Structure of Polyzetas and Explicit Representation on Transcendence Bases of Shuffle and Stuffle Algebras

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Structure of Polyzetas and Explicit Representation on Transcendence Bases of Shuffle and Stuffle Algebras

V Bui, G Duchamp, Ngoc Minh

To cite this version:

V Bui, G Duchamp, Ngoc Minh. Structure of Polyzetas and Explicit Representation on Transcen- dence Bases of Shuffle and Stuffle Algebras. Journal of Scientific Computing, Springer Verlag, 2016,

�10.1016/j.jsc.2016.11.007�. �hal-01427741v2�

Structure of Polyzetas and Explicit Representation on Transcendence Bases of Shuffle and Stuffle Algebras

V.C. Bui

, G. H. E. Duchamp

, V. Hoang Ngoc Minh

♭Hue University of Sciences, 77 - Nguyen Hue street - Hue city, Vietnam

♯Institut Galilée, LIPN - UMR 7030, CNRS - Université Paris 13, F-93430 Villetaneuse, France,

♢Université Lille II, 1, Place Déliot, 59024 Lille, France

Abstract

Polyzetas, indexed by words, satisfy shuffle and quasi-shuffle identities. In this respect, one can explore the multiplicative and algorithmic (locally finite) proper- ties of their generating series. In this paper, we construct pairs of bases in duality on which polyzetas are established in order to compute local coordinates in the infinite dimensional Lie groups where their non-commutative generating series live. We also propose new algorithms leading to the ideal of polynomial relations, homogeneous in weight, among polyzetas (the graded kernel) and their explicit representation (as data structures) in terms of irreducible elements.

Keywords: Poincaré-Birkhoff-Witt basis; transcendence basis; Schützenberger’s factorization; noncommutative generating series; shuffle algebra; polyzetas.

1. Introduction

This paper will provide transparent arguments and proofs for results presented at the International Symposium on Symbolic and Algebraic Computation confer- ence, Bath, 6-9 July, 2015 [6].

For any composition, s = ( s

, . . . , s

) , the polyzetas [10] (also called multiple zeta values [33]) are defined by the following convergent series

ζ ( s

, . . . , s

) ∶= ∑

n1>...>nr>0

n

. . . n

, for s

> 1. (1) Any composition s ∈ ( N

)

can be associated to words of the form w = x

x

. . . x

x

∈ X

x

(resp. w = y

. . . y

∈ Y

) [20, 21], where X

(resp.

Y

) is the free monoid generated by the alphabet X = { x

, x

} (resp. Y = { y

}

)

admitting 1

(resp. 1

) as unit. The weight of composition s is defined the pos- itive integer s

+ . . . + s

which correspond to the length, denoted by ∣ w ∣ (resp. the sum of indexes, denoted by ( w ) ), of its associated word on X (Y ).

Using concatenation, shuffle and quasi-shuffle products, in Section 2,

1. we will recall the definition of Hopf algebras ( Q ⟨ X ⟩ , ● , 1

, ∆

, e ) and ( Q ⟨ Y ⟩ , ● , 1

, ∆ , e ) .

2. we will equip X with the (total) ordering x

< x

and denoting L ynX , the set of Lyndon words over X, the

PBW-basis { P

}

will be ex- panded over the basis { P

}

, of the free Lie algebra L ie

⟨ X ⟩ . Its dual basis { S

}

contains the pure transcendence basis of the algebra ( Q ⟨ X ⟩ ,

, 1

) denoted by { S

}

[29].

3. similarly, equipping Y with the (total) ordering y

> y

> y

> . . . and denoting L ynY the set of Lyndon words over Y , the basis { Π

}

, of the free Lie algebra of primitive elements

, and its associated PBW-basis { Π

}

will be proposed. The dual basis { Σ

}

is polynomial and contains also a pure transcendence basis of the algebra ( Q ⟨ Y ⟩ , , 1

) denoted by { Σ

}

[5, 23, 24].

4. we then establish the two following expressions of the diagonal series D

∶= ∑

w∈X^∗

w ⊗ w = ∏

l∈LynX

exp ( S

⊗ P

) , (2) D

∶= ∑

w∈Y^∗

w ⊗ w = ∏

l∈LynY

exp ( Σ

⊗ Π

) . (3) From these, in Section 3,

1. we will consider two generating series of polyzetas

[21, 20, 23, 24]:

Z

∶= ∏

l∈LynX X

exp ( ζ ( S

) P

) and Z ∶= ∏

l∈LynY {y1}

exp ( ζ ( Σ

) Π

) .(4)

1PBW : Poincaré-Birkhoff-Witt.

2P is a primitive element if∆ (P) =1Y^∗⊗P+P⊗1Y^∗.

3The coefficients ofZ⊔⊔ (resp.Z ) represent the finite parts of the asymptotic expansions of the polylogarithms{Liw}w∈X^∗ (resp. the harmonic sums{Hw}w∈Y^∗), at1(resp. at+∞), in the scale of comparison{(1−z)^−alog^b((1−z)⁻¹)}a∈Z,b∈N(resp.{N^aH^b1(N)}a∈Z,b∈N) [23].

2. we have also defined a third one

, Z

[11], which satisfies, via Schützen- berger’s factorization

on the completed Hopf algebra [23, 24] ,

Z

= e

Z . (5)

3. in order to identify the local coordinates of Z

(and Z ), on a group of associators [23, 24], we will rely on the following comparison (see [11])

Z

= B ( y

) π

( Z

) , where B ( y

) = exp ( γy

− ∑

k≥2

(− 1 )

ζ ( k )

k y

) . (6) Here, π

∶ Q ⊕ Q ⟪ X ⟫ x

Ð→ Q ⟪ Y ⟫ is the linear projection

mapping x

x

. . . x

x

to y

. . . y

.

By cancellation

[23, 24], (5) and (6) yield the following identity Z = B

( y

) π

( Z

) , where B

( y

) = exp (∑

k≥2

( − 1 )

ζ ( k )

k y

) . (7) 4. simultaneously, algorithms will be also implemented in Maple to represent polyzetas

in terms of irreducible polyzetas producing algebraic relations among the local coordinates

{ ζ ( S

)}

(and { ζ ( Σ

)}

) [7].

To end this section, let us point out some crucial points of our purposes : 1. Similar tables

for { ζ ( l )}

have been obtained up to weight 10 [20],

12 [1] and 16 [31]. These differ from the zig-zag relation among the moulds of formal polyzetas, due to Ecalle [15], i.e. the commutative generating se- ries of symbolic polyzetas (Boutet de Monvel [4] and Racinet [17] have also given equivalent relations for the noncommutative generating series of symbolic polyzetas, see also [10]) producing linear relations and base them- selves on regularized double shuffle relation [2, 25, 27] and from identities among associators, due to Drinfel’d [12, 13, 19].

4The coefficients ofZγ represent the finite parts of the asymptotic expansions of{Hw}w∈Y^∗, in the scale of comparison{N^alog^b(N)}a∈Z,b∈N[11]. Here,γdenotes the Euler’s constant.

5Also called MRS factorization after Mélançon, Reutenauer and Schützenberger.

6It can be extended toQ⟪X⟫with the conventionπY(w) =0for eachwending byx⁰.

7Not byspecificationγto0.

8running on a computer Core(TM)i5-4210U CPU @ 1.70GHz, up to weight12[7].

9Since, in (4), onlyconvergentpolyzetas arise then we do not need any regularization process.

10They form a Gröbner basis of the ideal of polynomial relations among the convergent polyze- tas and the ranking of this basis is based mainly on the order of Lyndon words [1, 20, 31]. For that, this basis is also called Gröbner-Lyndon basis.

2. In the classical theory of finite-dimensional Lie groups, every ordered basis of the Lie algebra provides a system of local coordinates in a suitable neigh- borhood of the unity of the group via an ordered product of one-parameter groups corresponding to the ordered basis [32]. In this work, we get a per- fect analogue of this picture for Hausdorff groups, through Schützengerger’s factorization, this doesn’t depend on regularization (see the next remark).

Moreover, through the bridge equation (6) relating two elements on these groups and by identification of local coordinates, in infinite dimension, of their L.H.S. and R.H.S. (which involve only convergent polyzetas) we get again a confirmation of Zagier’s conjecture, up to weight 12. This is not a consequence of regularized double-shuffle relation (see the next remarks).

3. Of course, the generating series given in (4) and (5) induce, as already shown in [23, 24], three morphisms of (shuffle and quasi-shuffle) algebras, studied earlier in [21, 22, 11] and constructed in [23, 24]

ζ

∶ ( Q ⟨ X ⟩ ,

, 1

) Ð→ ( R, ., 1

) , (8) ζ ∶ ( Q ⟨ Y ⟩ , , 1

) Ð→ ( R, ., 1

) , (9) γ

∶ ( Q ⟨ Y ⟩ , , 1

) Ð→ ( R, ., 1

) , (10) which satisfy, for any u = x

x

. . . x

x

∈ x

X

x

and v = π

( u ) ,

ζ

( u ) = ζ ( v ) = γ

= ζ ( s

, . . . , s

) (11) and the algebraic generators of length 1 satisfy (see Footnotes 3, 4)

ζ

( x

) = ζ

( x

) = ζ ( y

) = 0 and γ

= γ. (12) Hence, ζ

, ζ and γ

are characters of (shuffle and quasi-shuffle) Hopf algebras, and their graphs, written as series, respectively read [22, 11]

∑

w∈X^∗

ζ

( w ) w = Z

, ∑

w∈Y^∗

ζ ( w ) w = Z , ∑

w∈Y^∗

γ

w = Z

(13) and

Z

= ( ζ

⊗ Id

)D

, Z = ( ζ ⊗ Id

)D

, Z

= ( γ

⊗ Id

)D

. 4. By (4), for any u, v ∈ L ynX X and u

= π

( u ) , v

= π

( y ) , one has

ζ

( u ) ζ

( v ) = ζ

( u

v ) and ζ ( u

) ζ ( v

) = ζ ( u

v

) . (14) By (7), for any l ∈ L ynX X and l

= π

( l ) , one has on the other hand

11They are group-like :∆⊔⊔(Z⊔⊔) =Z⊔⊔⊗Z⊔⊔,∆ (Z ) =Z ⊗Z ,∆γ(Zγ) =Zγ⊗Zγ.

i) ζ

( x

l − x

l ) = − ζ

( x

l ) = − ⟨ Z

∣ x

l ⟩ , ii) ζ ( y

l

− y

l

) = − ζ ( y

l

) = − ⟨ Z ∣ y

l

⟩ , iii) ⟨ B

( y

) ∣ y

⟩ = 0.

This means that since (7) is equivalent to (6) then, for the quasi-shuffle product, the regularization to γ is equivalent to the regularization to 0 [23, 24] and this yields immediately the family of regularized double shuffle relations considered in [1, 2, 17, 20, 27, 31] (see also [4, 10, 25, 26, 30]).

Our method is then quite different with [4, 10, 30] for which their authors suggest the simultaneous regularization the divergent polyzetas ζ

( 1 ) to the indeterminate

T . But by this way, they obtain relations among polyzetas which are formal because they depend mainly on numerical values

of T . 2. Background

2.1. Generalities

Let w = y

. . . y

∈ Y

, the length and the weight of the word w are defined respectively by the numbers ∣ w ∣ = k and ( w ) = s

+ . . . + s

.

Let us define the commutative product on

QY , denoted by µ (see [9, 16]),

∀ y

, y

∈ Y, µ ( y

, y

) = y

, (15) or its dual coproduct, ∆

, defined by

∀ y

∈ Y, ∆

y

=

∑

i=1

y

⊗ y

(16)

satisfying,

∀ x, y, z ∈ Y, ⟨ ∆

x ∣ y ⊗ z ⟩ = ⟨ x ∣ µ ( y, z )⟩ . (17) Let Q ⟨ Y ⟩ be equipped by

1. The concatenation ● (or by its associated coproduct, ∆

).

12i.e. ζ⊔⊔(x⁰) =ζ⊔⊔(x¹) = ζ (y¹) = T (to compare with (12)) and sinceT is transcendent overRthen it can be suitable to be specialized to0, as effectively done in [17, 27].

13Since theR-algebra of polyzetas is not aR[T]-algebra then how to precise these values ?

14QY denotes theQ-vector space generated by the alphabetY, as a basis.

2. The shuffle product, i.e. the commutative product defined by [29], for any y

, y

∈ Y and u, v, w ∈ Y

w

1

= 1

w = w,

y

u

y

v = y

( u

y

v ) + y

( y

u

v ) (18) or by its associated coproduct, ∆

, defined, on the letters by,

∀ y

∈ Y, ∆

y

= y

⊗ 1

+ 1

⊗ y

(19) and extended so as to make it a homomorphism for the concatenation prod- uct. It satisfies

∀ u, v, w ∈ Y

, ⟨ ∆

w ∣ u ⊗ v ⟩ = ⟨ w ∣ u

v ⟩ . (20) 3. The quasi-shuffle product, i.e. the commutative product defined by [25], for

any y

, y

∈ Y and u, v, w ∈ Y

, w 1

= 1

w = w,

y

u y

v = y

( u y

v ) + y

( y

u v ) + µ ( y

, y

)( u v ) (21) or by its associated coproduct, ∆ , defined, on the letters by,

∀ y

∈ Y, ∆ y

= ∆

y

+ ∆

y

(22) and extended so as to make it a homomorphism for the concatenation prod- uct. It satisfies

∀ u, v, w ∈ Y

, ⟨ ∆ w ∣ u ⊗ v ⟩ = ⟨ w ∣ u v ⟩ . (23) Note that ∆

and ∆ are morphisms from Q ⟨ Y ⟩ for the concatenation but

∆

is not (for example ∆

( y

) = y

⊗ y

, whereas ∆

( y

)

= 0).

Hence, with the counit e defined by, for any P ∈ Q ⟨ Y ⟩ , e ( P ) = ⟨ P ∣ 1

⟩ , we get two pairs of mutually dual bialgebras

H

= ( Q ⟨ Y ⟩ , ● , 1

, ∆

, e ) , H

= ( Q ⟨ Y ⟩ ,

, 1

, ∆

, e ) , (24) H = ( Q ⟨ Y ⟩ , ● , 1

, ∆ , e ) , H

= ( Q ⟨ Y ⟩ , , 1

, ∆

, e ) . (25) Let us then consider the following diagonal series

D

= ∑

w∈Y^∗

w ⊗ w and D = ∑

w∈Y^∗

w ⊗ w. (26)

15Of course, we have (set theoretically)D⊔⊔ = D , but their structural treatments will be different.

Here, in D

and D , the operation on the right factor of the tensor product is the concatenation, and the operation on the left factor is the shuffle and the quasi- shuffle, respectively.

By the Cartier-Quillen-Milnor and Moore (CQMM in the sequel) theorem [9], the connected N-graded, co-commutative Hopf algebra H

is isomorphic to the enveloping algebra of the Lie algebra of its primitive elements which is L ie

⟨ Y ⟩ : H

≅ U(L ie

⟨ Y ⟩) and H

≅ U(L ie

⟨ Y ⟩)

. (27) Hence, denoting ( l

, l

) the standard factorization

of l ∈ L ynX X, let us con- sider

1. the PBW-Lyndon basis { P

}

constructed recursively as follows [29]

⎧⎪⎪⎪⎪

⎨⎪⎪⎪ ⎪⎩

P

= y

for y

∈ Y,

P

= [ P

, P

] for l ∈ L ynY Y, st ( l ) = ( l

, l

) , P

= P

. . . P

k

for

with

l1,...,l_k∈LynY, l1>...>l_k.

(28)

Example 1. i ) Considering on the alphabet Y ∶

P

= y

, P

= y

, P

= y

y

− y

y

, P

= y

y

y

− y

y

y

+ y

y

y

− y

y

y

. ii ) Considering on the alphabet X = { x

, x

} , x

< x

∶

P

= x

, P

= x

x

− x

x

, P

= x

y

− 2x

x

x

+ y

x

, P

0x²₁x0x1

= x

x

x

x

− x

x

x

+ 2x

x

x

x

x

+ 2x

x

x

x

x

x

− x

x

x

+ x

x

x

x

− x

x

x

x

− 2 x

, 1 x

x

x

x

+ x

x

x

+ x

x

x

− 2x

x

x

x

x

− x

x

x

x

. 2. and, by duality

, the basis { S

}

of ( Q ⟨ Y ⟩ ,

) , i.e.

∀ u, v ∈ Y

, ⟨ P

∣ S

⟩ = δ

. (29)

16A couple of Lyndon words(l1, l2)is called the standard factorization oflifl=l1l2andl2is the smallest nontrivial proper right factor ofl(for the lexicographic order).

17The dual family,i.e.the set of coordinates forming a basis in the algebraic dual which is here the space of noncommutative series, but as the enveloping algebra under consideration is graded in finite dimensions (by the multidegree), these series are in fact multi-homogeneous polynomials.

This linear basis can be computed recursively as follows [29]

⎧⎪⎪⎪⎪

⎪⎪ ⎨⎪⎪⎪

⎪⎪⎪⎩

S

= y

, for y

∈ Y,

S

= y

S

, for l = y

u ∈ L ynY, S

= S

. . .

S

k

i

! . . . i

!

for w = l

. . . l

, with

l

, . . . , l

∈ L ynY, l

> . . . > l

.

(30)

Example 2. i ) Considering on the alphabet Y ∶

S

= y

, S

= y

, S

= y

y

, S

= y

y

y

+ y

y

y

. ii ) Considering on the alphabet X ∶

S

= x

, S

= x

x

, S

= x

x

, S

0x²₁x0x1

= x

x

x

x

+ 3x

x

x

x

+ 6x

x

.

Similarly, by CQMM theorem, the connected N-graded, co-commutative Hopf algebra H is isomorphic to the enveloping algebra of its primitive elements:

Prim (H ) = Im ( π

) = span

{ π

( w )∣ w ∈ Y

} , (31) where, for any w ∈ Y

, π

( w ) is obtained as follows [23, 24]

π

( w ) = w +

(w)

∑

k=2

( − 1 )

k ∑

u1,...,uk∈Y⁺

⟨ w ∣ u

. . . u

⟩ u

. . . u

. (32) Note that (32) is equivalent to the following identity

w = ∑

k≥0

1

k! ∑

u1,...,u_k∈Y^∗

⟨ w ∣ u

. . . u

⟩ π

( u

) . . . π

( u

) . (33)

In particular, for any y

∈ Y , the primitive polynomial π

( y

) is given by π

( y

) = y

+

∑

( −1 )

l ∑

j1,...,ji≥1,j1+...+ji=s

y

. . . y

. (34) Example 3. π

( y

) = y

, π

( y

) = y

−

y

, π

( y

) = y

−

( y

y

+ y

y

) +

y

. As previously, the expressions (34) are equivalent to

y

= ∑

i≥1

1 i! ∑

s1+...+si=s

π

( y

) . . . π

( y

) , y

∈ Y . (35)

Example 4.

y

= π

( y

) , y

= π

( y

) + 1

2! π

( y

)

, y

= π

( y

) + 1

2! ( π

( y

) π

( y

) + π

( y

) π

( y

)) + 1

3! π

( y

)

.

Now let us consider the (endo-)morphism of algebras φ ∶ ( Q ⟨ Y ⟩ , ● ) → ( Q ⟨ Y ⟩ , ● ) verifying φ ( y

) = π

( y

) , it can be shown that it this an automorphism of Q ⟨ Y ⟩ . Then [24],

i) φ realizes an isomorphism from the bialgebra ( Q ⟨ Y ⟩ , ● , ∆

, e ) to the bial- gebra ( Q ⟨ Y ⟩ , ● , ∆ , e ) .

ii) we have the following commutative diagram Q ⟨ Y ⟩

φ

Q ⟨ Y ⟩ ⊗ Q ⟨ Y ⟩

φ⊗φ

Q ⟨ Y ⟩

//

Q ⟨ Y ⟩ ⊗ Q ⟨ Y ⟩ .

iii) ∆ ○ φ = φ ⊗ φ ○ ∆

.

iv) H ≅ U ( Prim (H )) and H

≅ U( Prim (H ))

.

v) the dual bases { Π

}

and { Σ

}

of respectively U ( Prim (H )) and U ( Prim (H ))

can be obtained as images, by respectively φ and φ ˇ

, of respectively { P

}

and { S

}

.

More precisely,

1. the PBW-Lyndon basis { Π

}

for U( Prim (H )) constructed recur- sively as follows

[5, 23, 24]

⎧⎪⎪⎪⎪

⎨⎪⎪⎪ ⎪⎩

Π

= π

( y

) for y

∈ Y,

Π

= [ Π

, Π

] for l ∈ L ynY Y, st ( l ) = ( l

, l

) , Π

= Π

1

. . . Π

k

for

with

l1,...,l_k∈LynY, l1>...>l_k.

(36) Example 5.

Π

= y

, Π

= y

−

y

, Π

= y

y

− y

y

,

Π

= y

y

y

−

y

y

− y

y

y

+

y

y

− y

y

y

+

y

y

y

+

y

y

−

y

y

y

− y

y

y

+

y

y

+ y

y

y

+

y

y

y

−

y

y

+

y

y

.

18In other words, the family{Πw}w∈Y^∗ is the images of the family{Pw}w∈Y^∗ by the isomor- phism of bialgebrasφ[24].

2. and, by duality, the basis { Σ

}

of (Q ⟨ Y ⟩ , ), i.e.

∀ u, v ∈ Y

, ⟨ Π

∣ Σ

⟩ = δ

. (37) This linear basis can be computed recursively as follows

[5, 23, 24]

⎧⎪⎪⎪⎪

⎪⎪⎪⎪ ⎨⎪⎪⎪

⎪⎪⎪⎪⎪

⎩

Σ

= y

, for y

∈ Y, Σ

= ∑

(☆)

1

i! y

Σ

, for l = y

. . . y

∈ L ynY, Σ

= Σ

1

. . . Σ

k

i

! . . . i

! , for w = l

. . . l

, with l

> . . . > l

∈ L ynY.

(38)

In ( ☆ ) , the sum is taken over all { k

, . . . , k

} ⊂ { 1, . . . , k } and all l

≥ . . . ≥ l

such that ( y

, . . . , y

) ⇐

( y

, . . . , y

, l

, . . . , l

) , where ⇐

denotes the transitive closure of the relation on standard sequences, denoted by ⇐ [5].

Example 6.

Σ

= y

, Σ

= y

, Σ

= y

y

+

y

,

Σ

= y

y

y

+ y

y

y

+ y

+

y

y

+

y

+

y

y

.

We insist on the fact that, the families { Σ

}

and { S

}

are basically different in the sense of they make two systems of local coordinate and two lists of irreducible elements (like Groebner-Lyndon basis, see below Table 1. In fact we have in general π

( Σ

) ≠ S

, i.e. almost l ∈ L ynY and π

( S

) ≠ Σ

i.e.

almost l ∈ L ynX . This does not occur with the Lyndon words themselves, which was provided in [20], because π

( l ) ∈ L ynY for any l ∈ L ynX.

2.2. Local coordinates

Following Wei-Norman’s theorem [32], we know that for a given (finite di- mensional) Lie group

G, its Lie algebra g, and a basis B = ( b

)

of g, there exists a neighbourhood W of 1

(in G) and n local coordinate k -valued

analytic functions

W → k , ( t

)

19In other words, the family{Σw}w∈Y^∗ is the image of the family{Sw}w∈Y^∗ by the (linear) automorphismφˇ⁻¹[24].

20Real (withk=R) or complex (withk=C).

21See footnote above.

such that, for all g ∈ W , g =

→

∏

1≤i≤n

e

= e

. . . e

.

The proof relies on the fact that, ( t

, . . . , t

) → e

. . . e

is a local diffeo- morphism from k

to G at a neighbourhood of 0.

Example 7 (Wei-Norman in finite dimensions). Let M ∈ Gl

( 2, R ) (Gl

( 2, R ) denote the connected component of 1 in the Lie group Gl ( 2, R ) (it is the group of matrices with positive determinant)

M = ( a

a

a

a

)

In order to perform the decomposition, we will “go back to identity” by computing M T DU = I, where I stands for the identity matrix and T is upper unitriangular, D diagonal strictly positive and U unitary, then M = U

D

T

will be the Iwasawa [3] decomposition of M.

1. (orthogonalization) we perform block-computation on the columns of M to obtain an orthogonal matrix

M Ð→ ( a

a

a

a

)( 1 t

0 1 ) = M T = ( a

a

a

a

) = ( C

C

) = M

.

the both of columns are orthogonal if t

= −

a²₁₁+a²₂₁

. 2. (normalization) We normalize M

,

M

= ( C

C

) ⎛

⎝

1

∣∣C₁⁽¹⁾∣∣

0

0

∣∣C₂⁽¹⁾∣∣

⎞

⎠ = M

D

= M

e

−log(∣∣C1⁽¹⁾∣∣)⎛

⎜⎝

1 0 0 0

⎞⎟

⎠−log(∣∣C2⁽¹⁾∣∣)⎛

⎜⎝

0 0 0 1

⎞⎟

⎠

.

3. (unitarization) As the columns of M

form an orthogonal basis and as det ( M

) > 0, one can write

M

= ( a

a

a

a

) = ( cos ( t

) − sin ( t

) sin ( t

) cos ( t

) ) = e

t2

⎛⎜

⎝

0 1

− 1 0

⎞⎟

⎠

,

and as M

is in a neighbourhood of I

, one has t

= arctan (

) .

4. (summary)

M T D = M

= e

arctan(^a21_a11)⎛

⎜⎝

0 1

− 1 0

⎞⎟

⎠

, hence

M = e

a21

a11)(₋⁰1 ¹0)

D

T

= e

a21

a11)(₋⁰1 ¹0)

e

1 0

0 0)

e

(1)

2 ∣∣)(⁰0 ⁰1)

e

⟨C1∣C2⟩

∣∣C1∣∣2 (⁰0 ¹0)

. One then gets a Wei-Norman decomposition of M with respect to the basis of the Lie algebra gl ( 2, R ) : ( 0 1

− 1 0 ) , ( 1 0

0 0 ) , ( 0 0

0 1 ) , ( 0 1 0 0 ) .

Now, in infinite dimensions, i.e. here within the algebra of double series (whose support is Y

⊗ Y

) endowed with the law

⊗● ˆ , we have Schützenberger’s factorization(s) [29] as a perfect analogue of Wei-Norman’s theorem for the group of group-like series. For D

D

=

↳ l∈LynY

∏

exp ( S

⊗ P

) ∈ H

⊗ ˆ H

;

or with the law ⊗● ˆ , we also have the extension of Schützenberger’s factorization for D is then [5, 23, 24]

D =

↳

∏

l∈LynY

exp ( Σ

⊗ Π

) ∈ H

⊗ ˆ H .

These can be used to provide a system of local coordinates on the Hausdorff group (i.e. group of group-like elements

). Applying these factorizations to the multiple zeta functions ζ

, ζ , or to Z

and Z (which are all group-like), we have the representations

Z

=

↳

∏

l∈LynX∖X

e

, and Z =

↳

∏

l∈LynY∖{y1}

e

.

It means that, all relations among polyzetas which can be seen here will be taken from relations among their local coordinates. Our method is due to identity (7) to reduce relations between the two systems of local coordinates { ζ ( S

)}

and { ζ ( Σ

)}

.

22These series are, in fact, characters for^⊔⊔(resp. )

23The dual bases{Pl}l∈LynX and{Sl}l∈LynX are computed by (28), (30).

3. Structure of polyzetas

3.1. Representations of polynomials on bases

The aim of this subsection is to provide a method to represent any polynomial of Q ⟨ Y ⟩ in terms of each base { P

}

, { S

}

, { Π

}

or { Σ

}

.

Recall that the bases { P

}

and { Π

}

are homogeneous and upper tri- angular, the bases { S

}

and { Σ

}

are homogeneous and lower triangu- lar

. Without loss of generality we can assume that P ∈ Q ⟨ Y ⟩ is a homogeneous polynomial of weight n, we now represent P in terms of the basis { Σ

}

by the following algorithm.

Algorithm 1

INPUT: A homogeneous polynomial P of weight n.

OUTPUT: The representation of P in terms of the basis { Σ

}

.

Step 1. We choose the leading term

of P , assumed λ

w

. Expressing the word w

as follows

w

= Σ

+ ∑

v<w1,(v)=n

α

v. (39)

The polynomial P can now be rewritten in the form P = λ

Σ

+ ∑

v<w1,(v)=n

β

v. (40)

Step 2. We repeat Step 1 with P now understood as the polynomial ∑

β

v , and so on until the last monomial which admits the smallest word of weight n, y

, and we really have y

= Σ

. At last, by re-putting the coefficients, we will obtain the representation of the original in form that

P = ∑

v≤w1,(v)=n

λ

Σ

. (41)

Example 8. P ∶= 2y

y

− 1 / 2y

.

Step 1. Since Σ

= y

y

+ y

y

+ y

, we replace y

y

with Σ

− y

y

− y

in P P = 2Σ

− 2y

y

− 5 / 2y

.

24w.r.t the words and the lexicographic ordering, for example,Σw=w+∑v<w,(v)=(w)αvv.

25This term includes the greatest word in the support ofPand its coefficient.