LynX \ X, S w = S ¡i 1 l 1 ¡ . . . ¡ S ¡i k l k i 1 ! . . . i k ! for w = l i 1 1 . . . l i k k , l 1 > . . . > l k . (3.13)
A table of the first elements, up to length 6, can be found in Appendix 5.5.1.

The polylogarithmic function is a shuffle character on (C⟨X⟩, ¡,1

X * ) (X = {x 0 , x 1 })
with values in H(Ω) 13(C⟨X⟩, ¡,1 and this, by (3.7), automatically splits the product as

X * ) Li (H(Ω), ×, 1 Ω ) (3.14) 3.3
L = ↘ l∈LynX |l|x 1 ≥1 exp(Li S l P l ) × ↘ l∈LynX |l|x 1 =0 exp(Li S l P l ) = L + × exp(x 0 log(z)). (3.16)
Now, using recursion (3.13), one checks that all terms Li S l in the exponent of L + tend to zero as z ∈ Ω tends to zero. This entails that lim z∈Ω,z→0

L(z)e -x 0 log(z) = 1 H(Ω)⟨⟨X⟩⟩ which is precisely the asymptotic initial condition of (3.38).

Bases in duality: Zinbiel algebra and Magnus

basis.

Here, we deal with Magnus basis on the free associative algebra arising from Lazard's elimination in k-AAU. We then focus on Zinbiel algebras and present a systematic method to study Magnus duality in Theorem 3.9. In the last part of this section, there are several applications of the Magnus duality introduced in Remark 3.2.

For convenience, in this section we suppose that k is a commutative ring with unit and X = B + Z is a finite set partitioned in two blocks, where

B = {b 1 , • • • , b M } and Z = {z 1 , • • • , z N }.
Now we formulate and prove the following corollary:

Corollary 3.6. (Magnus basis) Let us consider the free associative algebra

k⟨X⟩ = k⟨b 1 , • • • , b M , z 1 , • • • , z N ⟩.
The collection of polynomials (called by Magnus polynomials, cf. [START_REF] Magnus | Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations[END_REF] Ch V §5.6)

rn(w 1 z i 1 ) • • • rn(w k z i k ).w, (3.17)
where

k ≥ 0, w 1 , • • • , w k , w ∈ B * ; i 1 , • • • , i k ∈ [1, N ] (if k = 0 then rn(w 1 z i 1 ) • • • rn(w k z i k )
will be denoted by 1 X * ), is a k-linear basis of k⟨X⟩.

Proof. It follows from Example 2.8 points (ii) and (iii) that the natural algebra homomorphism

f * : k⟨B * Z⟩♯k⟨B⟩ → k⟨X⟩, u 1 ⊗ u 2 → f * 1 (u 1 )f * 2 (u 2 )
is an isomorphism in k-AAU. Moreover, the families of monomials {w

1 z i 1 • • • w k z i k | k ≥ 0, w 1 , • • • , w k ∈ B * ; i 1 , • • • , i k ∈ [1, N ]}
* (u 1 ⊗ u 2 ) = f * 1 (u 1 )f * 2 (u 2 )
, we thus deduce that the collection of Magnus polynomials (3.17) is a k-linear basis of k⟨X⟩. We verified our corollary by applying Lazard's elimination in k-AAU.

Our next aim is to describe the dual of Magnus basis under the standard pairing

⟨• | •⟩ : k⟨X⟩ ⊗ k⟨X⟩ → k (3.18)
classically defined by, for T ∈ k⟨X⟩ and P ∈ k⟨X⟩,

⟨T | P ⟩ = w∈X * ⟨T | w⟩⟨P | w⟩,
where, when w is a word, ⟨T | w⟩ stands for the coefficient of w in T (see Subsection 1.2.4). More precisely, in this case, there is a unique dual of these polynomials because the Magnus basis (3.17) is multi-homogeneous with respect to the N (X) -grading 14 .

We start with a notion of Zinbiel algebras that were introduced by Jean-Louis Loday in [START_REF] Loday | Cup-product for Leibniz cohomology and dual Leibniz algebras[END_REF] (see also in Loday [START_REF] Loday | Generalized bialgebras and triples of operads[END_REF]) as the Koszul dual operad to Leibniz algebras (hence the name coined by the topologist J.M. Lemaire), which is more than adequate for our purpose which means to construct the dual of Magnus basis.

Definition 3.1. A (left) Zinbiel algebra over k (a unital commutative ring) is a kmodule A equipped with a bilinear map ≺ satisfying the following relation

(x ≺ y) ≺ z = x ≺ (y ≺ z) + x ≺ (z ≺ y), for all x, y, z ∈ A. (3.19)
Moreover, an element 1 A of a (left) Zinbiel algebra A is called the unit if for any

1 A ̸ = x ∈ A, 1 A ≺ x = 0, x ≺ 1 A = x,
and 1 A ≺ 1 A is not define. 14 We recall that a polynomial

T ∈ k⟨X⟩ is called multi-homogeneous if T ∈ k α ⟨X⟩, for α = (α x ) x∈X ∈ N (X) (see also Subsection 1.2.4).
The behavior of ≻ with respect to the unit 1 A ∈ A is given by (for any 1 A ̸ = x ∈ A)

1 A ≻ x = x, x ≻ 1 A = 0, note that 1 A ≻ 1 A is not defined.
Unital (right) Zinbiel algebras form a category denoted by k-Zinb (the category of unital (right) Zinbiel k-algebras). Moreover, for any (right) Zinbiel algebra A, we can construct a corresponding commutative associative algebra by using the symmetrized product

x * y = x ≻ y + y ≻ x. (3.21)
In other word, (A, * ) is an associative and commutative algebra.

Starting from the paper of M. P. Schüzenberger [START_REF] Schüzenberger | Sur une propriété combinatoire des algèbres de Lie libres pouvant être utilisée dans un problème de mathématiques appliquées, Séminaire Dubreil-Jacotin Pisot (Algèbre et théorie des nombres)[END_REF] in 1958, the author originally introduced the concept of (left) half-shuffle on the free associative algebra k⟨X⟩, it also was studied in the recent works of E. Burgunder [START_REF] Burgunder | A symmetric version of Kontsevich graph complex and Leibniz homology[END_REF], L. Foissy and F. Patras [START_REF] Foissy | Natural endomorphisms of shuffle algebras[END_REF], H.

Nakamura [START_REF] Nakamura | Demi-shuffle duals of Magnus polynomials in free associative algebra[END_REF]. In our context, we now introduce a notion of (right) half-shuffle on k⟨X⟩ as the linear extension of the binary product on words given by

(x 1 • • • x p ) r ¡ (x p+1 • • • x n ) = (x 1 • • • x p ¡ x p+1 • • • x n-1 )x n , 1 X * r ¡ (x p+1 • • • x n ) = x p+1 • • • x n , (x 1 • • • x p ) r ¡ 1 X * = 0.
In terms of the (right) half-shuffle product, we can easily verify without difficulty that (k⟨X⟩, r ¡,1 X * ) is a unital (right) Zinbiel algebra. Moreover, by the symmetric product

x * y = x r ¡ y + y r ¡ x = x ¡ y, the commutative algebra associated to the unital CHAPTER 3. CHARACTERS Zinbiel algebra (k⟨X⟩, r ¡,1 X * ) is the unital shuffle algebra (k⟨X⟩, ¡,1 X * ).

Before introducing the Magnus's duality theorem on the free associative algebra (k⟨X⟩, conc, 1 X * ) and the Zinbiel algebra (k⟨X⟩, r ¡,1 X * ), we proceed to the following propositions that are necessary for describing our general picture how they arise.

Proposition 3.7. Let us consider k⟨X⟩Z ⊕ k.1 X * ∼ = k⟨B * Z⟩ (resp. k⟨X⟩ B ∼ = k⟨B⟩)
the subalgebra of the free associative algebra k⟨X⟩ generated by B * Z ∪ {1 X * } (resp.

B * ). Then i) k⟨X⟩ B and k⟨X⟩Z ⊕ k.1 X * are unital shuffle subalgebras of (k⟨X⟩, ¡,1 X * ).
ii) k⟨X⟩ B and k⟨X⟩Z ⊕ k.1 X * are unital Zinbiel subalgebras of (k⟨X⟩, r ¡,1 X * ).

iii) We have k⟨X⟩ = (k⟨X⟩Z ⊕ k.1 X * ) ¡ k⟨X⟩ B , as shuffle algebras.

Proof. It is straightforward to show that k⟨X⟩ B and k⟨X⟩Z ⊕ k.1 X * are close under the shuffle (resp. Zinbiel) product ¡ (resp. r ¡). One derives (i) and (ii). To prove (iii), as a consequence of Radford's results [START_REF] Radford | A natural ring basis for shuffle algebra and an application to group schemes[END_REF], one can write down a basis of any free shuffle algebra in terms of Lyndon words over X, where X = B → + Z is an ordinal sum. This implies that any polynomial P ∈ k⟨X⟩ can be written uniquely as a linear combination of the shuffle of words in B * with polynomials in k⟨X⟩Z ⊕ k.1

X * P = k≥0 P k ¡ w k , P k ∈ k⟨X⟩Z ⊕ k.1 X * and w k ∈ B * , (3.22)
where the families (P k ) k≥0 being finitely supported. We verified our claim and then we omit the obvious proof of the point (iii).

Let us consider multi-homogeneous polynomials of k⟨X⟩Z ⊕k.1 X * arising by Zinbiel product of B * Z, so-call Zinbiel polynomials, defined by

(• • • ((w 1 z i 1 r ¡ w 2 z i 2 ) r ¡ w 3 z i 3 ) r ¡ • • • r ¡ w k-1 z i k-1 ) r ¡ w k z i k , (3.23)
where

k ≥ 0, w 1 , • • • , w k ∈ B * and i 1 , • • • , i k ∈ [1, N ] (if k = 0 then (3.23) will be
denoted by 1 X * ). Henceforth we write simply

w 1 z i 1 r ¡ • • • r ¡ w k z i k instead of (3.23)
(please don't mistake this for an associative expression).

The above formulas lead to the next assertion 3.3. BASES IN DUALITY: ZINBIEL ALGEBRA AND MAGNUS BASIS.

Proposition 3.8. The Zinbiel algebra k⟨X⟩Z ⊕ k.1 X * is a free k-module with a natural monomial basis formed

w 1 z i 1 • • • w k z i k , where w 1 , • • • , w k are words in B * and i 1 , • • • , i k ∈ [1, N ].
Moreover, the Zinbiel algebra has a second k-linear basis formed as

(3.23).
Proof. The first item is trivial. To prove the next statement, one claims that any monomial basis element

w 1 z i 1 • • • w k z i k of the free k-module k⟨X⟩Z ⊕k.1 X * is a k-linear
combination of multi-homogeneous polynomials formed as (3.23). In fact, we now prove the claim by induction on k : if k = 1 then we first observe that w 1 z i 1 = w 1 z i 1 satisfies our claim. Now, for any word

w 1 , • • • , w k , w k+1 in B * , where i 1 , • • • , i k , i k+1 ∈ [1, N ]
then, by the formula (3.22), there is u l ∈ B * and

P l ∈ k⟨X⟩Z ⊕ k.1 X * (∀l ∈ N) such that w 1 z i 1 • • • w k z i k w k+1 = l≥0 P l ¡ u l ,
where (P l ) l≥0 being finitely supported and each

P l ∈ k⟨X⟩Z ⊕ k.1 X * is k-linear com-
bination of multi-homogeneous polynomials arising by Zinbiel product (the right half-

shuffle) of B * Z v 1 z i 1 r ¡ • • • r ¡ v k z i k , where v 1 , • • • , v k are words in B *
by inductive hypothesis. Therefore, let us consider the concatenation product of the above equation with the letter z i k+1 on the right, one has

w 1 z i 1 w 2 z i 2 • • • w k z i k w k+1 z i k+1 = l≥0 P l ¡ u l z i k+1 = l≥0 P l r ¡ u l z i k+1
and so is a k-linear combination of multi-homogeneous polynomials formed as (3.23) because in general in terms of the non associative identity (3.2) on Zinbiel product, it is [START_REF] Chen | Iterated path integrals[END_REF]. We verified our claim. On the other hand, as may be easily verified, under the totally ordered set X splits as

straightforward that for any v 1 , • • • , v k , v k+1 , • • • , v n ∈ B * and i 1 , • • • , i n ∈ [1, N ] then (v 1 z i 1 r ¡ • • • r ¡ v k z i k ) r ¡ (v k+1 z i k+1 r ¡ • • • r ¡ v n z in ) is a k-linear combination of elements (3.
X = B → + Z, the largest monomial in w 1 z i 1 r ¡ • • • r ¡ w k z i k
will be the following

w 1 z i 1 • • • w k z i k , (3.24)
CHAPTER 3. CHARACTERS they will occur with coefficient 1. It is obvious that distinct elements in (3.23) have distinct largest terms (3.24), hence the linearly independent of the elements (3.23) is then verified by taking and evaluating any k-linear expression of elements (3.23). We then deduce our result.

As a consequence of Proposition 3.7 and Proposition 3.8, the collection of Zinbiel polynomials (w

1 z i 1 r ¡ • • • r ¡ w k z i k ) ¡ w (3.25) (where k ≥ 0, w 1 , • • • , w k , w ∈ B * and i 1 , • • • , i k ∈ [1, N ]) is a k-linear basis of (k⟨X⟩, r ¡,1 X * ).
The next theorem shows that for the descriptions of the above propositions it is sufficient to consider the duality of the Magnus basis in the free associative algebra (k⟨X⟩, conc, 1 X * ).

Theorem 3.9. (Magnus duality, cf. Nakamura [START_REF] Nakamura | Demi-shuffle duals of Magnus polynomials in free associative algebra[END_REF]) The collections of Magnus polynomials (3.17 Remark 3.1. It is interesting to consider the following useful discussions i). Our main technical idea to solve the above theorem is similar to the tool of constructing and proving the PBW basis P w and its dual S w (w words on X) presented in Reutenauer's book [START_REF] Reutenauer | Free Lie Algebras[END_REF] §5.2. But, our approach is in generalized graded bialgebra type structures studied in Appendix 5.4.5.

ii). We survey briefly in the Reutenauer's book [START_REF] Reutenauer | Free Lie Algebras[END_REF] §1.5 (more precisely, see Appendix 5.4.4) that the k-module k⟨X⟩ of the noncommutative polynomials has two natural graded Hopf algebra structures which are dual to each other (in the graded sense), that is H conc (X) = (k⟨X⟩, conc, 1 X * , ∆ ¡ , ϵ) and its graded dual

H ¡ (X) = (k⟨X⟩, ¡,1 X * , ∆ conc , ϵ).
iii) Our approach appeared in Appendix 5.4.5 is indeed the (right) Zinbiel bialgebra Z r ¡ (X) := (k⟨X⟩, r ¡,1 X * , ∆ conc , ϵ) and its graded dual the Zinb c -As-bialgebra CHAPTER 3. CHARACTERS Lemma 3.11. For any polynomials T 1 , T 2 ∈ k⟨X⟩Z and double polynomial P =

m i=1 P i 1 ⊗P i 2 ∈ k⟨X⟩⊗k⟨X⟩, then ⟨T 1 ⊗T 2 | P.∆ r ¡ [rn(wz)]⟩ = ⟨T 1 ⊗T 2 | P.[1⊗rn(wz)]⟩
for each word w over the alphabet B.

Proof. If w = 1 X * then ⟨T 1 ⊗ T 2 | P.∆ r ¡ [rn(wz)]⟩ = ⟨T 1 ⊗ T 2 | P.∆ r ¡ (z)⟩ = ⟨T 1 ⊗ T 2 | P.(1 ⊗ z)⟩ by the definition of ∆ r
¡ in Appendix 5.4.5. Assume that for any word w

over B of length k, then ⟨T 1 ⊗ T 2 | P.∆ r ¡ [rn(wz)]⟩ = ⟨T 1 ⊗ T 2 | P.[1 ⊗ rn(wz)]⟩ for any polynomials T 1 , T 2 ∈ k⟨X⟩Z and double polynomial P = m i=1 P i 1 ⊗P i 2 ∈ k⟨X⟩⊗k⟨X⟩. Now, in case w = b m 1 • • • b m k b m k+1 ∈ B * , where m 1 , • • • , m k+1 ∈ [1, M ], it follows from the compatible relation of ∆ r ¡ in Appendix 5.4.5 that one has ∆ r ¡ [rn(wz)] = ∆ r ¡ [b m 1 .rn(b m 2 • • • b m k+1 z) -rn(b m 2 • • • b m k+1 z).b m 1 ] = ∆ ¡ (b m 1 )∆ r ¡ [rn(b m 2 • • • b m k+1 z)] -∆ ¡ [rn(b m 2 • • • b m k+1 z)]∆ r ¡ (b m 1 ) = (b m 1 ⊗ 1 + 1 ⊗ b m 1 )∆ r ¡ [rn(b m 2 • • • b m k+1 z)] -∆ ¡ [rn(b m 2 • • • b m k+1 z)](1 ⊗ b m 1 ) = (b m 1 ⊗ 1 + 1 ⊗ b m 1 )∆ r ¡ [rn(b m 2 • • • b m k+1 z)] -[rn(b m 2 • • • b m k+1 z) ⊗ b m 1 + 1 ⊗ rn(b m 2 • • • b m k+1 z)b m 1 ].
Thus, by the above formula and the inductive hypothesis, we arrive at the following

⟨T 1 ⊗ T 2 | P.∆ r ¡ [rn(wz)]⟩ = ⟨T 1 ⊗ T 2 | P.(b m 1 ⊗ 1 + 1 ⊗ b m 1 )∆ r ¡ [rn(b m 2 • • • b m k+1 z)]⟩ -⟨T 1 ⊗ T 2 | P.[rn(b m 2 • • • b m k+1 z) ⊗ b m 1 ]⟩ -⟨T 1 ⊗ T 2 | P.[1 ⊗ rn(b m 2 • • • b m k+1 z)b m 1 ]⟩ = ⟨T 1 ⊗ T 2 | P.(b m 1 ⊗ 1 + 1 ⊗ b m 1 )[1 ⊗ rn(b m 2 • • • b m k+1 z)]⟩ -⟨T 1 ⊗ T 2 | P.[rn(b m 2 • • • b m k+1 z) ⊗ b m 1 ]⟩ -⟨T 1 ⊗ T 2 | P.[1 ⊗ rn(b m 2 • • • b m k+1 z)b m 1 ]⟩ = ⟨T 1 ⊗ T 2 | P.[b m 1 ⊗ rn(b m 2 • • • b m k+1 z)]⟩ -⟨T 1 ⊗ T 2 | P.[rn(b m 2 • • • b m k+1 z) ⊗ b m 1 ]⟩ + ⟨T 1 ⊗ T 2 | P.[1 ⊗ rn(b m 1 b m 2 • • • b m k+1 z)]⟩ = ⟨T 1 ⊗ T 2 | P.[1 ⊗ rn(b m 1 b m 2 • • • b m k+1 z)]⟩,
where we obtained

⟨T 1 ⊗ T 2 | P.[b m 1 ⊗ rn(b m 2 • • • b m k+1 z)]⟩ = m i=1 ⟨T 1 ⊗ T 2 | P i 1 .b m 1 ⊗ P i 2 .rn(b m 2 • • • b m k+1 z
)⟩ = 0 by using T 1 ∈ k⟨X⟩Z, and similarly one has

⟨T 1 ⊗ T 2 | P.[rn(b m 2 • • • b m k+1 z) ⊗ b m 1 ]⟩ = 0.
We verified our lemma.

BASES IN DUALITY: ZINBIEL ALGEBRA AND MAGNUS BASIS.

End of the proof of Proposition 3.10. -By the above lemma, one has

⟨[w 1 z r ¡ • • • r ¡ w k-1 z] ⊗ w k z | ∆ ¡ [rn(u 1 z) • • • rn(u k-1 z)]∆ r ¡ [rn(u k z)]⟩ = ⟨[w 1 z r ¡ • • • r ¡ w k-1 z] ⊗ w k z | ∆ ¡ [rn(u 1 z) • • • rn(u k-1 z)][1 ⊗ rn(u k z)]⟩.
Thus, from the equation above and a standard formula ⟨w k z | rn(u k z).T ⟩ = 0 for each non empty monomial T of the set {rn(u i z)} 1≤i≤k-1 , we arrive at the following computational processes

⟨w 1 z r ¡ • • • r ¡ w k z | rn(u 1 z) • • • rn(u k z)⟩ = ⟨[w 1 z r ¡ • • • r ¡ w k-1 z] ⊗ w k z | ∆ ¡ [rn(u 1 z) • • • rn(u k-1 z)][1 ⊗ rn(u k z)]⟩ = ⟨[w 1 z r ¡ • • • r ¡ w k-1 z] ⊗ w k z | [rn(u 1 z) ⊗ 1 + 1 ⊗ rn(u 1 z)] • • • [rn(u k-1 z) ⊗ 1 + 1 ⊗ rn(u k-1 z)][1 ⊗ rn(u k z)]⟩ = ⟨[w 1 z r ¡ • • • r ¡ w k-1 z] ⊗ w k z | rn(u 1 z) • • • rn(u k-1 z) ⊗ rn(u k z)⟩ = ⟨w 1 z r ¡ • • • r ¡ w k-1 z | rn(u 1 z) • • • rn(u k-1 z)⟩.⟨w k z | rn(u k z)⟩ = ⟨w 1 z r ¡ • • • r ¡ w k-1 z | rn(u 1 z) • • • rn(u k-1 z)⟩.δ u k w k = δ u 1 w 1 • • • δ u k-1 w k-1 δ u k w k = δ (u 1 ,••• ,u k ) (w 1 ,••• ,w k )
(by the inductive progresses) .

Moreover, we now state to the general case

B = {b 1 , • • • , b M } and Z = {z 1 , • • • , z N }.
Let k be a positive integer. For convenience, we will borrow and extend some of the notations and techniques appeared in Nakamura's paper [START_REF] Nakamura | Demi-shuffle duals of Magnus polynomials in free associative algebra[END_REF] which the author used to prove the Magnus duality in case X = B + Z where B = {x 0 } and Z = {x λ } λ∈Λ (Λ: a nonempty index set). We now denote ι

= (i 1 , • • • , i k ) ∈ [1, N ] k and for w 1 , • • • , w k , w ∈ B * = {b 1 , • • • , b M } * we shall denote P (ι) (w 1 ,••• ,w k ) = rn(w 1 z i 1 ) • • • rn(w k z i k ), S (ι)
(w 1 ,••• ,w k ) = w 1 z i 1 r ¡ • • • r ¡ w k z i k , w (ι)
(w 1 ,••• ,w k ,w) = w 1 z i 1 • • • w k z i k .w.
We now prove that the pairing ⟨S

(ι) (w 1 ,••• ,w k ) | P (λ) (u 1 ,••• ,u h ) ⟩ is equal to the Kronecker delta δ (λ;u 1 ,••• ,u h ) (ι;w 1 ,••• ,w k ) , where ι = (i 1 , • • • , i k ) ∈ [1, N ] k , w 1 , • • • w k ∈ B * and λ = (λ 1 , • • • , λ h ) ∈ [1, N ] h , u 1 , • • • , u h ∈ B * . Given a fix ι = (i 1 , • • • , i k ) ∈ [1, N ] k . Let us consider V ι a
submodule of k⟨X⟩ generated by the monomials {w

(ι) (w 1 ,••• ,w k ,w) : w 1 , • • • , w k , w ∈ B * }. We easily observe that if ι = (i 1 , • • • , i k ) ̸ = λ = (λ 1 , • • • , λ h ) then V ι and V λ are orthogonal under the standard pairing ⟨• | •⟩. One also observes that • Given a fix λ, then P (λ) (u 1 ,••• ,u h ) ∈ V λ , for any u 1 , • • • , u h ∈ B * . • Given a fix ι, then S (ι) (w 1 ,••• ,w k ) ∈ V ι , for any w 1 , • • • , w k ∈ B * . Thus, if ι ̸ = λ, then ⟨S (ι) (w 1 ,••• ,w k ) | P (λ) (u 1 ,••• ,u h ) ⟩ = 0 = δ (λ;u 1 ,••• ,u h ) (ι;w 1 ,••• ,w k ) . We only consider the case ι = λ = (i 1 , • • • , i k )
. We now introduce the following notations

P (w 1 ,••• ,w k ) = rn(w 1 z) • • • rn(w k z), S (w 1 ,••• ,w k ) = w 1 z r ¡ • • • r ¡ w k z, w (w 1 ,••• ,w k ,w) = w 1 z • • • w k z.w,
and then V k a submodule of k⟨B, {z}⟩ generated by the monomials {w (w 1 ,••• ,w k ,w) :

w 1 , • • • , w k , w ∈ B * }. The mapping ϕ ι : V k → V ι , w (w 1 ,••• ,w k ,w) → w (ι) (w 1 ,••• ,w k ,w
) is an isomorphism of k-modules and preserves the standard pairing ⟨• | •⟩ (for this tool, see [START_REF] Nakamura | Demi-shuffle duals of Magnus polynomials in free associative algebra[END_REF] Thm 3.2). Thus, one has the following

⟨S (ι) (w 1 ,••• ,w k ) | P (ι) (u 1 ,••• ,u k ) ⟩ = ⟨ϕ ι (S (w 1 ,••• ,w k ) ) | ϕ ι (P (u 1 ,••• ,u k ) )⟩ = ⟨S (w 1 ,••• ,w k ) | P (u 1 ,••• ,u k ) ⟩ = ⟨w 1 z r ¡ • • • r ¡ w k z | rn(u 1 z) • • • rn(u k z)⟩ = δ (u 1 ,••• ,u k ) (w 1 ,••• ,w k ) = δ (ι;u 1 ,••• ,u k ) (ι;w 1 ,••• ,w k ) .
Consequently, we verified the "Conc & Half shuffle duality" that means ⟨S

(ι) (w 1 ,••• ,w k ) | P (λ) (u 1 ,••• ,u h ) ⟩ = δ (λ;u 1 ,••• ,u h ) (ι;w 1 ,••• ,w k ) . The proof is complete. QED Proof of Theorem 3.9 : We recall the graded set X = B + Z, where B = {b 1 , • • • , b M } and Z = {z 1 , • • • , z N }.
Let k be a positive integer. To approach the notations already used when proving Proposition 3.10, although in a more sophisticated

way, if ι = (i 1 , • • • , i k ) ∈ [1, N ] k and for each w 1 , • • • , w k , w ∈ B * = {b 1 , • • • , b M } * , we consider useful notations P (ι) (w 1 ,••• ,w k ,w) = rn(w 1 z i 1 ) • • • rn(w k z i k ).w,
(ι) (w 1 ,••• ,w k ,w) = (w 1 z i 1 r ¡ • • • r ¡ w k z i k ) ¡ w, w (ι) (w 1 ,••• ,w k ,w) = w 1 z i 1 • • • w k z i k .w.
We now prove that ⟨S

(ι) (w 1 ,••• ,w k ,w) | P (λ) (u 1 ,••• ,u h ,u) ⟩ is equal to the Kronecker delta δ (λ;u 1 ,••• ,u h ,u) (ι;w 1 ,••• ,w k ,w) . Given a fix ι = (i 1 , • • • , i k ) ∈ [1, N ] k .
According to the proof of Proposition 3.10, let us recall V ι that is a submodule of k⟨X⟩ generated by the monomials {w

(ι) (w 1 ,••• ,w k ,w) : w 1 , • • • , w k , w ∈ B * }. We also recall that if ι = (i 1 , • • • , i k ) ̸ = λ = (λ 1 , • • • , λ h ) then
V ι and V λ are orthogonal under the standard pairing ⟨• | •⟩. Then, we observe certain properties w) . We finally consider the case ι = λ = (i 1 , • • • , i k ). According to the adjoint ⟨T ¡ P | Q⟩ = ⟨T ⊗ P | ∆ ¡ (Q)⟩ (T, P, Q are polynomials) successfully achieved from constructing the graded Hopf algebra H conc (X) = (k⟨X⟩, conc, 1 X * , ∆ ¡ , ϵ) and its graded dual H ¡ (X) = (k⟨X⟩, ¡,1 X * , ∆ conc , ϵ) (see Appendix 5.4.4), we shall present

• Given a fix λ, then P (λ) (u 1 ,••• ,u h ,u) ∈ V λ , for any u 1 , • • • , u h , u ∈ B * . • Given a fix ι, then S (ι) (w 1 ,••• ,w k ,w) ∈ V ι , for any w 1 , • • • , w k , w ∈ B * . As a consequence, if ι ̸ = λ, one has ⟨S (ι) (w 1 ,••• ,w k ,w) | P (λ) (u 1 ,••• ,u h ,u) ⟩ = 0 = δ (λ;u 1 ,••• ,u h ,u) (ι;w 1 ,••• ,w k ,
⟨S (ι) (w 1 ,••• ,w k ,w) | P (ι) (u 1 ,••• ,u k ,u) ⟩ = ⟨(w 1 z i 1 r ¡ • • • r ¡ w k z i k ) ¡ w | rn(u 1 z i 1 ) • • • rn(u k z i k ).u⟩ = ⟨w 1 z i 1 r ¡ • • • r ¡ w k z i k ⊗ w | ∆ ¡ rn(u 1 z i 1 ) • • • rn(u k z i k ).u ⟩ = ⟨w 1 z i 1 r ¡ • • • r ¡ w k z i k ⊗ w | [rn(u 1 z i 1 ) ⊗ 1 + 1 ⊗ rn(u 1 z i 1 )] • • • [rn(u k z i k ) ⊗ 1 + 1 ⊗ rn(u k z i k )](b m 1 ⊗ 1 + 1 ⊗ b m 1 ) • • • (b m |u| ⊗ 1 + 1 ⊗ b m |u| )⟩,
where we putted

u = b m 1 • • • b m |u| ∈ B * . It follows from the fact that ⟨T | P ⟩ = 0 if
T ∈ k⟨X⟩Z and P ∈ k⟨X⟩B, after expressing the right-hand side of the last pairing above as a linear sum of

Q 1 ⊗ Q 2 (Q i ∈ k⟨X⟩), we clearly have that the last equality is equal to ⟨w 1 z i 1 r ¡ • • • r ¡ w k z i k ⊗ w | rn(u 1 z i 1 ) • • • rn(u k z i k ) ⊗ u⟩. Then, Proposition
3.10 is used to verify the following equation

⟨w 1 z i 1 r ¡ • • • r ¡ w k z i k ⊗ w | rn(u 1 z i 1 ) • • • rn(u k z i k ) ⊗ u⟩ = ⟨w 1 z i 1 r ¡ • • • r ¡ w k z i k | rn(u 1 z i 1 ) • • • rn(u k z i k )⟩⟨w | u⟩ = δ (u 1 ,••• ,u k ) (w 1 ,••• ,w k ) δ u w = δ (u 1 ,••• ,u k ,u) (w 1 ,••• ,w k ,w) = δ (ι;u 1 ,••• ,u k ,u) (ι;w 1 ,••• ,w k ,w) ,
and then ⟨S w) . We give a complete proof of the Magnus's duality and then our theorem. □

(ι) (w 1 ,••• ,w k ,w) | P (ι) (u 1 ,••• ,u k ,u) ⟩ = δ (ι;u 1 ,••• ,u k ,u) (ι;w 1 ,••• ,w k ,

Extension of characters: A theory of Domains

for Harmonic Functions and its Symbolic Counterpart.

In this section, we begin by reviewing the calculus induced by the framework of [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF].

In there, we extended Polylogarithm functions over a subalgebra of noncommutative rational power series, recognizable by finite state (multiplicity) automata over the alphabet X = {x 0 , x 1 }. The stability of this calculus under shuffle products relies on the nuclearity of the target space [START_REF] Schaefer | Topological Vector Spaces[END_REF]. We also concentrated on algebraic and analytic aspects of this extension allowing to index polylogarithms, at non positive multi-indices, by rational series and also allowing to regularize divergent polyzetas, at non positive multi-indices [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF]. As a continuation of works in [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF] and in order to understand the bridge between the extension of this "polylogarithmic calculus" and the world of harmonic sums, we propose a local theory, adapted to a full calculus on indices of Harmonic Sums based on the Taylor expansions, around zero, of polylogarithms with index x 1 on the rightmost end. This theory is not only compatible with Stuffle products but also with the Analytic Model. In this respect, it provides a stable and fully algorithmic model for Harmonic calculus. Examples by computer are also provided.

Introduction.

Riemann's zeta function is defined by the series

ζ(s) := n≥1 1 n s (3.26)
where s is a complex number. It is absolutely convergent for ℜ(s) > 1 (for any s ∈ C, ℜ(s) stands for the real part of s).

It can be extended to a meromorphic function on the complex plane C with a single pole at s = 1 [START_REF] Riemann | Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie[END_REF]

ζ(s 1 , • • • , s r ) := n 1 >•••>nr≥1 1 n s 1 1 • • • n sr r . (3.30)
Then H. Furusho et al. [START_REF] Furusho | Desingularization of multiple zetafunctions of generalized Hurwitz-Lerch type[END_REF] showed that the series [START_REF] Goncharov | Multiple polylogarithms and mixed Tate motives[END_REF][START_REF] Zhao | Analytic continuation of multiple zeta functions[END_REF]).

Then, for any r-uplet

(s 1 , • • • , s r ) ∈ N r + ∩ H r , r ∈ N + i.e.
with s 1 ≥ 2, the polyzeta ζ(s 1 , • • • , s r ) is also the limit at z = 1 of the polylogarithmic function, defined by:

Li s 1 ,••• ,sr (z) := n 1 >•••>nr>0 z n 1 n s 1 1 • • • n sr r (3.32)
for any z ∈ C such that | z |< 1. It is easily seen that, for any 16 In fact, in Euler's formula, s 1 , s 2 ∈ N + . This identity appeared under the name "Prima Methodus ..." (see [START_REF] Euler | Meditationes circa singulare serierum genus[END_REF] pp 141-144). 17 Multiple Zeta Values.

s i ∈ N + , r > 1, z d dz Li s 1 ,••• ,sr (z) = Li s 1 -1,••• ,sr (z) if s 1 > 1

EXTENSION OF CHARACTERS:

A THEORY OF DOMAINS FOR HARMONIC FUNCTIONS AND ITS SYMBOLIC COUNTERPART.

(

1 -z) d dz Li 1,s 2 ,••• ,sr (z) = Li s 2 ,••• ,sr (z) if r > 1 (3.33)
and this formulas will be ended at the "seed" Li 1 (z) = log 1 1 -z .

Moreover, if X * is the free monoid of rank two (generators, or the alphabet, X = {x 0 , x 1 } and the neutral 1 X * ) then the polylogarithms indexed by a list

(s 1 , • • • , s r ) ∈ N r
+ can be reindexed by the word x s 1 -1

0 x 1 • • • x sr-1 0 x 1 ∈ X * x 1 (3.34)
In order to reverse the recursion introduced in Equations (3.33) (two equations), we introduce two differential forms

ω 0 (z) = z -1 dz and ω 1 (z) = (1 -z) -1 dz, (3.35)
on Ω 18 . We then get an integral representation 19 of the functions (3.32) as follows 20 (see Def 3.2 [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF] and Figure 3.1) This provides not only the analytic continuation of (3.32) to Ω but also extends the indexation to the whole monoid X * , allowing to study the complete generating series

Li w (z) =                    1 H(Ω) if w = 1 X * z 0 ω 1 (s) Li u (s) if w = x 1 u
L(z) = w∈X * Li w (z)w (3.37)
18 Ω is the simply-connected domain C \ (] -∞, 0] ∪ [1, +∞[). 19 In here, we code the moves z d dz (resp. z 0 f (s) 1 -s ds) -with x 0 (resp. x 1 ). 20 Given a word w ∈ X * , we note |w| x1 the number of occurrences of x 1 within w. and show that it is the solution of the following first order noncommutative differential system (see [START_REF] Drinfeld | On quasitriangular quasi-hopf algebra and a group closely connected with Gal(Q/Q)[END_REF]) This differential system allows to show that L is a ¡-character21 i.e. In [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF], it has been established that the polylogarithm, well defined locally by (3.32), could be extended to some series (with conditions) by the last part of formula (3.41) where the polynomial P is replaced by some series. A complete theory of global domains was presented in [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF], the present work concerns the whole project of extending H • [START_REF] Duchamp | Harmonic sums and polylogarithms at negative multi-indices[END_REF][START_REF] Hoffman | Multiple harmonic series[END_REF] over stuffle subalgebras of rational power series on the alphabet Y , in particular the stars of letters and some explicit combinatorial consequences of this extension.

1 X * x 0 x 2 0 x 3 0 x 1 x 2 0 x 1 x 0 x 0 x 1 x 0 x 2 1 x 0 x 1 x 0 x 1 x 2 0 x 1 x 1 x 0 x 1 x 2 1 x 0 x 2
   d(S) = (ω 0 (z)x 0 + ω 1 (z)x 1 )S, ( N
∀u, v ∈ X * , ⟨ L | u ¡ v⟩ = ⟨ L | u⟩⟨ L | v⟩ and ⟨ L | 1 X * ⟩ = 1 H(Ω) . (3.39) ⟨ L | x n 0 ⟩ = log(z) n n! ; ⟨ L | x n 1 ⟩ = (-log(1 -z)) n n! ⟨ L | x 0 x 1 ⟩ = Li 2 (z) = n≥1 z n n 2 ; ⟨ L | x 1 x 0 ⟩ = ⟨ L | x 1 ¡x 0 -x 0 x 1 ⟩(z) ⟨ L | x 2 0 x 1 ⟩ = Li 3 (z) = n≥1 z n n 3 ; ⟨ L | x 1 x 0 ⟩ = (-log(1 -z)) log(z) -Li 2 (z) ⟨ L | x r-1 0 x 1 ⟩ = Li r (z) = n≥1 z n n r ; ⟨ L | x 2 1 x 0 ⟩ = ⟨ L | 1 2 (x 1 ¡x 1 ¡x 0 ) -(x 1 ¡x 0 x 1 ) + x 0 x 2 1 ⟩.
In fact, we focus on what happens in (well chosen) neighborhoods of zero (see Section 3.4.3), therefore, the aim of this work is manyfold.

(a) Use the extension to local Taylor expansions 22 as in (3.32) and the coefficients of their quotients by 1 -z, namely the harmonic sums, denoted H • and defined, for any w ∈ X * x 1 , as follows 23 (and also related literature [3,[START_REF] Hoffman | Multiple harmonic series[END_REF])

Li w (z) 1 -z = N ≥0 H π Y (w) (N )z N , (3.42)
by a suitable theory of local domains which assures to carry over the computation of these Taylor coefficients and preserves the stuffle indentity, again true for polynomials over the alphabet Y = {y n } n≥1 , i.e. 24 ∀S, T ∈ C⟨Y ⟩,

H S T = H S H T and H 1 C⟨Y ⟩ = 1 C N , (3.43)
note that 1 C⟨Y ⟩ is identified with 1 Y * and 1 C N is the constant (to one) function 25 N → C. This means that

H • : (C⟨Y ⟩, , 1 Y * ) -→ (C{H w } w∈Y * , ×, 1)
22 Around zero. 23 Here, the conc-morphism π X : (C⟨Y ⟩, conc, 1 Y * ) → (C⟨X⟩, conc, 1 X * ) is defined by π X (y n ) =

x n-1 0

x 1 and π Y is its inverse on Im(π X ). See [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF] for more details and a full definition of π Y . 24 Here, stands for the stuffle product which will be recalled as in the subsection 3.4.5. 25 In fact, it could be Q but we will use afterwards C-linear combinations.

mapping any word w

= y s 1 • • • y sr ∈ Y * to H w = H s 1 ,••• ,sr = N ≥n 1 >•••>nr>0 1 n s 1 1 • • • n sr r (3.44)
is a (unital) morphism 26 .

(b) Extend these correspondences (i.e. Li • , H • ) to some series (over X and Y , respectively) in order to preserve the identity 27Li π X (S) (z)

1 -z ⊙ Li π X (T ) (z) 1 -z = Li π X (S T ) (z) 1 -z (3.45)
true for polynomials S, T ∈ C⟨Y ⟩.

To this end, we use the explicit parametrization of the conc-characters obtained in [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF] and the fact that, under the stuffle products, they form a group.

Polylogarithms: from global to local domains.

Now we are facing the following constraint:

In order that the results given by symbolic computation reflect the reality with complex numbers (and analytic functions), we have to introduce some topology 28 .

Let H(Ω) = C ω (Ω; C) be the algebra (for the pointwise product) of complex-valued functions which are holomorphic on Ω. Endowed with the topology of compact convergence 29 , it is a nuclear space is summable in H(Ω) in other words, due to the fact that the space is complete (see [START_REF] Schaefer | Topological Vector Spaces[END_REF]), if and only if one has

(∀ W ∈ B H(Ω) )(∃F ⊂ f inite N)(∀F ′ ⊂ f inite (N \ F )), j∈F ′ Li S j ∈ W (3.46)
where B H(Ω) is the set of neighbourhoods of 0 in H(Ω).

(ii) The set of these series will be noted Dom(Li) and, for S ∈ Dom(Li), the sum n≥0 Li Sn will be noted Li S .

Of course, the criterium (3.46) is only a theoretical tool to establish properties of the domain of Li. In further calculations (i.e. in practice), we will not use it but the stability of the domain under certain operations.

(z) = n≥1 z n n k = Li x k-1 0 x 1 (z) = ⟨ L(z) | x k-1 0 x 1 ⟩
(where L(z) is as in Equation (3.37)) but, one can check that, for t ≥ 0 (real), the series (tx 0 ) * x 1 belongs to Dom(Li • ) (see Definition 3.3. (ii)) iff 0 ≤ t < 1. In fact, in this case,

Li (tx 0 ) * x 1 (z) = n≥1 z n n -t .
This opens the door to Hurwitz polyzetas [START_REF] Hoang Ngoc Minh | De l'algèbre des ζ de Riemann multivariées à l'algèbre des ζ de Hurwitz multivariées[END_REF].

The map Li • is now extended to a subdomain of C⟨⟨X⟩⟩, called Dom(Li • ) (see Def.

3.3, and also [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF]).

Example 3.5. For any α, β ∈ C, (αx 0 ) * , (βx 1 ) * and (αx 0 + βx 1 ) * = (αx 0 ) * ¡ (βx 1 ) * .

We have

Li (αx 0 ) * (z) = z α ; Li (βx 1 ) * (z) = (1 -z) -β ; Li (αx 0 +βx 1 ) * (z) = z α (1 -z) -β
where z ∈ Ω.

Proposition 3.12. (i) The domain Dom(Li) is a shuffle subalgebra of (C⟨⟨X⟩⟩, ¡,1 X * ). (3.47)

Proof. This proof has been done in [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF].

The picture about Dom(Li) within the algebra (C⟨⟨X⟩⟩, ¡,1 X * ), the positioning of C rat ⟨⟨X⟩⟩ (rational series, see [START_REF] Berstel | Rational series and their languages[END_REF][START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF]) and shuffle subalegbras as, for example, A = C⟨X⟩ ¡ C rat ⟨⟨x 0 ⟩⟩ ¡ C rat ⟨⟨x 1 ⟩⟩ read as follows: 1. The following conditions are equivalent (i) f can be analytically extended around zero.

(ii

) P ∈ C⟨X⟩x 1 ⊕ C.1 X * .
2. In this case Ω itself32 can be extended to

Ω 1 = C \ (] -∞, -1] ∪ [1, +∞[).
Sketch. (ii) =⇒ (i) being straightforward, it remains to prove that (ii) =⇒ (i). Let then P ∈ C⟨X⟩ such that f (z) = ⟨ L | P ⟩ has a removable singularity at zero. As a consequence of Proposition 3.7 point (iii) for the set partitioned X = x 0 + x 1 , this implies that our polynomial reads We can see that only the term with k = 0 survives using monodromy, for example, as follows.

P = k≥0 P k ¡ x ¡k 0 with P k ∈ C⟨X⟩x 1 ⊕ C.1 X * (3.
We suppose that Li P (z) = k≥0 Li P k (z) log(z) k can be analytically extended in a neighbourhood of zero (say B r (0)). Let z ∈ Ω ∩ B r (0). Using the path γ n (t) = z.e 2inπ.t (starting and ending at z winding n times around zero), we get Li P (z) = Li P 0 (z) + k≥1 Li P k (z)(log(z) + 2inπ) k for all n ∈ Z which entails Li P k (z) = 0 for all k ≥ 1. This holds for all z ∈ Ω ∩ B r (0) and hence we must have P k = 0 for all k ≥ 1.

The second step will be provided by the following Proposition which says that, for appropriate series, the Taylor coefficients behave nicely. is unconditionally convergent in the same disk, we set

1 1 -z n≥0 Li [S]n (z) = N ≥0 a N z N .
Then, for all N ≥ 0, 33 With the definition given later (3.49) this amounts to say that

⟨S | w⟩ N ≥0 H π Y (w) (N )z N ( * ) = N ≥0 n≥0 w∈X * ,|w|=n ⟨S | w⟩H π Y (w) (N )z N = N ≥0 n≥0 H π Y ([S]n) (N )z N ,
S ∈ C⟨⟨X⟩⟩x 1 ⊕ C1 X * ∩ Dom R (Li) .
34 For this topology, unconditional and absolute convergence coincide [START_REF] Schaefer | Topological Vector Spaces[END_REF]. is finite. This implies that, for any N ≥ 0,

a N = n≥0 H π Y ([S]n) (N ).
To prepare the construction of the "symbolic local germ" around zero, let us set, in the same manner as in [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF],

Dom R (Li) := {S ∈ C⟨⟨X⟩⟩x 1 ⊕ C1 X * | n≥0 Li [S]n is unconditionally convergent in H(D <R )} (3.49)
and prove the following:

Proposition 3.15. With the notations as above, we have:

1. The map given by R → Dom R (Li) from ]0, 1] to 2 C⟨⟨X⟩⟩ (the target is the set of subsets 35 of C⟨⟨X⟩⟩ ordered by inclusion) is strictly decreasing 2. Each Dom R (Li) is a shuffle (unital) subalgebra of C⟨⟨X⟩⟩.

Proof.

1. For 0 < R 1 < R 2 ≤ 1 it is straightforward that Dom R 2 (Li) ⊂ Dom R 1 (Li).
Let us prove that the inclusion is strict. Take | z |< 1 and let us, be it finite or infinite, evaluate the sum

M (z) = n≥0 | Li [S]n(t) (z) |= n≥0 ⟨S(t) | x n 1 ⟩ | Li x n 1 (z) |
then, by means of Lemma 3.16, with

x + 1 = x 1 x * 1 = x * 1 -1 and S(t) = m≥0 t m (x + 1 ) ¡m , we have M (z) = n≥0 | S(t) | x n 1 | Li x n 1 (z) |= n≥0 m≥0 | t m (x + 1 ) ¡m | x n 1 | Li x n 1 (z) | = m≥0 m!t m n≥0 S 2 (n, m) | Li x 1 (z) | n n! ≤ m≥0 m!t m n≥0 S 2 (n, m) Li n x 1 (| z |) n! , due to the fact that | Li x 1 (z) |≤ Li x 1 (| z |) (Taylor series with positive coefficients).
Finally, in view of equation (3.52), we get, on the one hand, for | z |< (t + 1) -1 ,

M (z) ≤ m≥0 t m (e Lix 1 (|z|) -1) m = m≥0 t m ( | z | 1-| z | ) m = 1-| z | 1 -(t + 1) | z | . CHAPTER 3. CHARACTERS
This proves that, for all r ∈]0,

1 t + 1 [, n≥0 ∥ Li [S]n(t) (z)∥ r < +∞.
On the other hand, if (t+1) -1 ≤| z |< 1, one has M (|z|) = +∞, and the preceding calculation shows that, with t choosen such that

0 ≤ 1 R 2 -1 < t < 1 R 1 -1, we have S(t) ∈ Dom R 1 (Li) but S(t) / ∈ Dom R 2 (Li) whence, for 0 < R 1 < R 2 ≤ 1, Dom R 2 (Li) ⊊ Dom R 1 (Li).
2. One has (proofs as in [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF])

(a) 1 X * ∈ Dom R (Li) (because 1 X * ∈ C⟨X⟩) and Li 1 X * = 1 H(Ω) .
(b) Taking S, T ∈ Dom R (Li) we have, by absolute convergence, S ¡ T ∈ Dom R (Li). It is easily seen that S ¡ T ∈ C⟨⟨X⟩⟩x 1 ⊕ C1 X * and, moreover, that 36 Li S Li T = Li S¡T .

The combinatorial Lemma needed in the Theorem 3.18 is the following: Lemma 3.16. For a letter "a", one has

(a + ) ¡m |a n = m!S 2 (n, m), (3.50
m!S 2 (n, m) x n n! = (e x -1) m . (3.51)
Proof. The expression (a + ) ¡m is the specialization of

L m = a + 1 ¡ a + 2 ¡ • • • ¡ a + m to a j → a (for all j = 1, 2 • • • m).
The words of L m are in bijection with the surjections

[1 • • • n] → [1 • • • m],
therefore the coefficient ⟨(a + ) ¡m |a n ⟩ is exactly the number of such 36 Proof by absolute convergence as in [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF]. Then I(τ ) ̸ = ∅ and R(τ ) ≥ R.

Proof.

1. The fact that I(τ ) ⊂]0, +∞[ is straightforward from the Definition. If 2. Let 0 < r < r 1 < R and consider the path γ(t) = r 1 e 2iπt , by Cauchy's formula, we have

there exists z 0 ∈ C \ {0} such that n,N ≥0 | a n,N z N 0 |< +∞ then, for all r ∈]0, |z 0 |[, we have n,N ≥0 | a n,N r N |= n,N ≥0 | a n,N z N 0 | r | z 0 | N ≤ n,N ≥0 |a n,N z N 0 | < +∞ in particular I(τ ) ̸ = ∅
| a n,N |=| 1 2iπ γ T n (z) z N +1 dz |≤ 2π 2π r 1 ∥T n ∥ K r N +1 1 ≤ ∥T n ∥ K r N 1 with K = γ([0, 2π]), hence n,N ≥0 | a n,N r N |≤ n,N ≥0 ∥T n ∥ K ( r r 1 ) N ≤ r 1 r 1 -r n≥0 ∥T n ∥ K < +∞.
Remark 3.3. (i) First point says that every function analytic at zero can be represented around zero as Li S (z) for some S ∈ C⟨⟨x 1 ⟩⟩. a N z N be a Taylor series converging on some non-empty disk centered at zero i.e. such that lim sup

N →+∞ | a N | 1 N = B < +∞, then the series S = N ≥0 a N (-(-x 1 ) + ) ¡N (3.54) is summable in C⟨⟨X⟩⟩ (with sum in C⟨⟨x 1 ⟩⟩), S ∈ Dom R (Li) with R = (B + 1) -1
and Li S = T .

2. Let S ∈ Dom R (Li) and S = n≥0

[S] n (homogeneous decomposition), we define

N -→ H π Y (S) (N ) by 39 Li S (z) 1 -z = N ≥0 H π Y (S) (N )z N . (3.55) Then, ∀r ∈]0, R[, n,N ≥0 | H π Y ([S]n) (N )r N |< +∞. ( 3.56)
In particular, for all N ∈ N, the series (of complex numbers), n≥0

H π Y ([S]n) (N )
converges absolutely to H π Y (S) (N ). Proof.

1. The fact that the series (3.54) is comes from the fact that ω(a N (-(-x 1 ) + ) ¡N ) ≥ N.

Now from the Lemma 3.16, we get

(S) n = N ≥0 (a N (-(-x 1 ) + ) ¡N ) n = (-1) N +n a N N !S 2 (n, N )x n 1 N ≥0 (-1) N +n a N N !S 2 (n, N )x n 1 .
Then, with r = sup z∈K | z | (we have indeed r = ||Id|| K ) and taking into account that

∥ Li x 1 ∥ K ≤ log( 1 1 -r ), we have n≥0 ∥ Li (S)n ∥ K ≤ n≥0 N ≥0 | a N | N !S 2 (n, N )∥ Li x n 1 ∥ K ≤ n≥0 N ≥0 | a N | N !S 2 (n, N ) ∥ Li x 1 ∥ n K n! ≤ N ≥0 | a N | n≥0 N !S 2 (n, N ) ∥ Li x 1 ∥ n K n! ≤ N ≥0 | a N | (e log( 1 1-r ) -1) N = N ≥0 | a N | r 1 -r N . Now, if we suppose that r ≤ (B + 1) -1 , we have r(1 -r) -1 ≤ 1 B
and this shows that the last sum is finite. Moreover, by Lemma 3.16 and Li

x n 1 = log n ( 1 1-z ) n!
| H π Y ([S]n) (N ) |=| 1 2iπ γ Li [S]n (z) (1 -z)z N +1 dz |≤ 2π 2π ∥ Li [S]n ∥ K (1 -r 1 )r N +1 1 , K = γ([0, 1
]) being the circle of center 0 and radius r 1 . Taking into account that, for K ⊂ compact D <R , we have a decomposition

n∈N ∥ Li [S]n ∥ K = M < +∞, we get n,N ≥0 | H π Y ([S]n) (N )r N | = n,N ≥0 | H π Y ([S]n) (N )r N 1 | ( r r 1 ) N = N ≥0 ( r r 1 ) N n≥0 | H π Y ([S]n) (N )r N 1 | ≤ N ≥0 ( r r 1 ) N M (1 -r 1 )r 1 ≤ M (1 -r 1 )(r 1 -r) < +∞. The series n≥0 Li [S]n (z) converges to Li S (z) in H(D <R ) (D <R is the open disk defined by |z| < R).
For any N ≥ 0, by Cauchy's formula, one has,

H π Y (S) (N ) = 1 2iπ γ Li S (z) (1 -z)z N +1 dz = 1 2iπ γ n≥0 Li [S]n (z) (1 -z)z N +1 dz = 1 2iπ n≥0 γ Li [S]n (z) (1 -z)z N +1 dz = n≥0 H π Y ([S]n) (N )
the exchange of sum and integral being due to the compact convergence. The absolute convergence comes from the fact that the convergence of

n≥0 Li [S]n (z) is unconditional [97]. 3. Fixing N ∈ N, from inequation (3.57), we get n≥0 | H Qn (N ) |< +∞ which proves the absolute convergence. Remark now that (π X (Q)) n = π X (Q n ) and π Y (π X (Q n )) = Q n , one has, for all | z |≤ r | z |< r, | Li π X (Qn) (z) |=| 1 -z || N ∈N H Qn (N )z N |≤ 2 | N ∈N H Qn (N )r N | .
Thus, for all K ⊂ compact D <r and z ∈ K, we arrive at

| Li π X (Qn) (z) |≤ 2 | N ∈N H Qn (N )r N |, CHAPTER 3. CHARACTERS in other words, ∥ Li π X (Qn) ∥ K ≤ 2 | N ∈N H Qn (N )r N | and n∈N ∥ Li π X (Qn) ∥ K ≤ 2 | n,N ∈N H Qn (N )r N |< +∞ which shows that π X (Q) ∈ Dom r (Li). The equation (3.58) is a consequence of point 2, taking S = π X (Q).
(Li) = 0<R≤1 Dom R (Li); Dom(H • ) = π Y (Dom loc (Li)) and, for S ∈ Dom loc (Li), Li S (z) = n≥0 Li [S]n (z) and Li S (z) 1 -z = N ≥0 H π Y (S) (N )z N .

Applications.

We remark that formula (3.32), i.e., [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF]). Using Theorem 3.18, we have

Li s 1 ,••• ,sr (z) := n 1 >•••>nr>0 z n 1 n s 1 1 • • • n sr
(Li) ∩ C[x * 1 ] = C[x * 1 ] such that Li -s 1 ,••• ,-sr = Li P (see
Li P (z) 1 -z = N ≥0 H π Y (P ) (N )z N . 3
f (z) := Li S (z) 1 -z ⊙ Li T (z) 1 -z = N ≥0 H π Y (S) (N )H π Y (T ) (N )z N , (3.63)
where ⊙ stands for the Hadamard product [START_REF] Hadamard | Théorème sur les séries entières[END_REF]. Now, due to Theorem 3.18 point

(2), for all N , p≥0 H π Y (Sp) (N ) = H π Y (S) (N ) and q≥0 H π Y (Tq) (N ) = H π Y (T ) (N ) (
H π Y (S) (N )H π Y (T ) (N ) = p≥0 H π Y (Sp) (N ) q≥0 H π Y (Tq) (N ) = p,q≥0 H π Y (Sp) (N )H π Y (Tq) (N ) = n≥0 p+q=n H π Y (Sp) π Y (Tq) (N ) = n≥0 H (π Y (S) π Y (T ))n (N ). (3.64)
Remains to prove that condition of Theorem 3.18, i.e. inequation (3.57) is fulfilled.

To this end, we use the well-known fact that if

∈]0, R 1 .R 2 [, p,q,N ≥0 |H π Y (Sp) (N )H π Y (Tq) (N ) r N | < +∞,
and this assures, for |z| < R 1 R 2 , the convergence of

f (z) = n,N ≥0 H (π Y (S) π Y (T ))n (N )z N . (3.65) Applying Theorem 3.18 point (3) to Q = π Y (S) π Y (T ) (with any r < R 1 R 2 ), we get π X (Q) = π X (π Y (S) π Y (T )) ∈ Dom loc (Li) and f (z) = N ≥0 n≥0 H (π Y (S) π Y (T ))n (N ) z N = Li π X (π Y (S) π Y (T )) (z) 1 -z .
Hence we obtain (3.62) and then (3.61).

EXTENSION OF CHARACTERS:

A THEORY OF DOMAINS FOR HARMONIC FUNCTIONS AND ITS SYMBOLIC COUNTERPART.

• H • from (Dom(H • ), , 1 Y * ) to C N (arithmetic functions N → C).
In general, a character from a k-algebra

α i y i * j≥1 β j y j * = i≥1 α i y i + j≥1 β j y j + i,j≥1 α i β j y i+j * (3.70)
This suggests to take an auxiliary variable, say q, and code "the plane" C.Y , i.e. This shows that if one sets, for z ∈ C and

T ∈ C + [[q]], G(z) = (π Umbra Y (e zT -1)) * ,
we get a one-parameter stuffle group 46 such that every coefficient is polynomial in z.

Differentiating it we get

d dz (G(z)) = (π Umbra Y (T ))G(z) (3.72)
43 Here we will use k = Q or C. 44 In this context all algebras are associative and unital. 45 Its inverse will be naturally noted π Umbra q

. 46 i.e. G(z 1 + z 2 ) = G(z 1 ) G(z 2 ); G(0) = 1 Y * . Lie Group G L(G) (Lie algebra) H(z 0 ) H(z) G(z) c H ′ (z)
G(z) = exp (zπ Umbra Y (T )) (3.73)
where the exponential map for the stuffle product is defined, for any P ∈ C⟨⟨Y ⟩⟩ such that ⟨P | 1 Y * ⟩ = 0, is defined by

exp (P ) := 1 Y * + P 1! + P P 2! + • • • + P n n! + • • • .
In particular, from (3.73), one gets, for k ≥ 1, the identity,

(zy k ) * = exp - n≥1 y nk (-z) n n .
5.1 Miscellaneous facts.

About central extensions.

We have the following well known result about central extensions

= s (0) (since g 2 is abelian) .

Hence, [x, y] = [s (p (x)) , y] = 0.

Factorization of characters.

This section deals with applications of MRS formula (3.6) to the factorization of characters.

As a result of equations (3.4), we have seen that, a Lie k-algebra (free as a module and k being a Q-algebra) g together with a totally ordered basis B = (b i ) i∈I basis of it being given1 , the space span{B α } α∈N (I) is a unital ⋆-subalgebra of U * (g). Let us then set U ∨ (g) := span{B α } α∈N (I) .

(5.5)

We suppose now that 1. g = s∈S g s is a S-graded Lie algebra (where S is an additive commutative semigroup), 2. the basis B = (b i ) i∈I is S-graded (we set deg(b i ) := s for the unique s ∈ S such that b i ∈ g s ), 3. the basis B is graded "in finite ranks" which means that, for all s ∈ S, the set

I s := | {i ∈ I | deg(b i ) = s} | is finite i.e.
U(g), µ U , 1 U , ∆ U , ϵ U , S U ) (5.6) by setting (a) µ U ∨ = ⋆ (the convolution restricted to U ∨ (g)), (b) 1 U ∨ = ϵ U (the counit restricted to U ∨ (g)), (c) ∆ U ∨ : U ∨ (g) → U ∨ (g) ⊗ U ∨ (g) is the dual of µ U by ∆ U ∨ (B α ) = α 1 ,α 2 ∈N (I) ⟨B α | B α 1 B α 2 ⟩B α 1 ⊗ B α 2 (5.7)
due to the hypotheses about the S-grading one can check (exercise) that this sum is finitely supported, 5.1. MISCELLANEOUS FACTS.

(d) ϵ U ∨ = δ 1 the Dirac evaluation which means that, for f

∈ U ∨ (g), ϵ U ∨ (f ) = ⟨f | 1 U ⟩, (e) S U ∨ (f ) = α∈N (I) ⟨f | S U (B α
)⟩B α again, due to the hypotheses about the S-grading one can check (exercise) that this sum is finitely supported, 3. with this at hand, one can check that (U ∨ (g),

µ U ∨ , 1 U ∨ , ∆ U ∨ , ϵ U ∨ , S U ∨ ) is a Hopf algebra.
Remark 5.1. In the case when g is the free Lie algebra

L k (X) = α∈N (X) L k (X) α (1.31) (then the enveloping algebra U(g) = (k⟨X⟩, conc, 1 X * , ∆ ¡ , ϵ, S U )) and {B α } α∈N (I)
is a multihomogeneous basis, we have U ∨ (g) = (k⟨X⟩, ¡,1 X * , ∆ conc , ϵ, S U ∨ ) (due to the fine grading by S = N (X) (1.27)).

Now we have the following factorization of characters Then we can consider U ∨ (g)⊗U(g) as a topological subring of the complete tensor product U * (g) ⊗U(g)

1. If χ is a k-valued ⋆-character on U ∨ (g) i.e. χ ∈ Ξ(U ∨ (g), k), then the opera- tor χ ⊗ Id is continuous on U ∨ (g) ⊗ U(g)
or, through the standard embedding, of End(U(g)). As U * (g)⊗U(g), we remark that U ∨ (g)⊗U(g) is also a dense subset of U * (g) ⊗U(g), then we can complete the tensor product as U ∨ (g) ⊗U(g) which is equal to U * (g) ⊗U(g).

commutative, its action can be noted on the right)

(χ) gen = α∈N (I) B α χ(B α ) = ↗ i∈I exp(B e i χ(B e i )).
(5.9)

On the other hand, it is well known4 that, a Hopf algebra (H, µ H , 1 H , ∆, ϵ, S) 5 and a commutative (associative with unit) algebra (A, µ A , 1 A ) being given (all over the same commutative ring k), then the set Ξ(H, A) = Hom k-AAU (H, A) is a group under convolution (the inverse being performed through precomposition with S, proofs are essentially the same as for k-valued characters).

If H = U(g) (where the Lie algebra g satisfies all above assumptions), formulas (5.8) and (5.9) still hold true when χ is a A-valued character, A being a commutative (associative with unit) algebra.

We must warn the reader that this is no longer the case if A can be non-commutative as shows the following counterexample. By Appendix 5.4.4, let us recall the construction of a cocommutative Hopf algebra H conc (X) = (k⟨X⟩, conc, 1 X * , ∆ ¡ , ϵ) where X = {a, b} and of a non-commutative algebra (indeed its algebra part) A conc = (k⟨X⟩, conc, 1 X * ). We then define an algebra morphism f ∈ Ξ(H conc , A conc ) by the universal property (1.22) as follows:

f (a) = a, f (b) = b. One observes that • ∆ ¡ (a) = a ⊗ 1 + 1 ⊗ a, and ∆ ¡ (b) = b ⊗ 1 + 1 ⊗ b, then • ∆ ¡ (ab) = ∆ ¡ (a)∆ ¡ (b) = ab ⊗ 1 + a ⊗ b + b ⊗ a + 1 ⊗ ab.
Thus, we arrive at

f ⋆ f (a).f ⋆ f (b) = conc •(f ⊗ f ) • ∆ ¡ (a). conc •(f ⊗ f ) • ∆ ¡ (b) = f (a)f (1) + f (1)f (a) f (b)f (1) + f (1)f (b) = (a + a)(b + b) = 4ab and f ⋆ f (ab) = conc •(f ⊗ f ) • ∆ ¡ (ab) 5.2. APPENDIX A: LIMITS AND COLIMITS. = f (ab)f (1) + f (a)f (b) + f (b)f (a) + f (1)f (ab) = ab + ab + ba + ab = 3ab + ba. As above, f ⋆ f (a).f ⋆ f (b) ̸ = f ⋆ f (ab), therefore f ⋆ f is not an algebra homomorphism i.e. f ⋆ f / ∈ Ξ(H conc , A conc ).

Appendix A: Limits and Colimits.

In category theory [START_REF] Lane | Categories for the Working Mathematician[END_REF][START_REF] Kashiwara | Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften[END_REF], a limit of a diagram is called a product (resp. an equalizer ) of these objects (resp. morphisms). Moreover, a limit of a contravariant functor F : I op → C, where I is a directed set7 that is considered as a small category in which the morphisms consist of arrows α : i → j if and only if i ≤ j, is called an inverse limit of the inverse system ((F i ) i∈I , (F ij ) i≤j∈I ),

where This explains, in particular, why the sum of all variables x∈X x, which is a polynomial in the case when X is finite, does not even belong to the B-completion of k⟨⟨X⟩⟩ in the case when X is infinite (see discussion in [113]). In the latter case a finer topology has to be defined to recover the series as the completion of the polynomials. It is (in any cases, but when X is infinite, this topology is different) the topology of pointwise convergence for which a fundamental system of neighbourhoods of zero is given by the system of two-sided ideals (m F ) F ⊂ f inite X * m F = {S ∈ k⟨⟨X⟩⟩ | (∀w ∈ F )(∀u s.t. w ∈ X * uX * )(⟨S | u⟩ = 0)} .

F ij : F j → F i
Another example which will be in used for finding the generating function of an endomorphism is the following in particular (Id) gen for enveloping algebras, see Section 3.2). 2. The image of Φ is closed for composition or, when M is a bialgebra, for convolution.

3. The image of Φ is dense in End k (M ) endowed with the topology of pointwise convergence (which is compatible with the ring structure, be it for composition or, when M is a bialgebra, for convolution). (where (β i ) i∈I is the coordinate family) is summable (see [START_REF] Grinberg | Hopf algebras in Combinatorics, version of 27[END_REF], Def 1.7.2). Its sum realises the inverse of Φ, therefore we can state.

6. Φ : M * ⊗M → End k (M ) is an isomorphism and the inverse isomorphism of rings is given by

Φ -1 (f ) = i∈I β i ⊗ f (β i ) (5.13)
(we recall that (β i ) i∈I is the coordinate family of forms defined by ⟨β i | β j ⟩ = δ ij ).

7. We will denote Φ -1 (f ) by (f ) gen and call it the generating series of f . Completion by inverse limits.

In the following part, let us consider that R is a ring and B is a filter base of ideals of

Classical completion.

As above, assume that that R is a ring and B is a filter base of ideals of R (5.15), we then arrive to the fact that (R, T R ) is a topological ring in which B is a fundamental set of neighborhoods of zero for T R . In this case, R is said to be linearly topologized and T R is called a linear topology (see Bourbaki [START_REF] Bourbaki | Commutative Algebra[END_REF] Ch III §4.2).

In particular, if B = {m n } n≥0 where (m n ) n≥0 is a decreasing sequence of ideals, then in this case (decreasing sequence and, moreover, the topology is Hausdorff), setting d(s, t) = 2 -ω(s-t) where ω(r) := max n∈N (r ∈ m n ) and ω(0) := +∞ (5.17) (for all r, s, t ∈ R), we can prove that d is an ultrametric distance and the linear topology T R can be defined by the ultrametric d : R × R → R ≥0 . In this way, we can get the completion R of the ultrametric space (R, d). For example when R is the free associative algebra k⟨X⟩ (with X finite), the completion is k⟨⟨X⟩⟩, see Example 5.2.

5.4 Appendix C: Structures of Bialgebra type.

Gradings.

The idea of a graded set or structure [START_REF]Graded Sets[END_REF] is to combine an algebraic structure with a (simpler) discrete/combinatorial structure with another one which "follows" the computations within it.

Let A ∈ k-AA and (S, ×) be a (commutative or not) semigroup. We will say that A = u∈S A u , [S]-graded as a module, is a S-graded algebra if the usual condition A u A v ⊂ A uv holds for all u, v ∈ S. A morphism between S-graded algebras φ : A → B is just a morphism between the underlying algebra structures which preserves the gradation (i.e. for all u ∈ S, φ(A u ) ⊂ B u ). We then have the following proposition which generalizes the similar classical ones for usual (commutative) gradings. Proposition 5.2. Let J ⊂ A, TFAE i) J = Ker(φ) for some morphism between S-graded algebras.

ii) J is a two-sided ideal of A which is [S]-graded as a module.

Proof. The proof is mutatis mutandis the same as in classical treatises. We sketch it there.

(i)=⇒ (ii) being straightforward, remains to prove the converse, now (ii) being assumed, we consider the canonical surjection s : A → A / J . We set ( A / J ) u := s(A u ) and, as graded modules, we have

A / J = u∈S ( A / J ) u = u∈S A u J u
with J u = J ∩ A u . So A / J is naturally endowed with a structure such that s is a morphism of S-graded algebras and Ker(s) = J . Remark 5.2. The treatment here is very similar to what can be found in Bourbaki [START_REF] Bourbaki | Algebra[END_REF] Ch II §11 and [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch II §2.6 save that, for our purpose, we need that a Lie algebra be graded on a semigroup rather than a monoid. In particular, in [START_REF]Graded ring[END_REF], one reads the sentence "If we do not require that the ring have an identity element, semigroups may replace monoids" shows that additive semigroups of degrees is probably a good working notion for Lie algebras and their enveloping algebras.

Enveloping algebra of S-graded Lie algebras.

For this section, the semigroup of degrees of a Lie algebra is commutative (because it must "follow" antisymmetry). Now, given (S, +) an additive (that is, commutative and noted additively) semigroup and A = [S] its underlying set, we now pass to the construction of the enveloping algebra U(g) of a S-graded Lie algebra g = s∈S g s by the following: the tensor algebra inherits a A * -gradation (see paragraph 5 (5.20)

One can check easily that this constitutes a (yet non-commutative) grading as, for all u, v ∈ A * , T u (g).T v (g) ⊂ T uv (g).

(5.21)

In the sequel we will use the following regrading lemma.

Lemma 5.3. Let S, T be two semigroups (commutative or not) and φ : S → T be a morphism of semigroups. Let A = s∈S A s be a S-graded algebra (associative or not). We set, for t ∈ T , A φ,t := φ(s)=t A s . Then A φ,-:= t∈T A φ,t is a T -graded algebra.

Proof. The proof is left to the reader. Now, µ : A * → S ⊔ {0}, s 1 . . . s k → s 1 + . . . + s k , 1 A * → 0 is a morphism of monoids (S ⊔ {0} is the monoid with neutral 0 constructed from (S, +)), the tensor algebra can be regraded (through Lemma 5. The following theorem states two things. Firstly that T SI(A) is a section of the natural morphism T (M ) → U(g) and secondly that rearranging the tensors in increasing form converges towards the projector on T SI(A) parallel to the kernel of the natural morphism.

To this end, we must define what is "rearranging the tensors" and will use the structure of paths of computations through appropriate labeled graphs in the spirit of [START_REF] Hopcroft | Introduction to Automata Theory, Languages and Computation[END_REF] 12 .

We need the following remark and definitions We now have to build a transition structure similar to what is defined in [START_REF] Stanley | Enumerative Combinatorics[END_REF] p.200 ii) The canonical morphism, can, restricted to T SI(A) , is onto and T (M ) = T SI(A) ⊕ Ker(can) (5.30) in other words T SI(A) := w∈SI(A) T w (M ) is a section of can.

2. Let us call proj, the projection on T SI(A) parallel to Ker(can) and, for all F ⊂ A * , let us denote by proj| F the restriction proj| ⊕ w∈F Tw(M ) , then i) For any chain of tensors (t 1 , • • • , t n ) such that supp(t n ) ⊂ SI(A) and such that, for all 1 ≤ j < n, can(t j ) = can(t j+1 ), we have t n = proj(t 1 ).

ii) The evaluation 13 of the path (5.29) is proj ii) As, in the preceding preceding point, the steps are preserving can-evaluation i.e.

for any for 1 ≤ j < n, step F j Φ j F j+1 and tensor t ∈ w∈F j T w (M ), we have can(t) = can( φ∈Φ j φ(t)), by composition, we get that for the "sufficiently long" path P =(5.29) and t ∈ w∈F 1 T w (M ), can(t) = can( φ∈Ev(P ) φ(t)). As Then M = M ∞ ⊕ Ker(f ).

and one gets the result.

2. i) Condition can(t j ) = can(t j+1 ) means that, for every step t j -t j+1 ∈ Ker(can), then proj(t j ) = proj(t j+1 ). Hence proj(t 1 ) = proj(t n ) = t n .

ii) Is a direct consequence of (2.i) above.

Example 5.4. A computation scheme, starting from 3222 with a swap episode followed by reduction of powers, is as follows (note that last step is aperiodic with F = {23, 3}) such that, for all 1 ≤ j < n, t j -t j+1 ∈ Ker(can) and support(t n ) ∈ SI(A) gives the same result (from t 1 ) which is proj(t 1 ).

ii) As a consequence of Theorem 5.4 (1.ii), in case when g admits a A-graded linear basis (then as a A-graded module, the original theorem is more general), the canonical morphism can : T (M ) → U(g) preserves a linear basis of the tensor subspace T SI(A) = w∈SI(A) T w (M ) (recall that SI(A) ⊂ A * is the language of strictly increasing words) to the universal enveloping algebra U(g).

We now pass to the following application of the graded structure (5. Observing the positions of the letters on the table Let X = {x 0 , x 1 } with x 0 < x 1 .

T ∞ = T 2 .
l P l S l x 0 x 0 x 0 x 1 x 1 x 1 x 0 x [x 0 , x 1 ] x 0 x 1 x 2 0 x [x 0 , [x 0 , x 1 ]] x 2 0 x 1 x 0 x [[x 0 , x 1 ], x 1 ]
x 0 x 2 1

x 3 0 x [x 0 , [x 0 , [x 0 , x 1 ]]] x 3 0 x 1 x 2 0 x [x 0 , [[x 0 , x 1 ], x 1 ]] x 2 0 x 2 1 x 0 x [[[x 0 , x 1 ], x 1 ], x 1 ] x 0 x 3 1 x 4 0 x [x 0 , [x 0 , [x 0 , [x 0 , x 1 ]]]] x 4 0 x 1 x 3 0 x [x 0 , [x 0 , [[x 0 , x 1 ], x 1 ]]] x 3 0 x 2 1 x 2 0 x 1 x x 1 [[x 0 , [x 0 , x 1 ]], [x 0 , x 1 ]] 2x 3 0 x 2 1 + x 2 0 x 1 x 0 x 1 x 2 0 x [x 0 , [[[x 0 , x 1 ], x 1 ], x 1 ]] x 2 0 x 3 1 x 0 x 1 x x 2 1 [[x 0 , x 1 ], [[x 0 , x 1 ], x 1 ]] 3x 2 0 x 3 1 + x 0 x 1 x 0 x 2 1 x 0 x [[[[x 0 , x 1 ], x 1 ], x 1 ], x 1 ] x 0 x 4 1 x 5 0 x [x 0 , [x 0 , [x 0 , [x 0 , [x 0 , x 1 ]]]]] x 5 0 x 1 x 4 0 x 2 1 [x 0 , [x 0 , [x 0 , [[x 0 , x 1 ], x 1 ]]]] x 4 0 x 2 1 x 3 0 x 1 x x 1 [x 0 , [[x 0 , [x 0 , x 1 ]], [x 0 , x 1 ]]] 2x 4 0 x 2 1 + x 3 0 x 1 x 0 x 1 x 3 0 x [x 0 , [x 0 , [[[x 0 , x 1 ], x 1 ], x 1 ]]] x 3 0 x 3 1 x 2 0 x 1 x x 2 1 [x 0 , [[x 0 , x 1 ], [[x 0 , x 1 ], x 1 ]]] 3x 3 0 x 3 1 + x 2 0 x 1 x 0 x 2 1 x 2 0 x 2 1 x x 1 [[x 0 , [[x 0 , x 1 ], x 1 ]], [x 0 , x 1 ]] 6x 3 0 x 3 1 + 3x 2 0 x 1 x 0 x 2 1 + x 2 0 x 2 1 x 0 x 1 x 2 0 x [x 0 , [[[[x 0 , x 1 ], x 1 ], x 1 ], x 1 ]] x 2 0 x 4 1 x 0 x 1 x x 3 1 [[x 0 , x 1 ], [[[x 0 , x 1 ], x 1 ], x 1 ]] 4x 2 0 x 4 1 + x 0 x 1 x 0 x 3 1 x 0 x [[[[[x 0 , x 1 ], x 1 ], x 1 ], x 1 ], x 1 ] x 0 x 5 1

  me during our weekend seminars. -Thank you, Dr. Le Hoang Tri, my professor from the Department of Mathematics at Danang University of Education. "You were the first person to introduce me to the world of modern mathematics". -To Professors Nguyen Duy Tan (Hanoi University of Science and Technology), Nguyen Viet Dung and Phung Ho Hai (Hanoi Institute of Mathematics, VAST) for your wholehearted support during my studies in Hanoi. -I thank to the Fondation Sciences Mathématiques de Paris (FSMP) for awarding me a master's scholarship in fundamental mathematics, which enabled me to continue my studies at Université Sorbonne Paris Cité in 2018. During my time there, I received inestimable support from Professor Gregory Ginot, Dr. Hoang Manh Truong (LAGA, Université Sorbonne Paris Nord) and Professor Anne Micheli (IRIF, Université Paris Cité) in both my master's program and my search for doctoral scholarships. I am deeply grateful for their guidance and assistance. -Thank you to all my friends, colleagues and acquaintances (Thanh Loan, Mrs. Minh, Mr. Truong, Mr. Hoan, Mr. Chien, Tin, Tuan Anh, Ms. Phuong and Mr. Duc, Mr. Khanh, Mr. Quan, Duyen, Dat, Vuong, Manh Linh, Quyet, Hang, Mr. Tai, Mr. Hoang, Dung, Mr. Kha, Mr. Thai, Mrs. Van ...) who have always supported me throughout my PhD journey in Paris, France.

Introduction Chapter 1 .

 1 Preamble 1.1. The (endo)functor Seq. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Free objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1. General principle. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2. Presented structures. . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3. Free monoids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4. Free associative algebras. . . . . . . . . . . . . . . . . . . . . . . 1.2.5. Free Lie algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. An example of free object with respect to graphs. . . . . . . . . . . . . Chapter 2. Lazard's elimination 2.1. Groups and Lie algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. A generalization of Lazard's elimination theorem. . . . . . . . . . . . . 2.2.1. Classical Lazard's elimination. . . . . . . . . . . . . . . . . . . . 2.2.2. Quotients of Lazard's eliminations. . . . . . . . . . . . . . . . . 2.2.3. Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Lazard elimination as a free object. . . . . . . . . . . . . . . . . . . . . 2.3.1. Category of SESS in Lie algebras. . . . . . . . . . . . . . . . . . 2.3.2. An equivalence of categories and a Lazard elimination functor. . 2.3.3. Drinfeld-Kohno Lie algebra with infinite number of generators: from strange to generalized gradings. . . . . . . . . . . . . . . . 2.4. Smash product algebra and Lazard's elimination. . . . . . . . . . . . . 2.5. Table of Lazard elimination formulas. . . . . . . . . . . . . . . . . . . .CONTENTS Chapter 3. Characters 3.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Convolution algebra and factorization. . . . . . . . . . . . . . . . . . . 3.3. Bases in duality: Zinbiel algebra and Magnus basis. . . . . . . . . . . . 3.4. Extension of characters: A theory of Domains for Harmonic Functions and its Symbolic Counterpart. . . . . . . . . . . . . . . . . . . . . . . . 3.4.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Polylogarithms: from global to local domains. . . . . . . . . . . 3.4.3. From Polylogarithms to Harmonic sums. . . . . . . . . . . . . . 3.4.4. Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5. Stuffle products, usage of one-parameter subgroups within stuffle characters and their symbolic computations. . . . . . . . . . . . Chapter 4. Conclusions and future directions CONTENTS Chapter 6. Bibliography 193

Figure 1 :

 1 Figure 1: Similar lower diagram with algebras and k-Mod replacing Set.

8

 8 there exists a unique morphism φ : M → N such that φ = φ • s

Definition 1 . 1 .Figure 1 . 1 :

 1111 Figure 1.1: For D 5 (group of order 10). Coxeter presentation is with s 1 (symmetry w.r.t. the line passing through node 5) and s 2 (symmetry w.r.t. the line passing through node 2) and relations [s 2 i = 1 ; (s 1 s 2 ) 5 = 1].

Example 1 . 3 . 1 .

 131 by an example presented as follows: consider a homomorphism of monoids l : M = ⟨ {a, b} | b 3 = ab ⟩ Mon → N extending from a monoid homomorphism {a, b} * → N which sends a → 2, b → 1 by the coequalizer (1.12) over the relation b 3 = ab, then clearly l is a finitely additive proper length function on M and hence M is a finitely graded monoid. Assume that A := {a, b} is a set of two letters with a < b. Let us consider monoid SF 0 (A) := ⟨A | a 2 = b 2 = 1⟩ Mon = ⟨{a, b} | 1 = a 2 = b 2 ⟩ Mon (1.14) It is easy to show that the onto morphism s : {a, b} * → SF 0 (A) admits as a section the square-free language 12 L = {a, b} * \ ({a, b} * a 2 {a, b} * ∪ {a, b} * b 2 {a, b} * ) (1.15) = (ab) * (1 + a) + (ba) * (1 + b). (1.16) (the + in 1.16 is, in this case, a dijoint union). Alternatively, we can consider the monoid SF 1 (A) := ⟨A | a 2 = a; b 2 = b⟩ Mon (1.17)

  A) m = W m the set of all words of length m without any b 2 factor and then |

  w⟩w, a Lie bracket is as usual defined by [P, Q] = P Q -QP. (1.29) With this bracket, k⟨X⟩ is a Lie algebra. A Lie polynomial is an element of the smallest submodule of k⟨X⟩ containing X and closed under the Lie bracket. The set of all Lie polynomials in k⟨X⟩ forms a Lie algebra and we denote it by L k (X). It can be shown, through Lyndon bases 14 , that the Lie algebra L k (X) is the free Lie algebra over k generated by X. It means that, in view of Subsection 1.2.1, the pair (j X , L k (X)) is a solution of the universal problem corresponding to following diagram: Set k-Lie X g L k (X).

14

  See, with Hall bases [13] Ch 2 §2.11 and [94] Ch 4 Prop 4.9.

15

  For any Lie algebra b, h and an action by derivations of the Lie algebra b on h i.e. a Lie homomorphism α : b → Der(h), we can construct g, a split Lie algebra extension of b by h whose underlying k-module is the external direct sum of modules g := h ⊕ b and the Lie bracket is given by the following formula

CHAPTER 1 .

 1 PREAMBLE What has been said before the definition shows us that for C one of the categories in the list (1.1) we can assign to each M ∈ C an object F (M ) := (M, θ M ) of CommAlph where θ M is the set of all pairs (x, y) of commuting elements of M (where the meaning of "commuting" depends on C). It is not difficult to check that, in each case, F : C → CommAlph is a functor. Then, for each category C of the list (1.1), one can state an universal problem in the style of Subsection 1.2.1.

1

 1 b 1 , b 5 ), (b 1 , b 6 ), (b 2 , b 3 ), (b 2 , b 4 ), (b 2 , b 6 ), (b 2 , b 7 ), (b 3 , b 4 ), (b 3 , b 5 ), (b 3 , b 7 ), (b 4 , b 5 ), (b 4 , b 6 ), (b 4 , b 7 ), (b 5 , b 6 ), (b 5 , b 7 ). Then the pieces are B = {b 1 , b 2 , • • • , b 7 } and the relations are (not mentioning the relations of the form b i Rb i ; if a relation b i Rb j holds then also b j Rb i ) b Rb 3 , b 1 Rb 4 , b 1 Rb 7 , b 2 Rb 5 , b 3 Rb 6 , b 6 Rb 7 .

1. 3 . 7 Figure 1 . 2 :

 3712 Figure 1.2: A heap of pieces H t corresponding to the word

Figure 2 . 1 :

 21 Figure 2.1: A braid on 6 strands.

Figure 2 . 2 :

 22 Figure 2.2: Glueing of braids and the identity of the braid group B n .

Figure 2 . 4 :

 24 Figure 2.4: A pure braid on 4 strands.

Figure 2 . 5 :

 25 Figure 2.5: A loop γ(t) in π 1 (C 5 * ).

Figure 2 . 6 :

 26 Figure 2.6: A standard twist generator of the pure braid group PB n .

Definition 2 . 1 .

 21 For any Lie algebras b, h and an action by derivations of the Lie algebra b on h i.e. a Lie homomorphism α : b → Der(h), we can construct g, a split Lie algebra extension of b by h whose underlying k-module is the external direct sum of modules g := h ⊕ b and the Lie bracket is given by the following formula

For example with three•

  letters {a, b, z} and Q = [[a, b], z] one has P = abz -baz. Proof of Theorem 2.3 (Sketch of ) : For all b ∈ B, the left translation t (0) b : B * Z → B * Z defined by t (0) b (uz) := buz can be extended as a derivation t (1) b of L k (B * Z) (see Lemma 1.1 i.e. Corollary of Prop 8, [13] Ch II §2.8).

  3, we remark that what precedes establishes the semidirect product i.e. the top row of diagram 2.14; the bottom row follows by identifying L k (B * Z) and L k (B) with their images in L k (X) and the fact that any nested Lie bracket with a factor in X can be expressed as a linear combination of rightnormed B * Z-brackets. □ Remark 2.4. (Dynkin combs and their evaluations.) i) In an algebra (A, * ) (not necessarily associative), we define an operator ev

3 . 1 )

 31 the bases of the free Lie algebra (like Hall, Lyndon, Viennot, Schützenberger) coming from the bracketing of a family of binary trees (T i ) i∈I through evaluation (see (2.20)). Lazard elimination provides an algorithmic way to create new families of monomial bases in the following way • Partition the alphabet X as X = B + Z • Totally order the new alphabet B * Z • Choose any process (Hall, Lyndon, Viennot, Schützenberger) and get the associated family of trees (T i ) i∈I such that (rn(T i )) i∈I is a (linear) basis of L k (B * Z) and remark that each rn(T i ) is the evaluation of the tree obtained from T i by appending Dynkin combs (corresponding to each uz) to the leaves of T i . Let us now consider a situation where g = h ⊕ b, h being an ideal and b a Lie subalgebra (hence we have the -internal -semidirect product g = h ⋊ b). Implementation of the associated arrows is the following Theorem 2.4 (Ladder LET (internal version)). Let g = h ⋊ b be a semi-direct product of Lie algebras, h (resp. b) being an ideal (resp. a Lie subalgebra) of g 7 .

2 . 2 ) 4 . 2 . 5 (

 22425 , let us give an external version of Theorem 2.Theorem Ladder LET (external version)). Let g 3 = g 1 ⋊ g 2 be a semi-direct product of Lie algebras, constructed after a morphism of Lie k-algebras α : g 2 → Der(g 1 ).

  and |{i, j, k}| = 3.

  the above results, one thus derives a commutative diagram of Lie algebras with split short exact rows 0

CHAPTER 2 .

 2 LAZARD'S ELIMINATION About M.-P. Schützenberger's questions on the Partially Commutative Free Lie algebra.

  by φ 4 (y 2 ) := y 1 + J (R). A routine check shows that φ 3 • φ 4 = Id Dom(φ 4 ) and φ 4 • φ 3 = Id Dom(φ 3 ) . □ End of the proof of Theorem 2.10. -We now consider the following composition

  of the type considered in Lemma 2.15 by φ : B * Z → C ∪ {0} as follows φ(uz) = j θ • s θ (uz) if TAlph(s θ (uz)) = {z} and φ(uz) = 0 otherwise.(2.77)

Figure 2 . 7 :

 27 Figure 2.7: Diagram of the arrows involved in the proof of Theorem 2.10 (beware this diagram is not commutative in general).

CHAPTER 2 .

 2 LAZARD'S ELIMINATION where, ω 0 (z) = z -1 dz and ω 1 (z) = (1 -z) -1 dz are two differential forms on the complement of the union of the real half-lines ] -∞, 0] and [1, +∞[ in the complex plane i.e. the simply-connected domain Ω = C \ (] -∞, 0] ∪ [1, +∞[) and for any series S ∈ H(Ω)⟨⟨X⟩⟩ over H(Ω) the algebra (for the pointwise product) of complex-valued functions which are holomorphic on Ω and d stands for the term by term derivation d(S) = w∈X * d dz (⟨S | w⟩)w.

  ideal h := Ker(p) and a Lie subalgebra b := Im(σ) such that g can be uniquely decomposed as an internal direct sum of submodules g = h ⊕ b. Then p (resp. σ) induces a Lie isomorphism h ∼ = g l (resp. b ∼ = g r ) and then a module isomorphism g ∼ = g l ⊕g r (external direct sum of modules). Moreover, one clearly defines an action of the Lie algebra b (resp.g r ) on h (resp. g l ) by derivations. In other words, at the level of elements b (resp.g r ) acts on h by D b (h) = [σ(b), h] (resp.internal brackets i.e. the adjoint representation).

Proposition 2 . 18 .

 218 The functor F : k-SSLie → B-GrLie is• an essentially surjective functor (i.e. every object g = h⋊b ∈ B-GrLie, there exist an object (2.92) ∈ k-SSLie and an isomorphism F ((2.92))) ∼ = g in B-GrLie, see[START_REF] Kashiwara | Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften[END_REF] Def 1.2.11) and• a fully faithful functor.

17

 17

  Two categories C and D are called equivalent if there exists an equivalence between them i.e. there are functors F : C → D, G : D → C and natural isomorphisms α : 1

  This is due to the fact that, for all b ∈ B and z ∈ Z, we have [b, J B ] ⊂ J B and [z, J B ] ⊂ J BZ by definition, the other summands (J Z and J BZ ) being Lie ideals.

2. 3 .

 3 LAZARD ELIMINATION AS A FREE OBJECT. and g 0 (Z, B) := s B|Z (L k (B)) the Lie subalgebra of it. One notes that g(Z, B) = g 1 (Z, B)⊕g 0 (Z, B) is indeed an object in B-GrLie. Moreover, we define the morphism j (Z,B) = (j Z , j B ) : (Z, B) → (g 1 (Z, B), g 0 (Z, B)) = F (g(Z, B)) in Set 2 , where j Z : Z → g 1 (Z, B) and j B : B → g 0 (Z, B) are set as the composites

  by universal property of Diagram(1.30). Similarly, we can also find a Lie homomorphism L k (B) f 0 -→ L 0 as the unique extension of a map f B : B → L 0 . As a result, we arrive at Lie homomorphisms f 1 : L k (B * Z) → L and f 0 : L k (B) → L by such embedding f 1 and f 0 into L, respectively. Further, it is straightforward to check on generators that these Lie homomorphisms and the action α : L k (B) → Der(L k (B * Z)) induce the following commutative diagram

4 ( 2 . 107 )

 42107 and |{i, j, k, l}| = then the infinite Drinfeld-Kohno Lie algebra is indeed the direct limit of such structure homomorphisms (2.106) DK k,∞ = lim -→ DK k,n .

  these are clearly also Lie monomorphims between two presented Lie algebras. Then a pair (DK k,∞ , ϕ n ) is a target of such direct system {DK k,n fn -→ DK k,n+1 } n≥2 in category k-Lie because it satisfies the following property

  In particular, it is easy to verify that u is a unique Lie homomorphism such that u • ϕ n = ψ n for each n ≥ 2. By this universal property, we thus deduce that DK k,∞ = lim -→ DK k,n .

Further, we

  consider (N ≥2 , ∨) the upper semi-lattice on the classical supremum m ∨ n := sup{m, n} (for all m, n ∈ N ≥2 ), in the same way as above we now describe a (N ≥2 , ∨)-graded structure of the infinite Drinfeld-Kohno Lie algebra DK k,∞ by the following way: for all m ∈ N ≥2 , if we denote DK

  [[S]] is the completion of the Hausdorff topological semigroup algebra Q[S] (see Appendix 5.3.2), where Q[[S]

Proposition 2 . 24 .

 224 Suppose we are given two finitely S-graded algebras A = s∈S A s and B = s∈S B s . Then, so are direct sum and tensor product, moreover two Hilbert series Hilb(A ⊕ B) = Hilb(A) + Hilb(B) and Hilb(A ⊗ B) = Hilb(A).Hilb(B).

18 A

 18 semigroup (S, +) is said to satisfy "Condition (D)" if each s ∈ S admits only a finite number of factorizations s = s 1 + • • • + s k (the positive integer k is fixed). To be more precise, for any s ∈ S,the set D 2 (s) = {(s 1 , s 2 ) ∈ S × S | s 1 + s 2 = s} is finite i.e. the map µ 2 : [S] 2 → S, (s 1 , s 2 ) → s 1 + s 2has finite fibers, see Bourbaki[START_REF] Bourbaki | Algebra[END_REF] Ch III § 2.10.

Example 2 . 5 .DK

 25 In the same manner as above, the Drinfeld-Kohno Lie algebra DK k,n+1 can be equipped with a ([2, n + 1], ∨) × (N ≥1 , +)-graded Lie algebra structure DK k,n+1 = 2≤i≤n+1 enveloping algebra U(DK k,n+1 ) inherits a [([2, n + 1], ∨) × (N ≥1 , +)] ⊔ {0}graded structure (5.23)

Theorem 2 . 25 .

 225 If τ : B ⊗ A → A ⊗ B is an algebra cross then the tensor product of algebras A ⊗ B equipped with the multiplication

2. 4 .

 4 SMASH PRODUCT ALGEBRA AND LAZARD'S ELIMINATION. A ∈ k-AAU and B be a bialgebra i.e. (B, m B , 1 B , ∆ B , e B ) with the usual axioms. We suppose given a left B-module with action denoted by ▷ : B ⊗ A → A i.e. b 1 ▷ (b 2 ▷ a) = (b 1 b 2 ) ▷ a and 1 B ▷ a = a identically. The algebra A is said to be a left B-module algebra

[ 84 ]

 84 Thm 2.13, then the smash product algebra A♯B has a unique Hopf algebra structure in which A ⊗ k1 B and k1 A ⊗ B are Hopf subalgebras (where k1 A and k1 B are the group algebras), called the semi-direct product of Hopf algebras.

Example 2 . 8 .

 28 Let X = B + Z be a graded set and α : L k (B) → Der(L k (B * Z)) be a morphism in k-Lie defined immediately by the extension of the left translation t (0) b : B * Z → B * Z, uz → buz (for all b ∈ B) to the derivation t (1) b ∈ Der(L k (B * Z)) and then, by universal property, the map b → t (1) b : B → Der(L k (B * Z)) can be extended to α that is indeed a morphism in k-Lie between the free Lie algebra L k (B) and the usual Lie algebra Der(L k (B * Z)) (the reader can review the above construction in the proof of Theorem 2.3). We consider the pairs of Lie homomorphisms f 1 : L k (B * Z) → L k (X) and f 2 : L k (B) → L k (X) defined by f 1 (uz) = rn(uz) (where rn is the right-normed bracketing i.e. rn(uz) = ad

  and z ∈ Z) and f 2 (b) = b satisfying the equivariance condition of (2.13) with respect to α. Then, the basic results of Proposition 2.27 are applied and summarized in the following properties i) one has a morphism of Lie algebras f : L k (B * Z) ⋊ L k (B) → L k (X) extending f i (i = 1, 2) in the usual sense by Proposition 2.27 point (1).

  are two morphisms in k-Lie satisfying the equivariant property (2.13) w.r.t. [α].

1 .

 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.2. Convolution algebra and factorization. . . . . . . . . . . . . . . . . 111 3.3. Bases in duality: Zinbiel algebra and Magnus basis. . . . . . . . . 122 3.4. Extension of characters: A theory of Domains for Harmonic Functions and its Symbolic Counterpart. . . . . . . . . . . . . . . . . . 134 3.4.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.4.2. Polylogarithms: from global to local domains. . . . . . . . . 139 3.4.3. From Polylogarithms to Harmonic sums. . . . . . . . . . . . 141 3.4.4. Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . 151 3.4.5. Stuffle products, usage of one-parameter subgroups within stuffle characters and their symbolic computations. . . . . . 155 3.1 Introduction.

  1 . Taking a bialgebra (B, µ, 1 B , ∆, ϵ) it is known that the set Ξ(B) = Hom k-AAU (B, k) of characters of the algebra part (B, µ, 1 B ) is a monoid under convolution 2 and, if an antipode 3.1. INTRODUCTION.

. 1 ) 3 . 1 .

 131 Example In this case U(g) = k⟨X⟩ and the generating function of Id (a.k.a (Id End ) gen , see also[START_REF] Racinet | Series generatrices non-commutatives de polyzetas et associateurs de Drinfel'd[END_REF], Def 3.8 and Prop 5.25), expressed in the complete tensor product k⟨⟨X⟩⟩ ⊗k⟨X⟩ reads (as Equation (3.6)) (Id End ) gen = w∈X * w ⊗ w = w∈X * S w ⊗ P w = ↘ l∈LynX exp(S l ⊗ P l ), (3.2)

  Grinberg and Reiner [56] § 1.6 Exercise 1.6.1 (a)).

3. 2 .Example 3 . 2 .

 232 CONVOLUTION ALGEBRA AND FACTORIZATION. QED Two applications.

  . BASES IN DUALITY: ZINBIEL ALGEBRA AND MAGNUS BASIS. built from (3.38). Product formula (3.2) provides a factorization of Li (in fact of any shuffle character) L = w∈X * Li w w = w∈X * Li Sw P w = ↘ l∈LynX exp(Li S l P l ). (3.15) Using Lazard's elimination with X = {x 0 , x 1 } = B → + Z where B = {x 0 } and Z = {x 1 }

  and {w | w ∈ B * } form k-linear bases of, respectively, k⟨B * Z⟩ and k⟨B⟩. Under the algebra isomorphism f

123 3 . 3 .

 33 BASES IN DUALITY: ZINBIEL ALGEBRA AND MAGNUS BASIS. However, we can replace the usual (left) Zinbiel algebra by its opposite algebra, called (right) Zinbiel algebra. Namely, Definition 3.2. A k-module A equipped with a bilinear map ≻ is called (right) Zinbiel algebra if it satisfies the identity x ≻ (y ≻ z) = (x ≻ y) ≻ z + (y ≻ x) ≻ z, for all x, y, z ∈ A.

  ) and Zinbiel polynomials (3.25) are dual bases of, respectively (k⟨X⟩, conc, 1 X * ) and (k⟨X⟩, r ¡,1 X * ).

3. 3 .

 3 BASES IN DUALITY: ZINBIEL ALGEBRA AND MAGNUS BASIS.

S

Remark 3 . 2 .

 32 Our constructions and results above for the Magnus basis and their duality in case the finite graded set X = B + Z (where B = {b 1 , • • • , b M } and Z = {z 1 , • • • , z N }) can be automatically approached to any graded set X = B + Z where B = {b γ } γ∈Γ and Z = {z λ } λ∈Λ (Γ, Λ are nonempty index sets). In case X = B+Z where B = {x 0 } and Z = {x λ } λ∈Λ (Λ: a nonempty index set, for example N + ), the Magnus duality also appeared in [89] Thm 3.2 to derive a formula of Le-Murakami [75], Furushotype[START_REF] Furusho | p-adic multiple polylogarithms and the p-adic KZ equation[END_REF] that expresses arbitrary coefficients of a group-like series J ∈ k⟨⟨x 0 , x 1 ⟩⟩ (k is a field of characteristic zero) in terms of the "regular" coefficients of J ([89] Thm 4.1). On the other hand, images of the Magnus polynomial and its dual under the antiautomorphism Φ of k⟨X⟩ (which sends w → w for all words on X, where w reverses the order of letters in the word w) belong in the (left) Zinbiel bialgebra and its dualisation framework[START_REF] Burgunder | A symmetric version of Kontsevich graph complex and Leibniz homology[END_REF], Appendix: Associative-Zinbiel bialgebras. The images also appeared in[START_REF] Ngoc | On the solutions of universal differential equation with three singularities[END_REF] Prop 5.10 to describe the coefficients of the complete generating series (3.37)L(z) = w∈X * Li w (z)w(where X = {x 0 , x 1 }) in terms of the indeterminates are monomials of the set {rn(x k 0 x 1 ) = ad kx 0 x 1 } k≥0 , where L(z) is the group-like solution of the following first order noncommutative differential equation (3.38) (the Knizhnik-Zamolodchikov equation KZ 3 due to Drinfeld[START_REF] Drinfeld | Quantum group[END_REF][START_REF] Drinfeld | On quasitriangular quasi-hopf algebra and a group closely connected with Gal(Q/Q)[END_REF])   d(S) = (ω 0 (z)x 0 + ω 1 (z)x 1 )S, (N CDE) lim z∈Ω,z→0S(z)e -x 0 log(z) = 1 H(Ω)⟨⟨x 0 ,x 1 ⟩⟩ , asymptotic initial condition, where, ω 0 (z) = z -1 dz and ω 1 (z) = (1 -z) -1 dz are two differential forms on the simply-connected domain Ω = C \ (] -∞, 0] ∪ [1, +∞[) and for any series S ∈ H(Ω)⟨⟨x 0 , x 1 ⟩⟩ over H(Ω) the algebra (for the pointwise product) of complex-valued functions which are holomorphic on Ω (for more details see in[START_REF] Ngoc | On the solutions of universal differential equation with three singularities[END_REF] Prop 5.10 and Subsection 3.4.1).3.4. EXTENSION OF CHARACTERS:A THEORY OF DOMAINS FOR HARMONIC FUNCTIONS AND ITS SYMBOLIC COUNTERPART.

ζ(s 1 ,

 1 • • • , s r ) converges absolutely for s ∈ H r whereH r := {s = (s 1 , • • • , s r ) ∈ C r |∀m = 1, • • • , r; ℜ(s 1 ) + • • • + ℜ(s m ) > m} .(3.31) In the convergent cases, ζ(s 1 , • • • , s r ) they are called "polyzeta values" (or MZV 17 ) at multi-index s = (s 1 , • • • , s r ). Indeed s → ζ(s) is holomorphic on H r and has been extended to C r as a meromorphic function (see

z 1 ω

 1 0 (s) Li u (s) if w = x 0 u and |u| x 1 = 0, i.e. w ∈ x *0 z 0 ω 0 (s) Li u (s) if w = x 0 u and |u| x 1 > 0, i.e. w / ∈ x * 0 ,(3.36)the upper bound z belongs to Ω (asΩ = C \ (] -∞, 0] ∪ [1, +∞[) is a simply-connected domain, the integrals, which can be proved to be convergent in all cases, depend only on their bounds). The neutral element of the algebra of analytic functions H(Ω), a constant function, will be here denoted 1 H(Ω) .

  (1 -z) d dz) -or more precisely sections z 0 f (s) s ds (resp.

1 x 3 1Figure 3 . 1 :

 1331 Figure 3.1: Tree of addresses and corresponding coefficients below.

  CDE) lim z∈Ω,z→0 S(z)e -x 0 log(z) = 1 H(Ω)⟨⟨X⟩⟩ , asymptotic initial condition, (3.38) where, for any S ∈ H(Ω)⟨⟨X⟩⟩ and d stands for the term by term derivation d(S) = w∈X * d dz (⟨S | w⟩)w.

  Note that, in what precedes, we used the pairing ⟨• | •⟩ between series and polynomials, classically defined by, for S ∈ k⟨⟨X⟩⟩ and P ∈ k⟨X⟩ ⟨S | P ⟩ = w∈X * ⟨S | w⟩⟨P | w⟩, (3.40) 3.4. EXTENSION OF CHARACTERS: A THEORY OF DOMAINS FOR HARMONIC FUNCTIONS AND ITS SYMBOLIC COUNTERPART. where, when w is a word, ⟨S | w⟩ stands for the coefficient of w in S and k any commutative ring with unit (as here H(Ω)). With this at hand, we extend at once the indexation of Li from X * to C⟨X⟩ by Li P := w∈X * ⟨P | w⟩ Li w = n≥0 |w|=n ⟨P | w⟩ Li w . (3.41)

Example 3 . 4 (

 34 [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF]). For example, the classical polylogarithms: dilogarithm Li 2 , trilogarithm Li 3 , etc... are defined and obtained through the coding (3.34) by Li k

DomFigure 3 . 2 :

 32 Figure 3.2: Domain of Polylogarithms and the algebra A.

3. 4 . 3 From

 43 Polylogarithms to Harmonic sums. Definition of Dom(Li) has many merits 31 and can easily be adapted to arbitrary (open and connected) domains. However this definition, based on a global condition over a fixed domain Ω ⊂ C \ ([0, +∞[) with 0 ∈ Ω, does not provide a sufficiently clear interpretation of the stable symbolic computations around a point, in particular at z = 0. One needs to consider a sort of "symbolic local germ" worked out explicitely. Indeed, as the harmonic sums (or MZV) are the coefficients of the Taylor expansion at zero of the convergent polylogarithms divided by 1 -z, we only need to know locally these functions. In order to gain more indexing series and to describe the local situation at zero, we reshape and define a new domain of Li around zero to Dom loc (Li • ).

3. 4 .

 4 EXTENSION OF CHARACTERS: A THEORY OF DOMAINS FOR HARMONIC FUNCTIONS AND ITS SYMBOLIC COUNTERPART. The first step will be to characterize the polylogarithms having a removable singularity at zero. The following proposition helps us characterize their indices. Proposition 3.13. Let P ∈ C⟨X⟩ and f (z) = ⟨ L | P ⟩ = w∈X * ⟨P | w⟩ Li w .

  48) the family (P k ) k≥0 being unique and finitely supported. Using (3.48) and (3.39), we get Li P (z) = k≥0 Li P k (z) log(z) k .

CHAPTER 3 .

 3 CHARACTERS Proposition 3.14. Let S ∈ C⟨⟨X⟩⟩x 1 ⊕ C1 X * such that S = n≥0 [S] n where [S] n = w∈X * ,|w|=n ⟨S | w⟩w, ([S] n are the homogeneous components of S), we suppose that 0 < R ≤ 1 and that n≥0 Li [S]n is unconditionally convergent (for the standard topology) within the open disk | z |< R 33 . Remarking that 1 1 -z n≥0 Li [S]n (z)

1 n≥0

 1 n≥0 H π Y ([S]n) (N ) = a N .Proof. Let us recall that, for any w ∈ X * x 1 , the function (1 -z) -1 Li w (z) is analytic in the open disk |z| < R. Moreover, one has 1 1 -z Li w (z) = N ≥0 H π Y (w) (N )z N .Since [S] n = w∈X * ,|w|=n ⟨S | w⟩w and (1 -z) -Li [S]n absolutely converges (for the standard topology 34 ) within the open disk D <R , one obtains, for all |z| < R 1 1 -z n≥0 Li [S]n (z) = 1 1 -z n≥0 w∈X * ,|w|=n ⟨S | w⟩ Li w (z) = n≥0 w∈X * ,|w|=n ⟨S | w⟩ Li w (z) 1 -z = n≥0 w∈X * ,|w|=n

3. 4 .

 4 EXTENSION OF CHARACTERS: A THEORY OF DOMAINS FOR HARMONIC FUNCTIONS AND ITS SYMBOLIC COUNTERPART. ( * ) being possible because w∈X * ,|w|=n

3. 4 .

 4 EXTENSION OF CHARACTERS: A THEORY OF DOMAINS FOR HARMONIC FUNCTIONS AND ITS SYMBOLIC COUNTERPART. surjections namely m!S 2 (n, m). A classical formula 37 says that n≥0 m!S 2 (n, m) x n n! = (e x -1) m . (3.52) In Theorem 3.18 below, we study, for series taken in C⟨⟨X⟩⟩x 1 ⊕ C.1 X * , the correspondence Li • to some H(D <R ), first (point 1) establishes its surjectivity (in a certain sense) and then (points 2 and 3) examine the relation between summability of the functions and that of their Taylor coefficients. For that, let us begin with a very general Lemma on sequences of Taylor series which adapts, for our needs, the notion of normal families as in [85]. Lemma 3.17. Let τ = (a n,N ) n,N ≥0 be a double sequence of complex numbers. Setting I(τ ) := {r ∈]0, +∞[ | n,N ≥0 |a n,N r N | < +∞}, one has 1. I(τ ) is an interval of ]0, +∞[, it is not empty iff there exists z 0 ∈ C \ {0} such that n,N ≥0 |a n,N z N 0 | < +∞. (3.53) In this case, we set R(τ ) := sup(I(τ )) > 0 and one has, (a) For all N , the series n≥0 a n,N converges absolutely (in C). Let us note a Nwith one subscript -its limit. (b) For all n, the convergence radius of the Taylor series T n (z) = N ≥0 a n,N z N is at least R(τ ) and n∈N T n is summable for the standard topology of H(D <R(τ ) ) with sum T (z) = N ≥0 a N z N . 2. Conversely, we suppose that there exists R > 0 such that (a) Each Taylor series T n (z) = N ≥0 a n,N z N converges in H(D <R ).

37

  See[START_REF] Stanley | Enumerative Combinatorics[END_REF], the twelvefold way, formula (1.94b)(pp. 74) for instance. (b) The series n∈N T n converges unconditionnally in H(D <R ).

|

  and it is an interval of ]0, +∞[ with lower bound zero.(a) Take r ∈ I(τ ) (hence r ̸ = 0) and N ∈ N, then we get the expected result asr N n≥0 | a n,N |= n≥0 | a n,N r N |≤ n,N ≥0 | a n,N r N |< +∞.(b) Again, take any r ∈ I(τ ) and n ∈ N, thenN ≥0 | a n,N r N |< +∞ which proves that R(T n ) ≥ r, hence the result38 . We also have | a n,N r N |< +∞ and this proves that R(T ) ≥ r, hence R(T ) ≥ R(τ ).

3. 4 .

 4 EXTENSION OF CHARACTERS: A THEORY OF DOMAINS FOR HARMONIC FUNCTIONS AND ITS SYMBOLIC COUNTERPART.(ii) In point 2, the arithmetic functions H π Y (S) ∈ Q N , for S ∈ Dom R (Li) are quickly defined (and in a way extending the old definition) and we draw a very important bound saying that, in this condition, for some r > 0 the array H π Y ([S]n) (N )r N n,N converges (then, in particular, horizontally and vertically).

(

  iii) Point 3 establishes the converse. Theorem 3.18.1. Let T (z) = N ≥0

3 .H

 3 Conversely, let Q ∈ C⟨⟨Y ⟩⟩ with Q = n≥0 Q n (decomposition by weights), we suppose that there exists r ∈]0, 1] such that n,N ≥0 | H Qn (N )r N |< +∞, (3.57) in particular, for all N ∈ N, n≥0 Qn (N ) = ℓ(N ) ∈ C unconditionally (= absolutely) converges (in C). Under such circumstances, S := π X (Q) ∈ Dom r (Li) and, for all z ∈ C such that | z |< r,

  for all n ≥ 0, one clearly has Li S = T . 2. This point and next point are consequences of Lemma 3.17. Now, considering the homogeneous decomposition S = n≥0 [S] n ∈ Dom R (Li). We first establish 3.4. EXTENSION OF CHARACTERS: A THEORY OF DOMAINS FOR HARMONIC FUNCTIONS AND ITS SYMBOLIC COUNTERPART.inequation(3.56). Let 0 < r < r 1 < R and consider the path γ(t) = r 1 e 2iπt , we have

  Now, we have have a better understanding of what can (and will) be the domain, Dom(H • ), of harmonic sums. Definition 3.4. We set Dom loc

r,

  still makes sense for |z| < 1 and (s 1 , • • • , s r ) ∈ C r so that we will freely use the indexing list to get index lists withs i ∈ Z for any i = 1, • • • , r and r ∈ N + .Recall that for any s 1 , • • • , s r ∈ N, we can express Li -s 1 ,••• ,-sr (z) as a polynomial of 1 1 -z with integer coefficients. Then, using (3.47) and (kx 1 ) * = [(x 1 ) * ] ¡k , we get1 (1 -z) k = Li (kx 1 ) * (z), ∀k ∈ N +and we obtain a polynomial P ∈ Dom

m≥0 c m z m has

  radius of convergence R > 0, then m≥0 | c m | z m has the same radius of convergence (use 1/R = lim sup m≥1 | c m | 1/m ), then from the fact that S ∈ Dom R 1 (Li) (resp. T ∈ Dom R 2 (Li)), we have (3.56) for each of them and, using the Hadamard product of these expressions, we get ∀r

expressions like ( 3 .(( 1 +

 31 69), in the style of Umbral calculus by π Umbra Y : n≥1 α n q n -→ n≥1 α n y n which is linear and bijective 45 from C + [[q]] to C.Y . With this coding at hand and for S, T ∈ C + [[q]], identity (3.70) reads (π Umbra Y (S)) * (π Umbra Y (T )) * = (π Umbra Y S)(1 + T ) -1)) * . (3.71)

Figure 3 . 3 :

 33 Figure 3.3: A path z → H(z) with left multiplier H ′ (z) and the one-parameter group G(z) with infinitesimal generator c = π Umbra Y

1 )

 1 be a split (i.e. ps = Id g 2 ) and central (i.e. j(g1 ) ⊂ Z(g 3 ) = {x ∈ g 3 | [x, g 3 ] = 0})extension of Lie algebras. Then, if g 2 is commutative, so is g 3 .Proof. Let x, y ∈ g 3 . Then, since ps = Id g 2 , we have x -s (p (x)) ∈ Ker p = Im j ⊆ Z (g 3 ) , so that [x -s (p (x)) , y] = 0. In other words, [x, y] = [s (p (x)) , y]. Similarly, we find y -s (p (y)) ∈ Z (g 3 ), thus [s (p (x)) , y -s (p (y))] = 0. In other words, [s (p (x)) , y] = [s (p (x)) , s (p (y))] (5.2) = s ([p (x) , p (y)]) (since s is a Lie morphism)

  each g s is a free k-module of finite rank with basis (b i ) i∈Is , 4. the semigroup (S, +) is locally finite. This case encompasses all free partially commutative Lie algebras and, in particular, all free Lie algebras (with S = N (X) ) and Drinfeld-Kohno Lie algebras (with S = ([2, n + 1], ∨) × (N ≥1 , +) or S = (N ≥2 , ∨) × (N ≥1 , +)) and many other combinatorial Lie algebras. By Appendix 5.4.2 and the assumption that "(S, +) is locally finite", the universal enveloping U(g) = s∈S⊔{0} U s (g) (see formula (5.23)) is a finitely 2 S ⊔ {0}-graded Hopf algebra. Thus, we can consider U ∨ (g) as the graded dual of U(g) i.e. U ∨ (g) = s∈S⊔{0} U * s (g) (for the case when S = (N, +), see Grinberg and Reiner [56] § 1.6). Then one can check (as an exercise) that 1. the bases {B α } α∈N (I) of U(g) and {B α } α∈N (I) of U ∨ (g) are S ⊔ {0}-graded in finite ranks, 2. one can dualize the S ⊔ {0}-graded Hopf algebra structure of (

2 .

 2 for the limiting process due to finite pointwise convergence3 and then extends to U ∨ (g) ⊗U(g) i.e. one has a morphism of topological rings χ ⊗ Id : U ∨ (g) ⊗U(g) → (U ∨ (g)) * . Thus, by denoting(χ) gen = (χ ⊗ Id) α∈N (I) B α ⊗ B α , we deduce that (χ) gen = α∈N (I) χ(B α )B α = ↗ i∈I exp(χ(B e i )B e i ). (5.8) If χ is a k-valued µ U -character on U(g) i.e. χ ∈ Ξ(U(g), k), then the operator Id ⊗χ is continuous on U ∨ (g) ⊗ U(g) for the limiting process due to finite poinwise convergence and then extends to U ∨ (g) ⊗U(g) i.e. one has a morphism of topological rings Id ⊗χ : U ∨ (g) ⊗U(g) → U * (g). Thus, by denoting (χ) gen = (Id ⊗χ) α∈N (I) B α ⊗ B α , we then have (due to the fact that k is3 By Example 5.3, the pair U * (g)⊗U(g), Φ, End k (U(g)) is a completion triplet of topological rings.

Example 5 . 1 .

 51 If an index category I is discrete (resp. a category with two objects and two parallel morphisms from one object to the other), then a diagram F : I → C is a family of objects (resp. a pair of parallel morphisms) in C, a limit of the diagram F

  are morphisms in C. A colimit of a diagram F : I → C is the limit of the opposite diagram F op : I op → C op , where I op and C op are the opposite categories of I and C respectively, denoted by The coproducts, coequalizers and direct limits are respectively the dual concept of products, equalizers and inverse limits, they are examples of colimits in category theory.5.3 Appendix B: Topological rings, their completions and combinatorics. 5.3.1 Topological rings. Definition 5.1. A ring R endowed with a topology T R is called a topological ring if and only if sum and product are continuous operations. Precisely, the following maps R × R → R, (x, y) → x + y and (x, y) → x.y are continuous. The topology of a topological ring is uniquely determined by the filter of neighbourhoods of zero B(0) (or a base of it). For conditions on B(0), see [14] Ch III §6.3. A topological ring (R, T R ) is Hausdorff if and only if m∈B(0) m = {0} and is said to 5.3. APPENDIX B: TOPOLOGICAL RINGS, THEIR COMPLETIONS AND COMBINATORICS.be complete if every Cauchy net 8 converges to a unique limit (this entails in particular that (R, T R ) is Hausdorff).Topological rings (R, T R ) (resp. Hausdorff topological rings, complete topological rings) and continuous morphisms of rings form categories : TopRng, HausTopRng, CompHausTopRng.Let now (R, T R ) be a topological ring (i.e. (R, T R ) ∈ TopRng) and F be the inclusion functor CompHausTopRng → TopRng ("Complete Hausdorff Topological Rings" to "Topological Rings"), then, we can state Definition 5.2. Let (R, T R ) be a topological ring. A completion of R is any pair(j R , R) such that 1. ( R, T R ) ∈ CompHausTopRng2. and j R ∈ Hom TopRng (R, F ( R)) fulfilling the following property • for each morphism f ∈ Hom TopRng (R, F (S)), there exists a unique morphism f ∈ Hom CompHausTopRng ( R, S) such that f = F ( f ) • j R . In other words, (j R , R) is a solution of the following universal problem TopRng CompHausTopRng (R, T R ) (S, T S ) ( R, T R ).

Example 5 . 2 .

 52 ways to introduce a completion but in our case, we can use the following simpler characterization.Characterization. -Let (R, T R ) ∈ HausTopRng, (S, T S ) ∈ CompHausTopRng and i : R → S is a topological embedding (of type HausTopRng), which means that i is a homeomorphism between R (with the original topology) and i(R) endowed with the induced topology from S, such that i(R) is dense in S. Then, the pair (F • i, S) is a completion of R. Let us consider k⟨⟨X⟩⟩ (i.e. the k-total algebra of X * , see 5.3.2) the associative algebra of formal power series over X (with coefficients in k) and the decreasing sequence of idealsm n := {S ∈ k⟨⟨X⟩⟩ | (∀w ∈ X <n )(⟨S | w⟩ = 0)}.One can check that the ring k⟨⟨X⟩⟩ is topologized by the filter base B = {m n } n≥0 and that its topology is defined by the ultrametric distanced(T, S) = 2 -ω(T -S) ,where, for each non zero series R,ω(R) = max n∈N (R ∈ m n )is the length of the shortest words w such that ⟨R | w⟩ ̸ = 0 and we set ω(0) = +∞.With this distance, one can check that the completion of the Hausdorff topological ring k⟨X⟩ is k⟨⟨X⟩⟩ if and only if X is finite. The reason for this is that the closure of k⟨X⟩ (for this topology) is the set of series for which each isobaric component is a polynomial. In other words, writing a series S = n≥0 S n where S n := |w|=n ⟨S | w⟩ w we have (left as an exercise) k⟨X⟩ = {S ∈ k⟨⟨X⟩⟩ | (∀n)(S n ∈ k⟨X⟩)}

5. 3 .Example 5 . 3 .

 353 APPENDIX B: TOPOLOGICAL RINGS, THEIR COMPLETIONS AND COMBINATORICS. Let M be a free k-module with a fixed basis B = (β i ) i∈I . One can show the following facts 1. The usual morphism Φ : M * ⊗ M → End k (M ) is into.

4 . 5 .

 45 Therefore this topology induces a topological ring structure on M * ⊗ M , for which we can complete the tensor product as M * ⊗M and the pair (M * ⊗ M, Φ, End k (M )) is a completion triplet. For any basis B = (β i ) i∈I and every f ∈ End k (M ), the family (β i ⊗ f (β i )) i∈I

5. 3 . 2 m m and the convolution product m 1 ∈M α m 1 m 1 m 2 ∈M β m 2 m 2 m 1 ,m 2 ∈M m 1 m 2 =m α m 1 β m 2 

 321122122 Towards series: the threefold way.Total algebras (i.e. series without topology).Given k a commutative ring (with unit) and M a multiplicative semigroup which satisfies "Condition (D)" 9 i.e. for all m ∈ M the setD 2 (m) = {(m 1 , m 2 ) ∈ M × M | m 1 m 2 = m}is finite (for example, M = X * where X is a set). Let us define the total semigroup algebra k[[M ]] 10 which is the k-module of all infinite sums m∈M α product k[[M ]] is a k-AA and a k-AAU when M is a monoid.

R

  i.e. (∀m 1 , m 2 ∈ B)(∃m 3 ∈ B)(m 3 ⊆ m 1 ∩ m 2 ).(5.15)Now, due to(5.15) and as a consequence of Prop 2 §1.2[START_REF] Bourbaki | General Topology[END_REF], we can define a unique topology T R on R such that (R, T R ) is a topological ring in which B is a fundamental set of neighborhoods of zero for T R . Assume that the topology T R is Hausdorff i.e.m∈B m = {0}, then the inverse limitR = lim ← -m∈B R / m ⊆ m∈B R / min the category TopRng is a completion R of the Hausdorff topological ring (R, T R ).Case of the total algebras. -In the preceding case (here 5.3.2), for all m ∈ M , we denote the set of factors of M F act(m) := {u ∈ M | m ∈ M uM } ⊂ M. it is straightforward, using "Condition (D)", that F act(m) is a finite set and that for all m 1 , m 2 ∈ M F act(m 1 ) ∪ F act(m 2 ) ⊂ F act(m 1 m 2 ) (5.16) then m m := span k (M \ F act(m)) 5.3. APPENDIX B: TOPOLOGICAL RINGS, THEIR COMPLETIONS AND COMBINATORICS. is the largest two-sided ideal of the semigroup algebra k[M ] 11 which avoids m (i.e. J such that m / ∈ J). Now, due to (5.16), we can define a unique topology T M on k[M ] such that (k[M ], T M ) is a Hausdorff topological ring in which B = (m m ) m∈M is a fundamental set of neighborhoods of zero for T M (every set of two sided ideals closed by intersection does that, here finite intersections of elements picked in B, and moreover m∈M m m = {0}). See also discussion in [109]. Furthermore, we can make explicit the topology of this particular completion as follows, the total semigroup algebra k[[M ]] (endowed with a topology in which B = ( m m ) m∈M is a fundamental set of neighborhoods of zero, where m m = {all infinite sums u∈M \F act(m) α u u} is the largest two-sided ideal of the total semigroup algebra k[[M ]] avoiding m) is the completion k[M ] of the Hausdorff topological semigroup algebra (k[M ], T M ).

5. 4 .

 4 APPENDIX C: STRUCTURES OF BIALGEBRA TYPE.

  .4.1) from g in the standard way, for w= [s 1 ] • • • [s n ] ∈ A + , we set T w (g) = g s 1 ⊗ • • • ⊗ g sn (5.18)andT 1 A * = T (0) (g) = k.1 T (g)

  3) with a S ⊔ {0}-graded structureT (g) = s∈S⊔{0}T s (g), where T s (g) = w∈A * µ(w)=s T w (g).(5.22)Now, let us consider the ideal J of T (g) which is generated by all elements of the forma ⊗ b -b ⊗ a -[a, b] where a, b are homogeneous, say a ∈ g s , b ∈ g t . As [g s , g t ] ⊆ g s+t (s, t ∈ S), the ideal J is homogeneous with respect to the gradation(5.22). Therefore the enveloping algebra U(g) = T (g) / J can be equipped with an induced S ⊔ {0}gradationU(g) = s∈S⊔{0}U s (g), where U s (g) = T s (g) J s .(5.23)Moreover, it is easily checked that U(g) is a S ⊔{0}-graded Hopf algebra i.e. that ∆ U , ϵ U and 1 U : k → U(g) and the antipode S U are graded morphisms (k being S ⊔{0}-graded with (k) 0 = k and (k) s = {0 k } for s ∈ S + = S \ {0 S }).

5. 4 . 3

 43 Iterated smash products and sup-gradings.In order to formulate a theorem about iterated smash products, we start with (A, <) a totally ordered alphabet. Let S A := {A, ∨} be the corresponding max-semigroup (i.e. a ∨ b = max{a, b} for all a, b ∈ A) and g = a∈A g a a S A -graded Lie algebra (i.e. for all a, b ∈ A, [g a , g b ] ⊆ g max{a,b} ). Let us consider 1. the formal direct sum M = a∈A U + (g a ) (where U + (g a ) is the augmentation ideal of the universal enveloping algebra U(g a )). In this module, all U + (g a ) has degree α, i.e. (M ) α = U + (g a ).2. the language of strictly increasing words SI(A) ⊂ A * , formallySI(A) := {w ∈ A * | for all j <| w |, w[j] < w[j + 1]}where, for a word w and 1 ≤ j ≤| w |, w[j] is the letter of w at place j.

3 .

 3 the decomposition T (M ) = w∈A * T w (M ) 4. the space T SI(A) := w∈SI(A) T w (M ) where SI(A) ⊂ A * is the language of strictly increasing words 5. the language of (weakly) increasing words W I(A) ⊂ A * , formally W I(A) := {w ∈ A * | for all 1 ≤ j < |w|, w[j] ≤ w[j + 1]}.

5. 4 .

 4 APPENDIX C: STRUCTURES OF BIALGEBRA TYPE.

Remark 5 . 3 .

 53 Given a, b ∈ A with a < b, by using the fact that [g a , g b ] ⊂ g b , then the mapping α :g a → Der(g b ), x → (ad x : y → [x, y]) is a morphism in k-Lie.By Example 2.7, one has a left U(g a )-module algebra action ▷ :U(g a ) ⊗ U(g b ) → U(g b ), x ⊗ y → x ▷ y = α U (x)(y).Due to the proof of Lemma 2.26, the mappingτ : U(g a ) ⊗ U(g b ) → U(g b ) ⊗ U(g a ), x ⊗ y → (1)(2) x (1) ▷ y ⊗ x(2) is an algebra cross in k-Mod. Note that we haveτ (U + (g a ) ⊗ U + (g b )) ⊂ U + (g b ) ⊗ U + (g a ) + U + (g b ) ⊗ 1 k . Thus, if we denote τ 0 = (S a ⊗S b )•τ 12 •τ •τ 12 •(S b ⊗S a ) : U(g b )⊗U(g a ) → U(g a )⊗U(g b ),where τ 12 is the standard twist map which interchanges the two factors in the tensor product and S a (resp. S b ) is the antipode of the Hopf algebra U(g a ) (resp. U(g b )), then one has τ 0 (U + (g b ) ⊗ U + (g a )) ⊂ U + (g a ) ⊗ U + (g b ) + 1 k ⊗ U + (g b ).

Fig 6 . 4 ,Definition 5 . 3 .

 6453 Fig 6.4, here the set of states will be infinite.

F 1 .

 1 Proof. 1. i) It suffices to remark that, for every step (F, Φ, F ′ ) 1. either Φ ̸ ⊂ ES 3 and norm(F ′ ) < norm(F ) 2. or Φ ⊂ ES 3 and then (a) F = F ′ ⊂ SI(A) (b) any further step of the path is a loop.

5 . 5 .

 55 F n ⊂ SI(A), we can use the following Lemma Let f : M → N be a linear morphism of two modules. We suppose given (M i ) i∈I a directed family of submodules (of M ) such that, setting M ∞ := i∈I M i the submodule of M , 13 See the equation (5.27).

5. 4 .

 4 APPENDIX C: STRUCTURES OF BIALGEBRA TYPE. (a) For all i ∈ I, f | M i is injective. (b) f | M∞ is onto.

31 )

 31 Remark 5.4. i) Any computationt 1 → t 2 → • • • → t n

  23) of S ⊔ {0}graded Hopf algebras U(g) in case g being the infinite Drinfeld-Kohno Lie algebra DK k,∞ (for the definition of this Lie algebra, see Proposition 2.21) and moreover we have studied that DK k,∞ = m≥2 DK (m) k,∞ is a (N ≥2 , ∨)-graded Lie algebra over k (see the formula (2.110)).

5. 4 .

 4 APPENDIX C: STRUCTURES OF BIALGEBRA TYPE.

(5. 34 )

 34 we can model as ↖≡↘ or ↙≡↗ .(I 3 )5.5 Appendix D: Maple Outputs.5.5.1 Lyndon basis and its dual.

  The (endo)functor Seq. . . . . . . . . . . . . . . . . . . . . . . . .

	Chapter 1
	Preamble
	Contents
	1.1.

15 1.2. Free objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.1. General principle. . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.2. Presented structures. . . . . . . . . . . . . . . . . . . . . . . 17 1.2.3. Free monoids. . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.4. Free associative algebras. . . . . . . . . . . . . . . . . . . . . 25 1.2.5. Free Lie algebras. . . . . . . . . . . . . . . . . . . . . . . . . 27 1.3. An example of free object with respect to graphs. . . . . . . . . .

  1 , α) and (H 2 , H 1 , β) (resp. (g 2 , g 1 , α) and (h 2 , h 1 , β)) is an equivariant pairs of group morphisms f i : G i → H i

	(resp. Lie algebra morphisms f i : g i → h i ) i.e. such that (in both cases) the following
	diagram

  .58) 2.2. A GENERALIZATION OF LAZARD'S ELIMINATION THEOREM.Due to the fact that the homogeneous polynomials linearly generate L k (Z) by(1.31) and(1.32), it is sufficient to prove this for homogeneous (Lie) polynomials of degree n for all n ∈ N ≥1 , we will do it by induction. Let then Q ∈ L k (Z) n .

  where rn is the Lie isomorphism as in Diagram (2.14)), we have the diagram

  2.92) and look there for a formal definition of the category of SSES) will be called "Short Exact Sequence with Section (SESS)"). Thus, if the extension of Lie algebras is split, then there is a Lie

). The extension of Lie algebras is said to split if SES (2.89) is split i.e. there is a Lie algebra homomorphism σ : g r → g such that p • σ = Id g r (σ is called a section of p. In this case (2.89) can be pictured as below CHAPTER 2. LAZARD'S ELIMINATION (see (

  union structure (i.e. the coproduct) of a family of subsets {X s } s∈S . Furthermore, if S is an additive commutative semigroup, an object M (resp. Lie bracket respects this gradation [g s , g t ] ⊆ g s+t , the multiplication respects this gradation A s .A t ⊆ A s+t ).

	Commentary 4. We summarize these definitions by the following table
	Structure	Grading support	Formula	Internal structure Global structure
	Set	Set	X = s∈S X s	subsets	No
	Mon or Grp	Monoid

g, A) in one of categories

Mon or Grp (resp. k-Lie, k-AAU) is said to be S-graded if it can be written as M = s∈S M s a disjoint union of a family of subsets {M s } s∈S (resp. g = s∈S g s an internal direct sum of a family of ksubmodules {g s } s∈S , A = s∈S A s an internal direct sum of a family of k-submodules {A s } s∈S ) such that the binary operation respects this gradation M s .M t ⊆ M s+t (resp. the

Table 1 :

 1 S-graded structures for the list of categories, where "Internal structure"

means "Algebraic internal structures of the components".

Remark 2.8.

1. In case M ∈ Mon and (N, +, 0) the commutative monoid of nonnegative integers, if we assume that M 0

  A FREE OBJECT.3. The Hilbert series of direct sum and standard tensor product of two finitely graded modules A and B is Hilb(A⊕B, t) = Hilb(A, t)+Hilb(B, t) and Hilb(A⊗B, t) = Hilb(A, t).Hilb(B, t), respectively. Now we attempt to find a bivariate Hilbert series of U(DK k,n+1 ) the universal enveloping algebra of the Drinfeld-Kohno Lie algebra that has a nice description in the next subsection 2.3.3 (see Definition 2.3 and Example 2.5). Firstly, we say that the monovariate Hilbert series are known. Example 2.3. (cf. Kohno [70], Etingof et al. [44]) The Hilbert series of the universal enveloping algebra of the Drinfeld-Kohno Lie algebra is

  SSLie, for short we will designate SESS (2.97) by x 1 and SESS (2.98) by x 2 , then the function

	2.3. LAZARD ELIMINATION AS A FREE OBJECT.
	and
	0	g l 2	j 2	g 2	p 2 σ 2	g r 2	0	(2.98)
	in k-
								97)

  3. LAZARD ELIMINATION AS A FREE OBJECT. (then they are commutative diagrams in the functor categories [Set 2 , B-GrLie] and [B-GrLie, Set 2 ], respectively.) 2.3.3 Drinfeld-Kohno Lie algebra with infinite number of generators: from strange to generalized gradings. Now, for any integer n ∈ N, let ([2, n + 1], ∨) be the upper semi-lattice (where we use the classical supremum i ∨ j := sup{i, j} for all i, j ∈ [2, n + 1]). We remark that, ([2, n + 1], ∨) is also a commutative semigroup. We then claim that iterated decompositions of the Drinfeld-Kohno Lie algebras are naturally graded by supremum.

	More general semigroups and strange gradings.

In fact, we now describe a ([2, n + 1], ∨)-graded structure of the Drinfeld-Kohno Lie

algebra DK k,n+1 = L k (T n+1 ) J R[n+1]

as follows: for each j ∈ [2, n + 1], we introduce DK (j) k,n+1 the Lie subalgebra of DK k,n+1 generated by the set [T j

  Proposition 2.21. Consider the infinite Drinfeld-Kohno Lie algebra DK k,∞ , defined by the quotient of the free Lie algebra L k (T ∞ ) generated by an infinite set of noncommutative variables T ∞ = {t i,j } 1≤i<j<+∞ modulo the Lie ideal J R[∞] generated by infinitesimal pure braid relations

), then lim -→ DK k,n the direct limit with such structure homomorphisms (2.106) has a very simple and in a sense a tame description by the following proposition 2.3. LAZARD ELIMINATION AS A FREE OBJECT.

  otherwise. As a final consequence, we can translate this series 2.3. LAZARD ELIMINATION AS A FREE OBJECT. into the commutative algebra of formal power series of two variables Q[[t 1 , t 2 ]] (endowed with the convolution product that is with the supremum w.r.t. t 1 and ordinary Cauchy product w.r.t. t 2 20

Table 2 :

 2 Lazard's elimination for the list of categories.

  Proposition 3.3. The linear isomorphism Φ -1 : End(U(g)) → U * (g) ⊗U(g) sending any f to α∈N (I) B α ⊗ f (B α ). It is a morphism of rings from the convolution algebra End(U(g)) to the complete tensor product U * (g) ⊗U(g) as in Example 5.3. Proof. Proof omitted (general fact, see Example 5.3 or see

now work out what is written in Reutenauer's Prop 1.10 [94] by the following 3.2. CONVOLUTION ALGEBRA AND FACTORIZATION.

  15 .). In fact, the story began with Euler's works to find the solution The numbers ζ(s 1 , s 2 ) were called "double zeta values" at (s 1 , s 2 ). More generally, for any r ∈ N + and s 1 , • • • , s r ∈ C, we denote

					CHAPTER 3. CHARACTERS
	of Basel problem. In these works, Euler proved that [45]
	ζ(2) =	n≥1	1 n 2 =	π 2 6	.		(3.27)
	Moreover, for suitable s 1 , s 2 , Euler gave an important identity as follows 16 :
	ζ(s 1 )ζ(s 2 ) = ζ(s 1 , s 2 ) + ζ(s 1 + s 2 ) + ζ(s 2 , s 1 ),	(3.28)
	where
	ζ(s 1 , s 2 ) :=	n 1 >n 2 ≥1	1 n s 1 1 n s 2 2	.	(3.29)

  30 . ) is in the domain of Li if and only if the family (Li Sn ) n≥0

	3.4. EXTENSION OF CHARACTERS: A THEORY OF DOMAINS FOR
	HARMONIC FUNCTIONS AND ITS SYMBOLIC COUNTERPART.
	(so that S =
	Definition 3.3. (i) Let S ∈ C⟨⟨X⟩⟩ be a series decomposed in its homogeneous
	(w.r.t. the length) components
	S n =	⟨S | w⟩ w
	|w|=n

n≥0

S n

  .4. EXTENSION OF CHARACTERS: A THEORY OF DOMAINS FOR HARMONIC FUNCTIONS AND ITS SYMBOLIC COUNTERPART. Proof. For equation (3.60), we get, from Proposition 3.15 that Dom loc (Li) is the union of an increasing set of shuffle subalgebras of C⟨⟨X⟩⟩ (the map R → Dom R (Li) is strictly decreasing). It is therefore a shuffle subalgebra of the latter. For equation (3.61), suppose S ∈ Dom R 1 (Li) (resp. T ∈ Dom R 2 (Li)). By [59] and Theorem 3.18 point (3.55), for |z| < R 1 R 2 , one has

  43 (A, * 1 , 1 A ) with values in (B, * 2 , 1 B ) is none other than a morphism between the k-algebras A and a commutative algebra44 B. The algebra (A, * 1 , 1 A ) does not have to be commutative, for example characters of (C⟨X⟩, conc, 1 X * ) -i.e. conc-characters -can be easily proved to be all of the form

	*
	α x x	(3.69)
	x∈X
	They are closed under shuffle and stuffle and endowed with these laws, they form
	a group. Expressions like the infinite sum within brackets in (3.69) (i.e. homogeneous
	series of degree 1) form a vector space noted C.Y .
	As a consequence, given P =

i≥1 α i y i and Q = j≥1 β j y j , we know in advance that their stuffle is a conc-character i.e. of the form ( n≥1 c n y n ) * . Examining the effect of this stuffle on each letter (which suffices), we get the identity i≥1

  6 F : I → C, if exists, is an object

		lim ←-i∈I	F i	or lim ← -
					lim ← -I	F
						(5.10)
					ϕ i	ϕ j
		F i		Fα	F j
	commutes.
	ii) moreover, the limit lim ← -I	F is the (initial) universal object with this property
					X
					u
				ψ i	lim ← -I	F	ψ j	(5.11)
					ϕ i	ϕ j
		F i		Fα	F j .
	Notice that limits (lim ← -I	F, (ϕ

I

F for short in C together with morphisms

ϕ i : lim I F → F i such that

i) for all morphisms α : i → j in I the triangle i ) i∈I ) are (if they exist) unique up to isomorphism by the uniqueness requirement

(5.11)

of the point (ii) in the above definition.

  . . T j T j+1 . . . T k . . . t 1,2 . . . t 1,j t 1,j+1 . . . t 1,k . . . ⇐⇒ t i,j t j,k -t j,k t i,j + t i,k t j,k -t j,k t i,k ≡ 0 ⇐⇒ t j,k t i,j ≡ t i,j t j,k + t i,k t j,k -t j,k t i,k R 3 [∞]: For the case |{i, j, k, l}| = 4, [t i,j , t k,l ] ≡ 0 ⇔ t i,j t k,l ≡ t k,l t i,j

	. . . . . . . . .	. . . . . .	. . .
	. . . t i,j t i,j+1 . . . t i,k	. . .
	. . . . . .	. . . . . .	. . .
	t j,j+1 . . . t j,k	. . .
		. . . t j+1,k . . .
	we can model the last equality of (5.32) as follows
			(5.33)
	and then
	↖≡↘ + ↓ -↑ .	(I 2 )
	Replacing (I 2 ) to (I 1 ) we have that
	←-≡-→ + ↑ -↓ .	(I ′ 1 )
	3.

←-≡-→ + ↘ -↖ . (I 1 ) 2. R 2 [∞]: For 1 ≤ i < j < k < +∞.

Similarly as above, we have

[t i,j + t i,k , t j,k ] ≡ 0

  Table of families {P w } w∈X * (Lyndon basis) and its dual {S w } w∈X * .

Résumé. Ce mémoire est tout entier consacré à la réécriture des inversions dans certaines structures avec produit, leur réarrangements et les contreparties combinatoires de ces transformations pour les partitions d'alphabets c'est à dire l'élimination de Lazard de générateurs (LE) et les formules associées (en particulier dans leurs quotients). Les théorèmes du type (LE) donnent lieu à des formules uniformes pour tous les alphabets et ont des schémas similaires pour les groupes, les monoïdes, les algèbres de Lie et les algèbres associatives avec unité. Ces outils donnent lieu à de nombreux algorithmes implémentables. La forme la plus simple de (LE) se produit dans la catégorie des kalgèbres de Lie (k étant un anneau unitaire), nous nous concentrons sur les monoïdes et les algèbres de Lie et donnons des exemples sur des "smash-produits" itérés pour lesquels la réécriture des mots ("string rewriting") joue un rôle crucial non seulement dans la compréhension des formes normales, mais encore dans la façon dont on converge vers elles. La fin de cette thèse se concentre sur les utilisations supplémentaires de (LE) et de l'indexation de mots dans le contexte du half-shuffle, des algèbres de Zinbiel et de la dualité de Magnus. En outre, il aborde le sujet des hyperlogarithmes et de la théorie des caractères.

This rather vague sounding question as well as question

heading subsection 2.2.3 (i.e. "About M.-P. Schützenberger's questions on the Partially Commutative Free Lie algebra") have been reinterpreted in the language of (algebraic) categories as the "search of the source of a free (forgetful) functor".3 Of a Lie algebra linearly free, over a Q-algebra as announced in[START_REF] Reutenauer | Free Lie Algebras[END_REF].

Or pointwise finitely supported families, see[START_REF] Grinberg | Hopf algebras in Combinatorics, version of 27[END_REF] Def 1.7.2.

More rigorously, finite or denumerable sets, N-graded by finite blocks.

Freeness is here with respect to sets. To be more complete we will give in Section 1.3 an example of freeness with respect to graphs.

This means that, for all i ∈ I, s(u i ) = s(v i ).

This means that, for all i ∈ I, φ(u i ) = φ(v i ).

For abelian groups (noted additively)u i = v i is equivalent to u i -v i = 0. For groups noted multiplicatively u i = v i is equivalent to u i v -1 i = 1.

This means that, for all i ∈ I, s(u i ) = s(v i ).

This means that, for all i ∈ I, φ(u i ) = φ(v i ).

Description with regular expressions (extended with "set minus" for the second term, see[START_REF]Regular expression[END_REF]).

Usually, the initial conditions are 0, 1, • • • .

Classical for the categories Grp and k-Lie, smash product for k-AAU (see the book[START_REF] Montgomery | Crossed Products of Hopf Algebras and Enveloping Algebras[END_REF] by Susan Montgomery) and by left translations for Mon.

For two subgroups A, B ⊂ G, (A, B) is the subgroup generated the commutators (x, y) :=x -1 y -1 xy, x ∈ A, y ∈ B.

See the discussion after this question, especially Andreas Thom's answer.

For colimits, see Appendix 5.2. Moreover, we can survey the categorical framework of semi-direct products in the context of Grp by discussions in MO [108].

It is sufficient to test the equivariance on generators as stressed in the last part of remark (2.3).

This means that g = h ⋊ b is an internal semi-direct product.

The maps pr i , i = 1, 2 standing respectively for the first and second projections of the cartesian product.

With I = X.

For D-invariance, see Lemma 1.3.

Understated, but which can be unfolded on request.

As a type, R is a mere subset of L k (Y 1 ) but "relator" means that it is intended to be the generating set of an ideal.

It was originally invented and named such in Kohno[START_REF] Kohno | Série de Poincaré-Koszul associée aux groupes de tresses pures[END_REF] and Etingof et al.[START_REF] Etingof | The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points[END_REF] respectively.

Where the semi-direct decomposition of DK k,n+1 is also established.

In fact, B is a monoid.

In computer science, it means that each s ∈ S admits only a finite number of factorizationss = s 1 + • • • + s k (k ∈ N ≥1 ) i.e. the map µ : [S] + = [S] * \ 1 [S] * , (s 1 , . . . , s k ) → s 1 + . . . + s k has finitefibers, see Eilenberg[START_REF] Eilenberg | Automata, Languages and Machines[END_REF]. Remarkable that "locally finite" induces "Condition (D)", but the converse is not true in general.

If k is only a ring, embeddings may fail as shows the example of Z / 3Z = A and Z = B as two Z-algebras.

See Appendix 5.4.4, Remark 5.6 point (iii).

The readers can find definitions about bi-and Hopf algebras in[START_REF] Grinberg | Hopf algebras in Combinatorics, version of 27[END_REF].

Will be defined below Section

3.2.

In the sense of[START_REF] Duchamp | Deformations of Algebras: Twisting and Perturbations[END_REF].

Deformation is in the sense of[START_REF] Méliot | Representation theory of symmetric groups[END_REF] Ch 4 i.e. such that, at q = 0 we get the shuffle product and at q = 1 we get the stuffle product.

Ω ⊂ C is a simply-connected domain convenient for the system (3.38) as, for instance,Ω = C \ (] -∞, 0] ∪ [1, +∞[).

Whence the famous sum ζ(-1) = 1 + 2 + 3 + • • • = -1 12by which, among other "results", S.Ramanujan was noticed by G. H. H. Hardy (see[START_REF] Berndt | Ramanujan: Letters and Commentary[END_REF]).

Here, the shuffle product is denoted by ¡. Its definition is classical and recalled in the equation (3.66) of Subsection 3.4.5.

In fact, it was shown that this morphism is into, see[START_REF] Hoang Ngoc Minh | Shuffle algebra and polylogarithms[END_REF] Thm 4.

Here ⊙ stands for the Hadamard product[START_REF] Hadamard | Théorème sur les séries entières[END_REF].

Readers who are not keen on topology or functional analysis may skip the details of this section and hold its conclusions.

This topology is defined by the seminorms (whereK ⊂ Ω is compact) p K (f ) = sup s∈K |f (s)|.

Space where commutatively convergent and absolutely convergent series are the same. This will allow the domain of the polylogarithm to be closed by shuffle products (i.e. the possiblity to compute legal polylogarithms through shuffle products).

As the fact that, due to special properties of H(Ω) (it is a nuclear space[START_REF] Schaefer | Topological Vector Spaces[END_REF]), one can show that Dom(Li) is closed by shuffle products.

The domain, for z of Li P .

For any set E, the set of its subsets is noted 2 E .

For a Taylor series T , we note R(T ) the radius of convergence of T .

This definition is compatible with the old one when S is a polynomial.

Throughout this paragraph, notations will be those of Section 3.2.

It means that each k-module U s (g) is of finite rank.

See e.g. Kassel[START_REF] Kassel | Quantum groups[END_REF] Ch III §8 Exercise 11.

The order is always (space, product, unit, coproduct, counit, antipode).

i.e. a functor from a small category I to an arbitrary category C. We say that C is a diagram of the index category I.

i.e. a poset such that for any elements i, j ∈ I, there exists an element k ∈ I such that i ≤ k and j ≤ k.

It is a net[START_REF] Vermeeren | Sequences and nets in topology[END_REF] 110] (x α ) α∈A (A is a directed set) such that for all m ∈ B(0) there exists B ∈ A such that, for all α, β ≥ B, x α -x β ∈ m (this means that the set {x α } α≥B is m-small).

Bourbaki [10] Ch III § 2.10.

This associative algebra is unital if M admits a unit i.e. M is a monoid.

For this algebra, see Clifford and Preston[START_REF] Clifford | The Algebraic Theory of Semigroups[END_REF].

For a modern version, see[START_REF] Hopcroft | Introduction to Automata Theory, Languages, and Computation[END_REF].

Acknowledgments

I would like to express my sincere gratitude to my director, Professor Gérard H. E.

Duchamp, for his unwavering encouragement, guidance and support throughout my doctoral studies. Under his supervision, I was introduced to the vast and fascinating world of mathematics and was provided with invaluable approaches to solving problems in the field of "Combinatorics, Physics and their Interactions". Additionally, I am deeply grateful to my co-director, Professor Vincel Hoang Ngoc Minh, for his useful insights and ideas that greatly contributed to the success of my PhD thesis.

I thank to Professor Jean-Gabriel Luque (Université de Rouen) and Professor Darij Grinberg (Drexel University) for their valuable comments and reports which greatly contributed to the improvement of this manuscript.

Chapter 6

Bibliography

CHAPTER 3. CHARACTERS i) As was said, when q = 0, we get the shuffle product: y 2 y 1 ¡ y 3 y 1 y 2 = y 2 (y 1 ¡ y 3 y 1 y 2 ) + y 3 (y 2 y 1 ¡ y 1 y 2 ) = y 2 y 1 y 3 y 1 y 2 + 2y 2 y 3 y 2 1 y 2 + y 2 y 3 y 1 y 2 y 1 + 2y 3 y 2 y 2 1 y 2 +y 3 y 2 y 1 y 2 y 1 + y 3 y 1 y 2 y 1 y 2 + 2y 3 y 1 y 2 2 y 1 .

One can also see this product as indexed by paths (with North and East steps) from A to B as below

For instance, evaluation of the path

and evaluation of the path

is y 3 y 2 2 y 2 1 ; etc.

ii) When q = 1, one gets the stuffle product: y 2 y 1 y 3 y 1 y 2 = y 2 (y 1 y 3 y 1 y 2 ) + y 3 (y 2 y 1 y 1 y 2 ) + y 5 (y 1 y 1 y 2 )

= y 2 y 1 y 3 y 1 y 2 + 2y 2 y 3 y 2 1 y 2 + y 2 y 3 y 1 y 2 y 1 + y 2 y 3 y 1 y 3 + y 2 y 3 y 2 2 +y 2 y 4 y 1 y 2 + 2y 3 y 2 y 2 1 y 2 + y 3 y 2 y 1 y 2 y 1 + y 3 y 2 y 1 y 3 + y 3 y 3 2 +y 3 y 1 y 2 y 1 y 2 + 2y 3 y 1 y 2 2 y 1 + y 3 y 1 y 2 y 3 + y 3 y 1 y 4 y 1 + y 2 3 y 1 y 2 +y 2 3 y 2 y 1 + y 2 3 y 3 + 2y 5 y 2 1 y 2 + y 5 y 1 y 2 y 1 + y 5 y 1 y 3 + y 5 y 2 2 .

iii) In general, one can also see the q-stuffle product as indexed by paths (with North, North-East and East steps, each North-East path provides a q) from A to B as below

For instance, the path

y 5 evaluates as q.y 5 y 2 y 2 1 ;

and the path

evaluates as q 2 .y 3 y 4 y 2 ; etc.

iv) The complete q-stuffle product containing the two terms above is y 2 y 1 q y 3 y 1 y 2 = y 2 (y 1 q y 3 y 1 y 2 ) + y 3 (y 2 y 1 q y 1 y 2 ) + qy 5 (y 1 q y 1 y 2 )

= y 2 y 1 y 3 y 1 y 2 + 2y 2 y 3 y 2 1 y 2 + y 2 y 3 y 1 y 2 y 1 + qy 2 y 3 y 1 y 3 + qy 2 y 3 y 2 2 +qy 2 y 4 y 1 y 2 + 2y 3 y 2 y 2 1 y 2 + y 3 y 2 y 1 y 2 y 1 + qy 3 y 2 y 1 y 3 + qy 3 y 3 2 +y 3 y 1 y 2 y 1 y 2 + 2y 3 y 1 y 2 2 y 1 + qy 3 y 1 y 2 y 3 + qy 3 y 1 y 4 y 1 + qy 2 3 y 1 y 2 +qy 2 3 y 2 y 1 + q 2 y 2 3 y 3 + 2qy 5 y 2 1 y 2 + qy 5 y 1 y 2 y 1 + q 2 y 5 y 1 y 3 + q 2 y 5 y 2 2 .

In fact, q is the dual of the comultiplication ∆ q : k⟨Y ⟩ → k⟨Y ⟩ ⊗ k⟨Y ⟩ defined on letters by ∆ q (y s ) = y s ⊗ 1 Y * + 1 Y * ⊗ y s + q.

p+q=s p,q≥1 y p ⊗ y q and, with the grading ||w||

Z conc (X) := (k⟨X⟩, conc, 1 X * , ∆ r ¡ , ϵ). This mathematical framework provides useful computing techniques that will be presented to the reader as below.

Recall the Lazard's elimination in the category k-AAU meaning there exists the natural algebra isomorphism f * : k⟨B * Z⟩♯k⟨B⟩ → k⟨X⟩ which has been completely studied in Section 2.4. Let us denote k⟨X⟩ BZ ⊕ k.1 X * := f * (k⟨B * Z⟩) the subalgebra of the free associative algebra k⟨X⟩ generated by {rn(wz)} w∈B * ,z∈Z ∪{1 X * }. The following proposition will be the first step on the way of presenting our proof of Theorem 3.9.

Proposition 3.10. (Conc & Half shuffle duality) The families of Magnus polynomials rn(w

Proof. We assume that

over B, equal to 0 or 1 if the chains coincide not respectively), we now have computing processes

where, the first equality passing from the adjoint (for the scalar product

studied in Appendix 5.4.5, and the second equality passing from the compatible relation ∆ r ¡ (T P ) = ∆ ¡ (T )∆ r ¡ (P ) (where T, P polynomials) in Example 5.8 point (ii), Appendix 5.4.5. We also have the expression ∆ ¡ [rn(u i z)] = rn(u i z) ⊗ 1 + 1 ⊗ rn(u i z)

for each primitive elements rn(u i z) because we have the fragment {rn(u i z)} 1≤i≤k-1 ⊆ L k (X) ⊆ Prim H conc (X), see more details in Remark 5.6 point (ii), Appendix 5.4.4.

The following lemma is necessary for the proof.

Thus, for any N ∈ N, for readability, below 1 stands for 1

H π Y (10(6x 1 ) * -38(5x 1 ) * +55(4x 1 ) * -37(3x 1 ) * +11(2x 1 ) * -x * 1 ) (N ) = -

Observe that, from Definition 3. for n ≥ 1 and T 0 = 0. Direct calculation gives, for n ≥ 1,

However one can get unconditional convergence using a sommation by pairs (odd + even).

(ii) For all s ∈]1, +∞[, the series

We can now state the

3.4.5 Stuffle products, usage of one-parameter subgroups within stuffle characters and their symbolic computations.

It is well-known that, a Hopf algebra (H, µ, 1 H , ∆, ϵ, S) and A ∈ k-CAAU (a kcommutative and associative algebra with unit), the set Ξ(H, A)

is a group for convolution (and inverse performed through precomposition with S).

When k is a Q-algebra and under the usual condition that the reduced coproduct 40

, this group can be considered as a Lie group (an infinite-dimensional pro-unipotent 41 one) with a nice log-exp correspondence.

This feature is used in combinatorial physics [START_REF] Duchamp | One-Parameter Groups and Combinatorial Physics[END_REF] and one-parameter groups is a nice tool to get new combinatorial identities (see [START_REF] Gérard | Recipe theorem for the Tutte polynomial for matroids, renormalization group-like approach[END_REF]) as in the sequel.

For the some reader's convenience, we recall here the definitions of shuffle and stuffle products. As regards shuffle, the alphabet X is arbitrary and ¡ is defined by the following recursion (for x, y ∈ X and u, v

As regards stuffle, the alphabet is Y = Y N + = {y s } s∈N + and is defined by the following recursion

Be it for stuffle or shuffle, the noncommutative 42 polynomials equipped with this product form an associative commutative and unital algebra namely (C⟨X⟩, ¡,1 X * ) (resp.

As examples of characters, we have already seen

, where X = {x 0 , x 1 }. 40 Here, H + denote the kernel of ϵ and for all Bourbaki [13] Ch II §1.1. 41 In this context, this means that for any element g in this group, the family ((g -1 Ξ ) ⋆n ) n≥0 is summable in Ξ(H, A), where 1 Ξ = 1 A • ϵ denotes the unit element in the group Ξ(H, A). 42 For concatenation.

Chapter 4

Conclusions and future directions

Starting from groups and their associated Lie algebras in semi-direct product form, we investigated the presented premiages analogs which are precisely Lazard's elimination process. This study took place principally within the category k-Lie of Lie algebras over a ring k where a practical sufficient condition was found in order to recognize, at first sight, whether the presented structure could be split in semi-direct form. To this end was developed, successively, the notion of B = ({0, 1}, ∨)-graded structures and for iterated such, a larger notion of S-graded structures (Lie and their enveloping algebras) where (S, +) is a commutative semigroup.

Moreover, conditions on S where investigated in order to transfer the classical machinery to explicit formulas and the specialized notion of the direct sum and tensor products for the Hilbert series of any S-graded algebra in finite dimensions, S being an additive commutative semigroup. Essentially the links with "Condition (D)" (Bourbaki [START_REF] Bourbaki | Algebra[END_REF] Ch III § 2.10.), and locally finite semigroups of computer science (Eilenberg [START_REF] Eilenberg | Automata, Languages and Machines[END_REF]) are made precisely. Then, we allow ourselves to examine this definition by Example 2.4 and Example 2.5.

Questions (Q1) 1 and (Q2) 2 have been reformulated in terms of algebraic structures and "free functors", this elusive "free functor" has been completely worked out (source, 1 What are the expressions of Lazard elimination (LE) in several categories where there are sort of semi-direct products?

2 Are these universal? (i.e. is every semi-direct product the image of some Lazard eliminations?)

and is LE a free object?

target and formulas) in the case of the category of Lie k-algebras. This question is parallel to a similar one with "Free Partially Commutative Structures" [START_REF] Duchamp | Free partially commutative structures[END_REF] for which all targets of the functor are known 3 . The (unpublished) resolution of this "free functor" has been completely worked out and reformulated in Section 1.3. Remains to complete the same work for LE i.e.

1. Consider "Free looking" elimination formulas (see Table 2.5).

2. Find the correct enrichment for Monoids and Groups (because B-grading works also for k-AAU, the category of unital associative k-algebras).

3. Find the sources and functors (if possible, the source could be "double sets", as for partially commutative structures for which the source is unique whatever the target category).

The hope is that this technique based on a filtration of the alphabet of generators in conjunction with an appropriate filtration of the relators could apply to other algebras coming from combinatorics or geometry. Let us say a word about this last point. For many fibered spaces (in particular configuration spaces), the fundamental group of the base space acts on the fibers by automorphisms and then acts on the fundamental group of the fibers. Then appear natural semi-direct products of groups. In general, the lower central series (see Chapter 2) goes too fast for transferring these semi-direct products to Lie algebras by the associated graded algebra mechanism gr (see equation (2.4)), but other appropriate filtrations are usually considered by Nakamura and Takao [START_REF] Nakamura | Galois rigidity of pro-l pure braid groups of algebraic curves[END_REF] and recently by Sawada [START_REF] Sawada | Reconstruction of invariants of configuration spaces of hyperbolic curves from associated Lie algebras[END_REF]. We think that our elimination techniques can ease the understanding and calculations of these semi-direct products.

3 And in which LE plays a crucial tool.

where τ 0 is the "twist" of the smash product (see Remark 5.3). It can be shown that

therefore the result of this reduction process belongs to T uabv (M ) ⊕ T ubv (M ).

Second type (Reduction of powers) α = ({ua p v}, φ α , {uav}) with p ≥ 2, by (e) Composition of Arrows:

), we classically have t(P ) = t(α 1 ) and h(P ) = h(α n ).

5.4. APPENDIX C: STRUCTURES OF BIALGEBRA TYPE.

The evaluation of P , Ev(P ) is the composition of all the linear maps of its arrows i.e. with P = α 1 • • • α n ,

Norm: For all w ∈ A * , we set norm(w) = 3 (|w|+Inv(w)) (where Inv(w) = ♯{(i, j)|1 ≤ i < j ≤ |w| and w[i] > w[j]}). This definition is at once extended to finite subsets of F ⊂ A * by norm(F ) = w∈F norm(w). We remark that, for all elementary arrow α of the two first types, norm(t(α)) > norm(h(α)) and equality is got for the third type. Hence, for any general arrow α = (F 1 , Φ, F 2 ), norm(t(α)) > norm(h(α)) unless F 1 = F 2 ⊂ SI(A) in which case we have equality and all arrows of Φ are of third type.

(g) Aperiodic paths: An aperiodic path is a path whose last arrow has identical head and tail i.e. α n = (F, Φ, F ), this entails that F ⊂ SI(A) and that all arrows of Φ are of third type. i.e. can :

Then, with the notations and constructions above, one has 1. Every sufficiently long path in Γ trans with origin F 1 is aperiodic and ends with a subset F n ⊂ SI(A). More precisely, let F 1 be a finite subset of A * (i.e. F 1 ∈ 2 (A * ) )

and

(where, again,

We consider a path of Γ trans originating from F 1

(5.29) CHAPTER 5. APPENDIXES Let us recall that T ∞ = {t i,j } 1≤i<j<+∞ be a set of non commutative variables and then, the universal enveloping algebra of the infinite Drinfeld-Kohno Lie algebra is presented as

which is the quotient of the free associative algebra k⟨T ∞ ⟩ modulo the ideal J R[∞]

generated by infinitesimal pure braid relations

where for convenience, for all polynomials P, Q ∈ k⟨T ∞ ⟩, we will denote by

ii) A pair of adjacent letters t i,j t k,l is called an inversion if j > l.

iii) A word of form t i 1 ,j 1 t i 2 ,j 2 . . . t in,jn ∈ T * ∞ , for n ≥ 0 and 2 ≤ j 1 ≤ j 2 ≤ . . . ≤ j n (it means it has no inversion), is called in normal form, and the set of all such words named Inc(∞).

Due to this definition, we can expand the above relations as follows:

(5.32) CHAPTER 5. APPENDIXES Note that moving ↘, ↖ between two positions having row-index of this position equal to column-index of the other position, but↘, ↖ having different indices one by one.

The three modeling equalities (I ′ 1 ), (I 2 ) and (I 3 ) help to represent any word in combination of normal forms. Firstly we have the following lemma.

Lemma 5.6. Let w be a word having inversion t i,j t k,l (i.e. w = w 1 t i,j t k,l w 2 such that j > l). We can represent this inversion part (only the inverted factor) by a combination of normal forms.

Proof. In this case k < l < j. We then have the following cases.

• If |{i, j, k, l}| = 4, by (I 3 ) we have w ≡ w 1 t k,l t i,j w 2 .

• If |{i, j, k, l}| = 3, it is one in the two subcases:

we have w ≡ w 1 t i,l t i,j w 2 + w 1 t l,j t i,j w 2 -w 1 t i,j t l,j w 2 .

if i = l, it means w = w 1 t i,j t k,i w 2 . By (I 2 ) (note that k < i < j) we have w ≡ w 1 t k,i t i,j w 2 + w 1 t k,j t i,j w 2 -w 1 t i,j t k,j w 2 .

We have the following simple examples. Note that Lemma 5.6 can be used to remove an inversion between some two successive positions of a word w over T ∞ but, sometimes solving this inversion can make a new inversion, however this procedure terminates. In fact, at this stage we are going to study a linear basis in the universal enveloping algebra of the infinite Drinfeld-Kohno

Lie algebra DK k,∞ .

Proposition 5.7. The image of the set of normal forms, s ∞ (Inc(∞)), is a k-linear basis of the associative k-algebra U(DK k,∞ ).

Proof. This proposition is a direct application of Remark 5.4 (ii) to the alphabet A =

We now arrive to write an algorithm to implement that any element w = t i 1 ,j 1 . . . t in,jn of T * ∞ can be expressed as a linear combination of normal forms as follows.

Algorithm 1 Algorithm represents a word in form of combination of words in

Hopf structures of the k-module of noncommutative polynomials.

We first review in [START_REF] Reutenauer | Free Lie Algebras[END_REF] §1.5 that the k-module k⟨X⟩ of the noncommutative polynomials has two natural graded Hopf algebra structures which are dual to each other (in the graded sense).

The first one is H conc (X) = (k⟨X⟩, conc, 1 X * , ∆ ¡ , ϵ), where the cocommutative coproduct ∆ ¡ (so called the co-shuffle coproduct) defined as ∆ ¡ : k⟨X⟩ → k⟨X⟩ ⊗ k⟨X⟩ (5. [START_REF] Duchamp | Kleene Stars in Shuffle algebras, CAP Days[END_REF] which is a unique algebra homomorphism for which the words x ∈ X, are primitive elements

(5.36) More precisely, for any polynomial T ,

The counit ϵ : k⟨X⟩ → k is given by ϵ(T ) = ⟨T | 1 X * ⟩ for all polynomials T . The antipode S : k⟨X⟩ → k⟨X⟩ is the k-linear map defined, over words w ∈ X * , by 5.4. APPENDIX C: STRUCTURES OF BIALGEBRA TYPE. S(w) = (-1) |w| w, where w reverses the order of letters in the word w.

We now state to the following remarks Remark 5.5.

i) We recall that the universal enveloping algebra of a Lie k-algebra g is a pair (σ, U(g)), where U(g) is an object in k-AAU and σ : g → U(g) is a morphism in k-Lie, which is a solution of the following universal problem:

This arises that there exists the universal enveloping functor

which is a left-adjoint to the Liezation functor F .

ii) We also recall a noncommutative cocommutative Hopf structure of the enveloping algebra U(g) of a Lie k-algebra g. We construct a Lie algebra morphism δ : g → U(g) ⊗ U(g) by the formula

for any x ∈ g. By the universal property (5.37), δ extends to a unique algebra homomorphism ∆ U : U(g) → U(g) ⊗ U(g).

(5.40)

Moreover, there exists a k-linear map (called the antipode) S U : U(g) → U(g) characterized by S U (x) = -x for any x ∈ g, and an algebra homomorphism (that is the counit) ϵ U : U(g) → U(0) = k induced from the projection g → 0 (where 0 is the trivial Lie algebra). The reader can easily verify that (U(g), µ U ,

satisfying the axioms of a Hopf algebra. ii) In the case H = U(g) the universal enveloping algebra of a Lie k-algebra g, by the above Hopf structure of U(g) one observes that any element in g is primitive i.e.

g ⊆ Prim U(g) (and coincides if k is a field of characteristic zero, cf. Bourbaki [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch II §1.5 Corollary to Prop 9 or Cartier [START_REF] Cartier | A primer of Hopf algebras[END_REF] Thm 3.6.1).

iii) Let us consider now from the first part of Subsection 1.2.5, it provides that the free Lie algebra L k (X) is the Lie subalgebra of k⟨X⟩ generated by X. Together with its canonical map j : L k (X) → k⟨X⟩, it is not hard to see that the pair (j, k⟨X⟩) is a solution of the universal problem (5.37) in the above remark. We thus deduce that the free associative algebra k⟨X⟩ is the universal enveloping algebra U(L k (X)) of the free Lie algebra L k (X), these algebras are inherited the same Hopf structure H conc (X) = (k⟨X⟩, conc, 1 X * , ∆ ¡ , ϵ) (cf. Bourbaki [13] Ch II §3.1 Cor 1). Then, it is used to obtain the embedding L k (X) ⊆ Prim H conc (X) (they are equal if k is a field of characteristic zero, cf. Bourbaki [START_REF] Bourbaki | Lie groups and Lie algebras[END_REF] Ch II §3.1

Cor 2).

The second Hopf algebra structure will be the graded dual of H conc (X), namely H ¡ (X) = (k⟨X⟩, ¡,1 X * , ∆ conc , ϵ). That is a commutative graded Hopf algeba with the shuffle product and the deconcatenation coproduct ∆ conc : k⟨X⟩ → k⟨X⟩ ⊗ k⟨X⟩ given by, over the words w ∈ X * , ∆ conc (w) = w=uv u ⊗ v.

(5.41)

More precisely, for any polynomial T , we can write

The counit and the antipode are respectively given by ϵ and S as previously.

Zinbiel bialgebra and its dualisation.

According to [START_REF] Burgunder | A symmetric version of Kontsevich graph complex and Leibniz homology[END_REF], Burgunder constructed completely Zinbiel bialgebras and its dualisation in case the left (≺) Zinbiel product. In view of later developments, we will similarly study the fundamental results of Burgunder's framework in case the right (≻)

Zinbiel product with these notations translate into x ≺ y := y ≻ x.

We recall in Definition 3.2 that a k-module A over k a unital commutative ring equipped with a bilinear map ≻ is called (right) Zinbiel algebra (or dual Leibniz algebra) if it satisfies the identity

x ≻ (y ≻ z) = (x ≻ y) ≻ z + (y ≻ x) ≻ z, for all x, y, z ∈ A.

(5.42)

The behavior of ≻ with respect to the unit 1 A ∈ A is given by (for any 1 A ̸ = x ∈ A) (5.43)

The tensor product of a Zinbiel algebra (A, ≻) with itself will be a Zinbiel algebra equipped with following Zinbiel structure:

where (x

In particular, for unital setting, More precisely, ∆(x) ≻ ∆(y) = (x (1) ⊗ k x (2) ) ≻ (y (1) ⊗ k y (2) ) = (x (1) ≻ y (1) ) ⊗ k (x (2) * y (2) ), here we used a shorthand Sweedler's notation ∆(x) = (1)(2) x (1) ⊗ k x (2) ≡ x (1) ⊗ k x (2) , for any x ∈ Z (this is similar to Einstein's summation convention in mathematical physics, where the dummy summation index (1)(2) is dropped). We now study a dualisation of the above Zinbiel bialgebra. Let us consider the co-half-shuffle coproduct ∆ r ¡ : k⟨X⟩ → k⟨X⟩ ⊗ k⟨X⟩ which is a k-linear map defined over words as follows ∆ r

or, equivalently, for each polynomial T such that ⟨T | 1 X * ⟩ = 0, then ∆ r

here we used the notation ⟨w | 1 X * r ¡ 1 X * ⟩ = 0 for any non empty word w.

Example 5.8. We have the following i) It is easy to verify that (Id ⊗∆ r Thus, (k⟨X⟩, ∆ r ¡ , ϵ) is a counital (right) Zinbiel coalgebra. Furthermore, the cooperation ∆ * = τ ∆ r ¡ + ∆ r ¡ = ∆ ¡ .

ii) The k-module k⟨X⟩ of noncommutative polynomials on X has two natural generalized graded bialgebra type structures which are dual to each other (in the graded sense). The first one is the Zinbiel bialgebra (or As c -Zinb-bialgebra)

Z r ¡ (X) = (k⟨X⟩, r ¡,1 X * , ∆ conc , ϵ) discussed in Example 5.7, and its graded dual is a Zinb c -As-bialgebra Z conc (X) := (k⟨X⟩, conc, 1 X * , ∆ r ¡ , ϵ), where the co-half- shuffle ∆ r ¡ satisfies the compatible relation ∆ r

i.e. for any polynomials T, P then ∆ r ¡ (T P ) = ∆ ¡ (T )∆ r ¡ (P ).