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

Bùi Chiên, Gérard H.E. Duchamp, Vincel Hoang Ngoc Minh

To cite this version:

Bùi Chiên, Gérard H.E. Duchamp, Vincel Hoang Ngoc Minh. Structure of Polyzetas and Explicit

Representation on Transcendence Bases of Shuffle and Stuffle Algebras. Journal of Symbolic Compu-

tation, Elsevier, 2016. �hal-01403859�

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 November 28, 2016

Contents

1 Introduction 2

2 Background 5

2.1 Generalities . . . . 5

2.2 Local coordinates . . . 10

3 Structure of polyzetas 12 3.1 Representations of polynomials on bases . . . 12

3.2 Identifying the local coordinates . . . 13

3.2.1 Identifying with respect to the basis { Π

}

. . . 14

3.2.2 Identifying with respect to the basis { P

}

. . . 15

3.3 Algorithms to represent the structure of polyzetas . . . 16

3.4 Results . . . 18

3.4.1 Representation of polyzetas in terms of irreducible polyzetas 18 3.4.2 Conclusion of the results . . . 19

4 Conclusion 22

Abstract

Polyzetas, indexed by words, satisfy shuffle and quasi-shuffle identities.

In this respect, one can explore the multiplicative and algorithmic (locally

finite) properties 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 polyze-

tas (the graded kernel) and their explicit representation (as data structures) in

terms of irreducible elements.

1 Introduction

This paper will provide transparent arguments and proofs for results presented at the International Symposium on Symbolic and Algebraic Computation conference, Bath, 6-9 July, 2015 [7].

For any composition of positive integers, s = ( s

, . . . , s

) , the polyzetas [9]

(also called multiple zeta values [33]) are defined by the following convergent se- ries

ζ ( s

, . . . , s

) ∶= ∑

n1>...>nr>0

n

. . . n

, for s

> 1. (1) The Q-algebra generated by convergent polyzetas is denoted by Z .

Any composition s ∈ ( N

)

can be associated to words [26, 28] of the form x

x

. . . x

x

, defined on the alphabet X = { x

, x

} , or the form y

. . . y

, defined on the alphabet Y = {y

}

. The free monoids on these alphabets are respectively denoted by X

and Y

. In this respect, the weight of the composition s, determined by s

+ . . . + s

, is also the weight of the word y

. . . y

or the length of the word x

x

. . . x

x

.

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. Equipping X with the (total) ordering x

< x

and denoting by LynX , the set of Lyndon words over X, the Poincaré-Birkhoff-Witt (PBW) basis { P

}

will be expanded over the basis { P

}

, of the free Lie al- gebra Lie

⟨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 denot- ing by L ynY the set of Lyndon words over Y , the basis { Π

}

, of the 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, 24, 25].

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)

1P

is a primitive element if

. This Lie algebra is isomorphic (but

not equal) to the free Lie algebra.

From these, in Section 3,

1. We will consider two generating series of polyzetas

[24, 25, 26, 28]:

Z

∶=

↘

∏

l∈LynX X

exp ( ζ ( S

) P

) and Z ∶=

↘

∏

l∈LynY {y1}

exp ( ζ ( Σ

) Π

) (4) . The coefficients of Z

(resp. Z ) are obtained as the finite parts of the asymptotic expansions of the polylogarithms { Li

}

(resp. the harmonic sums {H

}

), at 1 (resp. at +∞ ), in the scale of comparison {(1 − z )

log

(( 1 − z )

)}

(resp. { N

H

( N )}

, where H

( N ) is the classic harmonic sum 1 + 1 / 2 + . . . + 1 / N ) [24].

2. We have also defined a third one, Z

[10], which satisfies, via the - extended Schützenberger’s factorization on the completed quasi-shuffle Hopf algebra [24, 25] ,

Z

= e

Z . (5)

The coefficients of Z

are obtained as the finite parts of the asymptotic ex- pansions of { H

}

, in the scale of comparison { N

log

( N )}

. In (5), γ denotes the Euler’s constant [10].

3. In order to identify the local coordinates of Z

(and Z ), on a group of associators [24, 25], we will rely on the following comparison (see [10]) Z

= B(y

)π

(Z

), where B(y

) = exp(γy

− ∑

k≥2

(−1)

ζ(k)

k y

). (6) Here, π

is a linear projection from Q ⊕ Q ⟪X⟫x

to Q ⟪Y ⟫ , mapping x

x

. . . x

x

to y

. . . y

, and π

denotes its inverse.

By cancellation [24, 25], (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

)}

1}

) [6].

To end this section, let us point out some crucial points of our purpose :

2

In (4), only

polyzetas arise then will not need any regularization.

3

The Maple program runs on a computer Core(TM)i5-4210U CPU @ 1.70GHz and obtains re-

sults up to weight

[6].

1. Similar tables

for {ζ(l)}

have been obtained up to weight 10 [28], 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 series of symbolic polyzetas (Boutet de Monvel [11] and Racinet [17] have also given equivalent relations for the noncommutative generating series of symbolic polyzetas, see also [9]) producing linear relations and which base themselves on regularized double shuffle relation [2, 20, 22] and different from identities among associators, due to Drinfel’d [12, 13, 19].

2. In the classical theory of finite-dimensional Lie groups, any ordered basis of Lie algebra provides a system of local coordinates in suitable neighborhood of the group unity via an ordered product of one-parameter groups corre- sponding to the ordered basis [32]. In this work, we get a perfect analogue of this picture for Hausdorff groups, through Schützenberger’s factorization, this doesn’t depend on regularization (see the next remark) [24, 25].

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 [24, 25], three morphisms of (shuffle and quasi-shuffle) algebras, studied earlier in [26, 27, 10] and constructed in [24, 25]

ζ

∶ ( Q ⟨X⟩, , 1

) Ð→ (Z, ×, 1), (8) ζ ∶ ( Q ⟨ Y ⟩ , , 1

) Ð→ (Z , × , 1 ) , (9) γ

∶ ( Q ⟨ Y ⟩ , , 1

) Ð→ (Z , × , 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 generators of length (resp. weight) one, for X

(resp. Y

), satisfy (see (4) and (5))

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 [10, 27]

∑

w∈X^∗

ζ

( w ) w = Z

, ∑

w∈Y^∗

ζ ( w ) w = Z , ∑

w∈Y^∗

γ

w = Z

(13)

4

They form a Gröbner basis of the ideal of polynomial relations among the convergent polyzetas

and the ranking of this basis is based mainly on the order of Lyndon words [1, 28, 31]. For that, this

basis is also called Gröbner-Lyndon basis.

and

Z

= (ζ

⊗ Id

)D

, Z = (ζ ⊗ Id

)D

, Z

= (γ

⊗ Id

)D

. 4. By (4), for any u, v ∈ LynX 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 ∈ LynX X and l

= π

(l) , one has, on the other hand

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), for the quasi-shuffle product, the regularization to γ is equivalent to the regularization to 0 [24, 25] and this yields immediately the family of regularized double shuffle relations considered in [1, 2, 17, 22, 28, 31] (see also [9, 11, 20, 21, 30]).

Our method is then different from [9, 11, 30] in which their authors suggest the simultaneous regularization of the divergent polyzeta ζ (1) to the inde- terminate T , i.e. ζ

( x

) = ζ

( x

) = ζ ( y

) = T (to compare with (12)).

Since T is transcendent over Q then it can be suitable to be specialized to 0, as effectively done in [17, 22] and, by this way, relations among polyzetas are formally obtained depending mainly on numerical values

of T .

2 Background

2.1 Generalities

Let Y = { y

}

be an infinite alphabet with the total order y

> y

> . . ..

Y

denotes the free monoid on Y which admits the empty word, denoted by 1

, as neutral element.

Let us define the commutative product on

Q Y , denoted by µ (see [4, 16]),

∀ y

, y

∈ Y, µ ( y

, y

) = y

, (15) or its dual coproduct, ∆

, defined by

∀ y

∈ Y, ∆

y

=

s−1

∑

i=1

y

⊗ y

(16)

satisfying,

∀ x, y, z ∈ Y, ⟨ ∆

x ∣ y ⊗ z ⟩ = ⟨ x ∣ µ ( y, z )⟩ . (17) Let Q ⟨ Y ⟩ denote the space of polynomials on the alphabet Y equipped by

5

They are group-like :

,

,

.

6

Since the

-algebra of polyzetas is not a

-algebra, how then can we determine these values

?

7QY

denotes the

-vector space generated by the alphabet

, as a basis.

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

).

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 [20], 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 e (P ) = ⟨P ∣ 1

⟩ (for any P ∈ Q ⟨Y ⟩ ). 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)

8

Of course, we have (set theoretically)

, but their structural treatments will be differ-

ent.

Here, for the algebras where live 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) theorem [4, 23], the con- nected N -graded, co-commutative Hopf algebra H

is isomorphic to the envelop- ing 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 by ( l

, l

) the standard factorization

of l ∈ L ynY Y , let us con- sider

1. The PBW 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

1

. . . P

k

for

with

l1,...,lk∈LynY, l1>...>lk.

(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

1

= 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

− 2x

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) 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

1

. . . S

k

i

! . . . i

!

for w = l

. . . l

, with

l

, . . . , l

∈ L ynY, l

> . . . > l

. (30)

9

A pair of Lyndon words

is called the standard factorization of

if

and

is the smallest nontrivial proper right factor of

(for the lexicographic order) or, equivalently its longest such.

10

The dual family,

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.

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

1

= 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 [24, 25]

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,...,uk∈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

+

s

∑

i=2

(− 1 )

l ∑

j1,...,ji≥1,j1+...+ji=s

y

. . . y

. (34) Example 3. π

( y

) = y

, π

( y

) = y

−

2

y

, π

( y

) = y

−

2

( y

y

+ y

y

) +

3

y

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

y

= ∑

i≥1

1

i! ∑

s1+...+s_i=s

π

(y

) . . . π

(y

), y

∈ Y . (35) Example 4.

y

= π

(y

), y

= π

( y

) +

2!

π

( y

)

,

y

= π

(y

) +

(π

(y

)π

(y

) + π

(y

)π

(y

)) +

π

(y

)

.

Now let us consider the (endo-)morphism of algebras φ ∶ ( Q ⟨Y ⟩, ●, 1) → ( Q ⟨ Y ⟩ , ● , 1 ) satisfying φ ( y

) = π

( y

) ; it can be shown that φ is an automor- phism of Q ⟨ Y ⟩ . Then we have [25],

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

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

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

//

φ

Q ⟨ Y ⟩ ⊗ Q ⟨ Y ⟩

Q ⟨Y ⟩

∆

// Q ⟨Y ⟩ ⊗ Q ⟨Y ⟩.

iii) H ≅ U (Prim(H )) and H

≅ U (Prim(H ))

.

iv) The dual bases { Π

}

and { Σ

}

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

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

, of respectively { P

}

and { S

}

.

More precisely,

1. The PBW basis { Π

}

for U ( Prim (H )) can be constructed recur- sively as follows [?, 5, 24, 25]

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎩

Π

= π

(y

) for y

∈ Y,

Π

= [ Π

, Π

] for l ∈ L ynY Y, st ( l ) = ( l

, l

) , Π

= Π

1

. . . Π

k

for w = l

. . . l

, with

l

, . . . , l

∈ L ynY, l

> . . . > l

.

(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

−

2

y

y

y

− y

y

y

+

2

y

y

+ y

y

y

+

2

y

y

y

−

2

y

y

+

4

y

y

. 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, 24, 25]

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

Σ

= y

, for y

∈ Y, Σ

= ∑

(☆)

1 i! y

1+...+s_ki

Σ

, for l = y

. . . y

∈ L ynY, Σ

=

Σ

1

. . . Σ

k

i

! . . . i

! , for w = l

. . . l

, with

l

, . . . , l

∈ L ynY, l

> . . . > l

.

(38)

In (☆) , the sum is taken over all {k

, . . . , k

} ⊂ {1, . . . , k} and all l

≥ . . . ≥ l

such that ( y

, . . . , y

)

∗

⇐ ( y

1

, . . . , y

, l

, . . . , l

) , where ⇐

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

Using Example 2.ii), we have in general, for any l ∈ L ynY, π

( Σ

) ≠ S

(resp. L ynX ∖ { x

} , π

( S

) ≠ Σ

) [24, 25]:

Example 6.

l ∈ LynY Σ

π

(l) ∈ LynX π

S

y

y

x

y

y

y

x

x

y

y

y

y

y

+

y

x

x

y

y

y

y

y

y

y

y

+ y

y

y

+ y

x

x

x

x

y

y

y

+ 3y

y

y

+

2

y

y

+

2

y

y

+

3

y

+ 6y

y

2.2 Local coordinates

Following Wei-Norman’s theorem [32], we know that, for a given (finite dimen- sional) k-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

)

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 dif- feomorphism 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 )

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, T is upper unitriangular, D diagonal strictly positive and U unitary, then M = U

D

T

will be the Iwasawa [3] decomposition of M . The decomposition algorithm goes in three steps as follows (step 4 is a summary)

11

Real (with

) or complex (with

).

12

It is the group of matrices with positive determinant.

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(∣∣C⁽¹⁾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(

11

).

4. (Summary)

M T D = M

= e

arctan(^a_a²¹

11)⎛

⎜⎝

0 1

−1 0

⎞⎟

⎠

, hence

M = e

a21

a11)(₋₁^{0 1}₀)

D

T

= e

a21

a11)(₋₁^{0 1}₀)

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 a subset of Y

⊗ Y

) endowed with the law ⊗● ˆ , we have Schützen- berger’s factorization(s) [14, 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

⊗H ˆ

;

or with the law ⊗● ˆ , we also have the extension of Schützenberger’s factorization for D which is then [5, 24, 25]

D =

↘

∏

l∈LynY

exp ( Σ

⊗ Π

) ∈ H

⊗H ˆ .

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 to use identity (7) to reduce relations between the two systems of local coordinates { ζ ( S

)}

and { ζ ( Σ

)}

.

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 basis {P

}

, {S

}

, {Π

}

or {Σ

}

.

Recall that the bases { P

}

and { Π

}

are homogeneous and upper triangular, the bases { S

}

and { Σ

}

are homogeneous and lower trian- gular

. 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)

13

In fact, these series are respectively characters for or .

14

w.r.t the words and the lexicographic ordering, for example,

15

This term includes the greatest word in the support of

and its coefficient.

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-expressing 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

.

Step 2. Since Σ

= y

y

+ 1 / 2y

, we replace y

y

with Σ

− 1 / 2y

in P P = 2Σ

− 2Σ

− 3 / 2y

.

Since y

= Σ

, we thus get P = 2Σ

− 2Σ

− 3/2Σ

. Corollary 1. For any w ∈ Y

, we can represent

w = P

+ ∑

u>w,∣u∣=∣w∣

α

P

= S

+ ∑

u<w,∣u∣=∣w∣

α

S

, w = Π

+ ∑

v>w,(v)=(w)

β

Π

= Σ

+ ∑

v<w,(v)=(w)

β

Σ

.

3.2 Identifying the local coordinates

We now use the alphabet X = {x

, x

} ordered by x

< x

. Returning to formula (7), with the bases { P

}

and { S

}

defined as (28) and (30), we will find relations among polyzetas by identifying on the bases as local coordinates. First, we expand B

, given in (7), in form of generating series of y

.

Lemma 1. We have

B

(y

) = 1 + ∑

m≥2

B

y

, with B

=

⌊m/2⌋

∑

i=1

∑

k1,...,ki≥2 k1+...+ki=m

(−1)

ζ(k

) . . . ζ(k

) k

. . . k

, where ⌊ m / 2 ⌋ is the largest integer not greater than m / 2.

16∣w∣

and

respectively denote the length and the weight of the word

Proof. Expanding the exponential, one has successively B

( y

) = ∑

n≥0

1 n! ( ∑

k≥2

(− 1 )

ζ ( k ) k y

)

n

= ∑

n≥0

1

n! ∑

k1,...,kn≥2

(− 1 )

ζ ( k

) . . . ζ ( k

) k

. . . k

y

= 1 + ∑

m≥2

⎛

⎜ ⎜

⎝

⌊m/2⌋

∑

n=1

1

n! ∑

k1,...,kn≥2 k1+...+kn=m

(− 1 )

ζ ( k

) . . . ζ ( k

) k

. . . k

⎞

⎟ ⎟

⎠ y

= 1 + ∑

m≥2

B

y

.

Example 9.

B

= −

2

, B

=

, B

= −

4

+

2

2²

, B

=

− 2

. 3.2.1 Identifying with respect to the basis {Π

}

Using the duality of the bases, we rewrite (7) as follows

∑

v∈Y^∗

ζ ( Σ

) Π

= B

( y

) ∑

v∈Y^∗

ζ

( π

( Σ

)) Π

. (42) Moreover, we see that B

( y

) is a series of a single letter (like a single variable), y

, and

y

Π

= Π

Π

= Π

1v

, ∀k ≥ 1, v ∈ Y

. We can then identify the coefficients in (42) and obtain:

Proposition 1. i) For any v ∈ Y

y

Y

, one has

ζ ( Σ

) = ζ ( π

Σ

).

ii) For any v = y

w ∈ Y

, k ≥ 1, w ∈ Y

y

Y

, one has ζ

(π

Σ

) +

k

∑

m=2

B

ζ

(π

Σ

) = 0.

Proof. From Lemma 1, we see that ⟨ B

( y

) ∣ y

⟩ = 1, ⟨ B

( y

) ∣ y

⟩ = 0 and

∀ m ≥ 2, ⟨ B

( y

) ∣ y

⟩ = B

.

Using the basis { Π

}

as a coordinate system, we identify the coefficients of the two sides in (42) and obtain the preceding statements.

17

As

and

are disjointed, the unique notation

is used here to replace

or

if the polynomial

only contains convergent words.

Example 10. 1. For v = y

, ζ(Σ

) = ζ(S

).

2. For v = y

y

, ζ ( Σ

) = ζ ( S

0x1

) − 2ζ ( S

0x1x0x1

) − 2ζ ( S

0x²₁

) + ζ(S

0x1

).

3. For v = y

, −

2

ζ ( S

1

) +

6

ζ ( S

0x1

) + B

= 0.

4. For v = y

y

, ζ(S

1

) − ζ(S

0x²₁

) +

ζ (S

0x1

) + B

= 0.

3.2.2 Identifying with respect to the basis {P

}

Let us denote by

{ P

}

the reductions of { P

}

on Q ⊕ Q ⟨ X ⟩ x

. By applying the mapping π

on the two sides of (42) and using the duality of the bases, we can rewrite the regularization as follows

B

(x

) ∑

u∈X^∗x1

ζ

(S

)P

= ∑

u∈X^∗x1

ζ (π

S

)P

. (43) Similarly, remarking that B

( x

) is a series of a single letter, x

,

x

P

= P

P

= P

1u

, ∀k ≥ 1, u ∈ X

. Proposition 2. i) For any u ∈ X

x

X

, ζ (S

) = ζ(π

S

).

ii) For any u ∈ x

X

x

X

, ζ ( π

S

) = 0.

iii) For any u = x

w ∈ X

, k ≥ 2, w ∈ X

x

X

, B

ζ ( S

) = ζ ( π

S

).

Proof. Similarly to Proposition 1, admitting the basis { P

}

as a coordinate system, we identify the coefficients of the two sides in (43) and then obtain the statements.

Example 11. 1. For u = x

x

, ζ ( S

) = ζ ( Σ

) . 2. For u = x

x

x

x

, ζ ( S

0x1

) = ζ ( Σ

) + 2ζ ( Σ

) + 6ζ ( Σ

) − 5ζ ( Σ

) .

3. For u = x

x

x

, ζ ( Σ

) −

2

ζ ( Σ

) = 0.

4. For u = x

x

x

, B

ζ ( S

) = 2ζ ( Σ

) − ζ ( Σ

)

− ζ ( Σ

).

18

They are defined by

. Note that

.

3.3 Algorithms to represent the structure of polyzetas

From Proposition 1 and 2, we really have relations among polyzetas represented on the bases { S

}

and { Σ

}

.

In fact, thanks to the formulas (30) and (38), we can easily represent these relations on the pure transcendence bases { S

}

or { Σ

}

respectively.

In the two following algorithms, one uses these relations and the other one (Algorithm 3) uses as well the structures of shuffle and stuffle products, we will eliminate these relations, in weight, to find the structure of polyzetas represented on the bases { S

}

and { Σ

}

. The following two algorithms will be proceeded by recurrence on the weight of the words.

The same result obtained will be shown in the next subsection.

Algorithm 2

This algorithm uses Proposition 1 and Algorithm 1 to establish polynomial relations among polyzetas on the basis { S

}

or uses Proposition 2 and Algorithm 1 to establish relations among polyzetas on the basis { Σ

}

.

We display here the second case.

INPUT: A positive integer n.

OUTPUT: The representations of polyzetas of weight n in terms of irreducible elements of polyzetas on the transcendence basis { Σ

}

.

Step 1. We set the list, denoted by X

, of all words

of weight

n of X

x

. Step 2. For each w ∈ X

, we set the polynomial P ∶= π

( S

) in Q ⟨ Y ⟩ and thanks

to Algorithm 1 we represent ζ (P ) in terms of { ζ ( Σ

)}

. By taking the representations of ζ (Σ

) ’s from the data of lower weights, we make repre- sentation in terms of irreducible elements for ζ (P ) and proceed to establish a polynomial relation as follows:

i) If w ∈ L ynX then we store ζ (P ) to the variable ζ ( S

) , ii) If w = x

u, u ∈ x

X

x

then we make the relation ζ (P ) = 0.

iii) If w ∈ x

X

x

LynX, we rewrite w in the form of Lyndon factoriza- tion, w = l

. . . l

. By taking ζ ( S

) , j = 1 . . . k from the data of lower weights, we make the relation

1

i

! . . . i

! ζ ( S

)

. . . ζ ( S

)

= ζ (P ) .

Step 3. We reduce the above relations to representations of polyzetas in terms of irreducible elements.

19

Note that, there are

words of weight

20

In the alphabet

, the weight of a word is understood as the length of that word.

The next lemma will give another way to find the relations among the family { ζ

( S

)}

and the family { ζ ( Σ

)}

.

Lemma 2. i) For any l

, l

∈ LynX X (resp. l

, l

∈ LynY {y

}), one has ζ ( S

S

) = ζ ( π

( S

) π

( S

)) ,

ζ ( Σ

Σ

) = ζ ( π

( Σ

) π

( Σ

))) .

ii) For any w ∈ x

X

x

or w ∈ x

x

X

x

(resp. w ∈ Y

y

Y

), one has ζ

( S

) = ζ ( π

( S

)) ,

ζ ( Σ

) = ζ

( π

( Σ

)) .

Proof. Remark that, for any w ∈ X

, S

= w + ∑

α

v and if l ∈ L ynX X then l ∈ x

X

x

.

Relying on properties of polyzetas on words, i.e. [24, 25]

ζ ( l

l

) = ζ ( π

( l

) π

( l

)) , ∀ l

, l

∈ L ynX X, ζ

( x

l ) = ζ ( y

π

( l )) , ∀ l ∈ L ynX X, we get the expected results.

Example 12. For l

= x

x

, l

= x

x

(in LynX) and l

= y

, l

= y

y

(in L ynY ):

ζ(S

)ζ (S

0x²₁

) = ζ (Σ

)ζ (Σ

) − 1

2 ζ(Σ

)ζ(Σ

), ζ ( Σ

) ζ ( Σ

) = ζ ( S

) ζ ( S

0x²₁

) + 1

2 ζ ( S

) ζ ( S

0x1

) . For w = x

x

x

(in x

x

X

x

) and w = y

y

(in y

Y

):

0 = 1

2 ζ ( Σ

)

+ ζ ( Σ

) − 2ζ ( Σ

) , 0 = − 1

2 ζ ( S

)

+ ζ ( S

0x²₁

) + ζ ( S

0x1

) . Algorithm 3

This algorithm uses Lemma 2 and Algorithm 1 to establish polynomial relations among polyzetas on the basis { S

}

or the basis { Σ

}

.

We display here the second case.

INPUT: A positive integer n.

OUTPUT: The representations of polyzetas of weight n in terms of irreducible elements of polyzetas on the transcendence basis {Σ

}

.

Step 1. We set a list, denoted by X

, all words of weight n in x

X

x

or x

x

X

x

.