Towards a noncommutative Picard-Vessiot theory

HAL Id: hal-02921131

https://hal.archives-ouvertes.fr/hal-02921131v2

Preprint submitted on 24 Oct 2021

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Towards a noncommutative Picard-Vessiot theory.

G. Duchamp, Viincel Hoang Ngoc Minh, Vu Nguyen Dinh, Pierre Simonnet

To cite this version:

G. Duchamp, Viincel Hoang Ngoc Minh, Vu Nguyen Dinh, Pierre Simonnet. Towards a noncommu-

tative Picard-Vessiot theory.. 2021. �hal-02921131v2�

Towards a noncommutative Picard-Vessiot theory

G.H.E. Duchamp

University Paris 13, Sorbonne Paris City, 93430 Villetaneuse, France,

V. Hoang Ngoc Minh

University of Lille, 1 Place Déliot, 59024 Lille, France,

V. Nguyen Dinh

Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Hanoi, Vietnam.

P. Simonnet

University of Corsica, 20250 Corte, France.

October 24, 2021 ^(Draft)

Abstract

A Chen generating series, along a path and with respect to m differential forms, is a non- commutative series on m letters and with coefficients which are holomorphic functions over a simply connected manifold in other words a series with variable (holomorphic) co- efficients. Such a series satisfies a first order noncommutative differential equation which is considered, by some authors, as the universal differential equation, i.e., in this case, univer- sality can be seen by replacing each letter by constant matrices (resp. holomorphic vector fields) and then solving a system of linear (resp. nonlinear) differential equations.

Via rational series, on noncommutative indeterminates and with coefficients in rings, and

their non-trivial combinatorial Hopf algebras, we give the first step of a noncommutative

Picard-Vessiot theory and we illustrate it with the case of linear differential equations with

singular regular singularities thanks to the universal equation previously mentioned.

Contents

1 Introduction 2

2 Combinatorial framework 4

2.1 Factorization in bialgebras 4

2.2 Representative series 9

3 Triangularity, solvability and rationality 13

3.1 Syntactically exchangeable rational series 13 3.2 Exchangeable rational series and their linear representations 15 4 Towards a noncommutative Picard-Vessiot theory 20

4.1 Noncommutative differential equations 20

4.2 First step of a noncommutative Picard-Vessiot theory 23

5 Conclusion 25

References 25

1 Introduction

Combinatorial Picard-Vessiot (PV for short) theory of bilinear systems ¹ was real- ized by Fliess and Reutenauer [29], as an application of differential algebra [42,47].

This theory allows to employ, with success, linear algebraic groups in control the- ory (i.e. as symmetry groups of linear differential equations), for which some ques- tions were solved thanks to the theory of Hopf algebras [11] and some combinato- rial and effective aspects were set in [46].

Let us, for instance, consider the following nonlinear dynamical system

q(z) = ˙ A ₀ (q)u 0 (z) + . . . + A _m (q)u m (z), q(z 0 ) = q ₀ , y(z) = f (q(z)), (1) where

(i) y is the output,

(ii) the vector state q = (q 1 , . . . , q _n ) belongs to a complex holomorphic manifold M of dimension n,

(iii) the observation f is defined within a fixed connected neighbourhood ² U of of the initial state q ₀ .

(iv) the vector fields (A i ) _i=0,...,m are defined with respect to the coordinates as fol- lows

1

Namely - locally - linear of the states q

1

, . . . , q

_N

and linear of the inputs u

0

, . . . , u

_m

[29].

2

In this introductive description the points are loosely identified with their coordinates through

some chart ϕ

U

: U → C

ⁿ

likewise, in [45], the space of holomorphic functions H (U) is described

by C

^cv

[[q

1

, . . . , q

_n

]].

A _i =

n

∑ j=1

A _i ^j (q) ∂

∂ q _j , with A _i ^j (q) ∈ H (U ), (2) (v) the inputs (u i ) i=0,...,m , as well as their inverses (u ⁻¹ _i ) i=0,...,m , belong to a sub- ring, C ₀ , of the ring of holomorphic functions H (Ω) with neutral element 1 H (Ω) over the simply connected manifold ³ Ω.

It is convenient (and possible) to separate the contribution of the vector fields (A i ) i=0,...,m and that of the differential forms ( ω _i ) i=0,...,m , defined by the inputs ω i (z) = u _i (z)dz, through the encoding alphabet X = {x i } _i=0,...,m which generates the monoid X ^∗ with neutral element 1 X

^∗

. Indeed, the output y can be computed by

y(z 0 , z) = hC z

₀

z k σ f _|

_q

0

i = ∑

w∈X

^∗

α _z ^z

₀

(w) Y (w)[ f ] _|

_q

0

, (3)

as the pairing (under suitable convergence conditions [31,34,36,45]) between the Chen series ⁴ of ( ω i ) _i=0,...,m along the path z ₀ z over Ω, C z

₀

z ∈ H (Ω)hhX ii [10], and the generating series of (1), σ f _|

_q

0

∈ H (U )hhXii [31], defined as follows C _z

₀

_z := ∑

w∈X

^∗

α _z ^z

₀

(w)w and σ f _|

_q

0

:= ∑

w∈X

^∗

Y (w)[ f ] _|

_q

0

w, (4)

where, in (3)–(4), the iterated integral α _z ^z

₀

(w) and the differential operator Y (w), are decoded, from the word w ∈ X ^∗ , recursively as follows

 



 

α _z ^z

₀

(w) = 1 H (Ω) and Y (w) = Id, for w = 1 X

^∗

, α _z ^z

₀

(w) =

Z _z

z

₀

ω i (s) α _z ^s

₀

(v) and Y (w) = A _i ◦ Y (v), for w = x i v , x _i ∈ X , v ∈ X ^∗ .

(5) In this work, following this route, considering the differential ring ( H (Ω), ∂ ) and equipping H (Ω)hhXii with the derivation defined, for any S ∈ H (Ω)hhX ii, by

dS = ∑

w∈X

^∗

( ∂ hS | wi)w , (6)

we can see that the Chen series satisfies the following noncommutative differential equation

dS = MS with M = u ₀ x ₀ + . . . + u _m x _m , (7)

considered by many authors as the universal differential equation [10,14,17,18,39].

Universality can be seen by specialization, i.e. replacing the letters by constant matrices (resp. holomorphic vector fields) and therefore obtaining linear (resp.

nonlinear) differential equations (see Remark 4.9 below) as well as their solutions.

3

This (usually one dimensional) manifold will be the support of the iterated integrals below.

4

By a Ree’s theorem [44], there is a primitive series L

_{z0 z}

= ∑

n≥1

L

_n

∈ H \ (Ω)hX i s.t. e

^{Lz0 z}

=

C

_{z0 z}

, meaning that C

_{z0 z}

is group-like and L

_n

is (homogenous of degree n ≥ 1) primitive series.

From equation (7), it follows (see, for example, [9]) that a PV theory of nonlinear systems (1) should be intimately connected with (7) (the reader may remark that, due to the connectedness of Ω, the constants of ( H (Ω)hhX ii, d) are

Const( H (Ω)hhX ii) = ker d = C.1 H (Ω) hhX ii. (8) This culminates with the fact that the coefficients of any suitable ⁵ solution is group-like, i.e. satisfies ⁶ , for any u, v ∈ X ^∗ and x _i ∈ X ,

∂ hS | x i ui = u i hS | vi and hS | u

^⊔⊔

vi = hS | uihS | vi ; hS | 1 X

^∗

i = 1 H (Ω) (9) Due to the fact that Ω is simply connected, the coordinate values of this series only depend on the endpoints and not on paths drawn on Ω. Denoting the subalgebra of ( H (Ω), ∂ ) generated by the family ( f _i ) i∈I and derivatives by C{{( f _i ) i∈I }} [49]

(i.e. the differential algebra generated by ( f _i ) i∈I ), it follows that [29]

span _C {hd ^l S | wi} _w∈X

^∗

_,l≥0 ⊂ span _C{{(u

_i

₎

_i=0,..,m

_}} {hS | wi} w∈X

^∗

(10)

⊂ span _C{{(u

±1

i

)

_i=0,..,m

}} {hS | wi} w∈X

^∗

(11)

and then, in Section 4, the isomorphism between span _C{{(u

±1

i

)

_i=0,..,m

}} { α _z ^z

₀

(w)} w∈X

^∗

and C{{(u ^±1 _i ) i=0,..,m }} ⊗ _C span _C { α _z ^z

₀

(w)} w∈X

^∗

will be examined (Theorem 4.4) via the PV-extension related to (7) and, on the other hand, the output of (1) will be com- puted (Theorem 4.8) by pairing the series given in (4). As example, this calculation will be achieved according to the algebraic combinatorics of rational series, estab- lished beforehand in Sections 2 (Theorems 2.2, 2.4) and 3 (Theorems 3.2, 3.7).

2 Combinatorial framework

In this section, coefficients are taken in a commutative ring ⁷ A and, unless explic- itly stated, all tensor products will be considered over the ambient ring (or field).

2.1 Factorization in bialgebras

In section 1, the encoding alphabet X was already introduced. In particular, for m = 1 (i.e. X = {x ₀ , x ₁ }), let us note that there are one-to-one correspondences

(s 1 , . . ., s _r ) ∈ N ^r ₊ ↔ x ^s ₀

¹

⁻¹ x ₁ . . . x ^s ₀

^r

⁻¹ x ₁ ∈ X ^∗ x ₁ ⇋ ^π

^Y

π

X

y _s

₁

. . . y _s

_r

∈ Y ^∗ , (12) where Y := {y k } k≥1 and π _X is the conc morphism, from AhY i to AhXi, mapping y _k to x ^k−1 ₀ x ₁ . This morphism π X admits an adjoint π Y for the two standard scalar

5

i.e. group-like at one - interior or frontier - point.

6

In the first identity, also called Friedrichs criterion, is involved the shuffle product (

⊔⊔

) [10,30,46].

7

although some of the properties already hold for a general commutative semiring [1].

products ⁸ which has a simple combinatorial description: the restriction of π _Y to the subalgebra (A1 X

^∗

⊕ AhY ix ₁ , conc), is an isomorphism given by π Y (x ^k−1 ₀ x ₁ ) = y _k (and the kernel of the non-restricted π Y is AhX ix 0 ). For all matters concerning finite (X and similar) or infinite (Y and similar) alphabets, we will use a generic model noted X in order to state their common combinatorial features. Let us recall also that the coproduct ∆ conc is defined, for any w ∈ X ^∗ , as follows

∆ conc w = ∑

u,v∈ X

^∗

,uv=w

u ⊗ v . (13)

As an algebra the A-module Ah X i is equipped with the associative unital con- catenation and the associative commutative and unital shuffle product. The latter being defined, for any x, y ∈ X and u, v, w ∈ X ^∗ , by the following recursion

w

^⊔⊔

1 X

^∗

= 1 X

∗ ⊔⊔

w = w and xu

^⊔⊔

yv = x(u

^⊔⊔

yv) + y(xu

^⊔⊔

v) (14) or, equivalently, by its dual comultiplication (which is a morphism for concatena- tions ⁹ ), defined, for each letter x ∈ X , by

∆

_⊔⊔

x = 1 X

^∗

⊗ x + x ⊗ 1 X

^∗

. (15) Once X has been totally ordered ¹⁰ , the set of Lyndon words over X will be denoted by L yn X . A pair of Lyndon words (l 1 , l ₂ ) is called the standard factor- ization of a Lyndon l (and will be noted (l 1 , l ₂ ) = st(l)) if l = l ₁ l ₂ and l ₂ is the longest nontrivial proper right factor of l or, equivalently, its smallest such (for the lexicographic ordering, see [43] for proofs and details). According to a theorem by Radford, the set of Lyndon words form a pure transcendence basis of the A-shuffle algebras (Ah X i,

^⊔⊔

, 1 X

^∗

).

It is well known that the enveloping algebra U ( L ie _A h X i) is isomorphic to the (connected, graded and co-commutative) bialgebra ¹¹ H

_⊔⊔

( X ) = (Ah X i, conc, 1 X

^∗

, ∆

_⊔⊔

, e) (the counit being here e(P) = hP | 1 X

^∗

i) and, via the pairing

Ahh X ii ⊗ A Ah X i −→ A, (16)

T ⊗ P −→ hT | Pi := ∑

w∈ X

^∗

hT | wihP | wi, (17)

we can, classically, endow Ah X i with the graded ¹² linear basis {P w } w∈ X

^∗

(ex- panded after any homogeneous basis {P _l } _l∈ L yn X of L ie _A h X i) and its graded

8

That is to say (∀p ∈ AhX i) (∀q ∈ AhY i) (h π

Y

p | qi

Y

= hp | π

X

qi

X

).

9

On Ah X i and Ah X i ⊗ Ah X i, respectively.

10

For technical reasons, the orders x

0

< x

1

(for X) and y

1

> . . . y

_n

> y

n+1

> . . . (for Y ) are usual.

11

In case A is a Q-algebra, the isomorphism U ( L ie

_A

h X i) ≃ H

_⊔⊔

( X ) can also be seen as an easy application of the CQMM theorem.

12

For X = X or = Y the corresponding monoids are equipped with length functions, for X we

consider the length of words and for Y the length is given by the weight ℓ(y

i₁

. . .y

in

) = i

₁

+. . . + i

_n

.

This naturally induces a grading of Ah X i and L ie

_A

h X i in free modules of finite dimensions. For

dual basis {S w } _w∈ X

^∗

(containing the pure transcendence basis {S l } _l∈ L yn X of the A-shuffle algebra). In the case when A is a Q-algebra, we also have the following factorization ¹³ of the diagonal series, i.e. [46] (here all tensor products are over A)

D _X := ∑

w∈ X

^∗

w ⊗ w = ∑

w∈ X

^∗

S _w ⊗ P _w =

ց l∈ L ∏ yn X

e ^S

^l

^⊗P

^l

(18)

and (still in case A is a Q-algebra) dual bases of homogenous polynomials {P w } w∈ X

^∗

and {S w } w∈ X

^∗

can be constructed recursively as follows

 

 

 



 

 

 

P x = x, S x = x for x ∈ X ,

P _l = [P _l

₁

, P _l

₂

], S _l = yS _l

^′

, for l = yl ^′ ∈ L yn X − X st(l) = (l 1 , l 2 ), P w = P _l ⁱ

¹

1

. . . P _l ⁱ

^k

k

, S w = S _l

^⊔⊔

ⁱ

¹

1 ⊔⊔

. . .

^⊔⊔

S

^⊔⊔

_l ⁱ

^k

k

i ₁ ! . . . i _k ! , for w = l ₁ ⁱ

¹

. . . l _k ⁱ

^k

, with l ₁ , . . . , l _k ∈ L yn X , l ₁ > . . . > l _k .

(19)

The graded dual of H

_⊔⊔

( X ) is H ^∨

⊔⊔

( X ) = (Ah X i,

^⊔⊔

, 1 X

^∗

, ∆ conc , ε ).

As an algebra, the module AhY i is also equipped with the associative commu- tative and unital quasi-shuffle product defined, for u, v , w ∈ Y ^∗ and y _i , y _j ∈ Y , by

w 1 _Y

^∗

= 1 _Y

^∗

w = w , (20)

y _i u y _j v = y _i (u y _j v) + y _j (y i u v) + y _{i+ j} (u v). (21) This product also can be dualized according to (y _k ∈ Y )

∆ y _k := y _k ⊗ 1 Y

^∗

+ 1 Y

^∗

⊗ y _k + ∑

i+ j=k

y i ⊗ y j (22)

which is also a conc-morphism (see [28]). We then get another (connected, graded and co-commutative) bialgebra which, in case A is a Q-algebra, is isomorphic to the enveloping algebra of the Lie algebra of its primitive elements,

H (Y ) = (AhY i, conc, 1 Y

^∗

, ∆ , e) ∼ = U (Prim( H (Y ))), (23) where Prim( H (Y )) = Im( π ₁ ) = span _A { π ₁ (w)|w ∈ Y ^∗ } and π ₁ is the eulerian pro- jector defined, for any w ∈ Y ^∗ , by [37,38]

π ₁ (w) = w +

(w) k=2 ∑

(−1) ^k−1

k ∑

u

₁

,...,u

_k

∈Y

⁺

hw | u ₁ . . . u _k iu ₁ . . . u _k , (24)

and, for any w = y _i

_i

. . .y i

_k

∈ Y ^∗ , (w) denotes the number i _i + . . . + i _k .

Remark 2.1 By (13) and (15), any letter x ∈ X is primitive, for ∆ conc and ∆

_⊔⊔

. By (22), the polynomials { π 1 (y _k )} _k≥2 and only the letter y ₁ are primitive, for ∆ .

general X , we consider the fine grading [46] i.e. the grading by all partial degrees which, as well, induces a grading of Ah X i and L ie

_A

h X i in free modules of finite dimensions.

13

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

Now, let {Π w } w∈Y

^∗

be the linear basis, expanded by decreasing Poincaré- Birkhoff-Witt (PBW for short) after any basis {Π l } l∈ L ynY of Prim( H (Y )) ho- mogeneous in weight ¹⁴ , and let {Σ w } _w∈Y

^∗

be its dual basis which contains the pure transcendence basis {Σ _l } _l∈ L ynY of the A-quasi-shuffle algebra. One also has the factorization of the diagonal series D _Y , on H (Y ), which reads ¹⁵ [37,38,39]

D _Y := ∑

w∈Y

^∗

w ⊗ w = ∑

w∈Y

^∗

Σ _w ⊗ Π _w =

ց l∈ L ∏ ynY

e ^Σ

^l

^⊗Π

^l

. (25)

We are now in the position to state the following

Theorem 2.2 ([38,39]) Let A be a Q-algebra, then the endomorphism of algebras ϕ π

₁

: (AhY i, conc, 1 Y

^∗

) −→ (AhY i, conc, 1 Y

^∗

) mapping y _k to π 1 (y k ), is an automor- phism of AhY i realizing an isomorphism of bialgebras between H

_⊔⊔

(Y ) and

H (Y ) ∼ = U (Prim( H (Y ))).

In particular, it can be easily checked that the following diagram commutes

AhY i AhY i ⊗ AhY i

AhY i AhY i ⊗ AhY i

∆

_⊔⊔

ϕ

π1

ϕ

_π1

⊗ϕ

_π1

∆

Moreover, the bases {Π w } w∈Y

^∗

and {Σ w } w∈Y

^∗

of U (Prim( H (Y ))) are im- ages by ϕ π

₁

and by the adjoint mapping of its inverse, ϕ ˇ _π ⁻¹

₁

of {P w } w∈Y

^∗

and {S w } w∈Y

^∗

, respectively.

Algorithmically, by Remark 2.1, the dual bases of homogenous polynomials {Π w } w∈Y

^∗

and {Σ w } w∈Y

^∗

can be constructed directly and recursively as follows

 

 

 

 

 

 

 



Π _y

_s

= π 1 (y s ), Σ _y

_s

= y _s for y _s ∈ Y, Π _l = [Π _l

₁

, Π _l

₂

], Σ _l = ∑

(∗)

y s

_k1

+...+s

_ki

i! Σ _l

₁

_...l

_n

, for l ∈ L ynY − Y st(l) = (l 1 , l ₂ ), Π _w = Π ⁱ _l

¹

1

. . . Π ⁱ _l

^k

k

, Σ _w = Σ _l ⁱ

¹

1

. . . Σ _l ⁱ

^k

k

i ₁ ! . . . i _k ! , for w = l ⁱ ₁

¹

. . . l _k ⁱ

^k

, with l ₁ , . . . , l _k ∈ L ynY, l ₁ > . . . > l _k .

(26)

In (∗), the sum is taken over all {k 1 , . . ., k _i } ⊂ {1, . . .,k} and l ₁ ≥ . . . ≥ l _n such that (y s

₁

, . . . , y _s

_k

) ⇐ ^∗ (y s

_k1

, . . .,y s

_ki

, l ₁ , . . . , l _n ), where ⇐ ^∗ denotes the transitive closure of

14

Factorization (25) will be true in particular for the basis (26) explicitly constructed there.

15

Again all tensor products will be taken over A. Note that this factorization holds for any en-

veloping algebra as announced in [46]. Of course, the diagonal series no longer exists and must be

replaced by the identity Id

U

(see [26], coda for details).

the relation on standard sequences, denoted by ⇐ [7,46].

To end this section, let us extend conc and

^⊔⊔

, for any series S, R ∈ Ahh X ii, by SR = ∑

w∈ X

^∗

u,v∈ X ∑

^∗

,uv=w

hS | uihR | vi

w, (27)

S

^⊔⊔

R = ∑

u,v∈ X

^∗

hS | uihR | viu

^⊔⊔

v, (28)

and , for any series S, R ∈ AhhY ii, by

S R = ∑

u,v∈Y

^∗

hS | uihR | viu v. (29)

Let us also extend the coproduct ∆ (resp. ∆ conc and ∆

_⊔⊔

) given in (22) (resp. (13) and (15)) over AhhY ii (resp. Ahh X ii), for any series S ∈ AhhY ii (resp. Ahh X ii), by linearity as follows

∆ S = ∑

w∈Y

^∗

hS | wi∆ w ∈ AhhY ^∗ ⊗ Y ^∗ ii, (30)

∆

_⊔⊔

S = ∑

w∈ X

^∗

hS | wi∆

⊔⊔

w ∈ Ahh X ^∗ ⊗ X ^∗ ii, (31)

∆ conc S = ∑

w∈ X

^∗

hS | wi∆ ^conc w ∈ Ahh X ^∗ ⊗ X ^∗ ii. (32) The series S is said to be

(i) group like, for ∆ conc , if hS | 1 X

^∗

i = 1 and ∆ conc S = S ⊗ S, (ii) primitive, for ∆ conc , if ∆ conc S = S ⊗ 1 X

^∗

+ 1 X

^∗

⊗ S.

Similarly for ∆

_⊔⊔

, ∆ and then, letting S ∈ Ahh X ii (resp. AhhY ii), the Ree’s theo- rem express that, for ∆

_⊔⊔

(resp. ∆ ), [44,46] (resp. [37,38])

S is primitive ⇐⇒ e ^S is group like, (33)

⇐⇒ e ^S satisfies the Friedrichs criterion, (34) i.e it satisfies, for any u, v ∈ X ^∗ (resp. Y ^∗ ), [44,46] (resp. [37,38])

he ^S | u

^⊔⊔

vi = he ^S | uihe ^S | vi (resp. he ^S | u vi = he ^S | uihe ^S | vi). (35) Or equivalently,

h∆

⊔⊔

e ^S | u ⊗ vi = he ^S | uihe ^S | vi (resp. h∆ e ^S | u ⊗ vi = he ^S | uihe ^S | vi). (36)

We are going to see how all these combinatorics will operate over rational se-

ries and will be suitable, as illustration, to describe solutions of linear differential

equations in Section 4 (see Theorems 4.4 and 4.8 bellow).

2.2 Representative series

Representative (or rational) series are the representative functions on the free monoid ¹⁶ [22] and their magic is that it rests on four (apparently distant) pillars:

• Separated coproduct (SC) ¹⁷ ,

• Finite orbit by shifts (FS),

• Result of a rational expression (RE),

• Linear representation (LR).

We first define what shifts, for (FS), and the Kleene star, for (RE) are, and then state the equivalence:

Definition 2.3 Let S ∈ Ahh X ii (resp. Ah X i) and P ∈ Ah X i (resp. Ahh X ii).

(i) The left (resp. right) shift ¹⁸ of S by P, is P ⊲ S (resp. S ⊳ P) defined by ¹⁹

∀w ∈ X ^∗ , hP ⊲ S | wi = hS | wPi (resp. hS ⊳ P | wi = hS | Pwi).

(ii) For any S ∈ Ahh X ii such that hS | 1 X

^∗

i = 0, the Kleene star of S is defined as ²⁰ S ^∗ = (1 − S) ⁻¹ .

(iii) In case A = K is a field, one can define also the Sweedler’s dual H ^◦

⊔⊔

( X ) of H

_⊔⊔

( X ) by S ∈ H ^◦

⊔⊔

( X ) ⇐⇒ ∆ conc (S) = ∑ _i∈I G _i ⊗ D _i [46], for some I finite, {G i } i∈I ; {D i } i∈I being series (as a matter of fact, it can be shown that they even can been choosen in H ^◦

⊔⊔

( X ), see [19,39])

Theorem 2.4 ([20,22,35,46]) For S ∈ Ahh X ii, the following assertions are equiv- alent ²¹

(i) The shifts {S ⊳ w} _w∈ X

^∗

(resp. {w ⊲ S} _w∈ X

^∗

) lie in a finitely generated shift-

16

These functions were considered on groups in [11,12].

17

Uniquely for fields.

18

Some schools (as Jacob one, see [40,32]) used to call this a residual. These actions are none other than the shifts of functions of harmonic analysis.

19

They are associative, commute with each other: S ⊳ (PR) = (S ⊳ P) ⊳ R, P ⊲ (R ⊲ S) = (P.R) ⊲ S and (P ⊳ S) ⊲ R = P ⊳ (S ⊲ R) and, for x,y ∈ X ,w ∈ X

^∗

, x ⊲ (wy) = (yw) ⊳ x = δ

x^y

w (Kronecker delta).

20

Using one of the topologies of section 4.2 (adapted with A replacing H (Ω)), we have S

^∗

=

∑

n≥0

S

ⁿ

. We also get the fact that the space A. [ X (used below) of series of degree 1, i.e. the set {∑

_x∈X

α (x)x}

_α∈AX

is the closure of the A-module A. X generated by letters. In the case of a finite alphabet however (here X = X ) [22], A. [ X = A. X .

21

When A is noetherian, first condition is equivalent to the fact that the module generated by {S ⊳

w}

w∈X∗

(resp. {w ⊲ S}

w∈X∗

) is finitely generated (and more precisely, in this case, by a finite

number of those shifts). Unfortunately we are not in this case here, but our ring being without zero

divisors (holomorphic functions), we can use the fraction field, here being realized by germs [15].

invariant A-module ²² .

(ii) The series S belongs to the (algebraic) closure of A. [ X by the operations {conc, +, ∗} (within Ahh X ii).

(iii) There is a linear representation ( ν , µ , η ), of rank n, for S with ν ∈ M _1,n (A), η ∈ M _n,1 (A) and a morphism of monoids µ : X ^∗ → M _n,n (A) such that

S = ∑

w∈ X

^∗

ν µ (w) η w.

A series satisfying one of the conditions of Theorem 2.4 is called rational.

The set of these series, a A-module ²³ , is denoted by A ^rat hh X ii and is closed by {conc, +, ∗}. We also have

Proposition 2.5 (see also [21,40]) The module A ^rat hh X ii (resp. A ^rat hhY ii) is closed by

^⊔⊔

(resp. ). Moreover, for i = 1, 2, let R i ∈ A ^rat hh X ii and ( ν i , µ i , η i ) be its representation of dimension n _i . Then the linear representation of ²⁴

R ^∗ _i is

0 1 ,

 





 µ i (x) + η i ν i µ i (x) 0 ν i η i 0





 



x∈ X

,



 η i

1





,

that of R ₁ + R ₂ is

ν 1 ν 2

,

 





 µ ₁ (x) 0 0 µ 2 (x)





 



x∈ X

,



 η ₁ η 2





,

that of R ₁ .R 2 is

ν 1 0 ,

 





 µ ₁ (x) η ₁ ν ₂ µ ₂ (x) 0 µ ₂ (x)





 



x∈ X

,



 η ₁ µ ₂ η ₂ η ₂





, that of R ₁

^⊔⊔

R ₂ is ( ν ₁ ⊗ ν ₂ , { µ ₁ (x) ⊗ I _n

₂

+ I _n

₁

⊗ µ ₂ (x)} x∈ X , η ₁ ⊗ η ₂ ), that of R ₁ R ₂ is ( ν 1 ⊗ ν 2 , { µ 1 (y k ) ⊗ I n

₂

+ I n

₁

⊗ µ 2 (y k )

+ ∑

i+ j=k

µ ₁ (y i ) ⊗ µ ₂ (y j )} k≥1 , η ₁ ⊗ η ₂ ).

Example 2.6 [Identity (−t ² x ₀ x ₁ ) ^∗

^⊔⊔

(t ² x ₀ x ₁ ) ^∗ = (−4t ⁴ x ² ₀ x ² ₁ ) ^∗ , [34,35]]

start 1 2

x ₀ , it

x ₁ , it

start I II

x ₀ ,t

x ₁ ,t

(−t ² x ₀ x ₁ ) ^∗ ↔ ( ν ₂ , { µ ₂ (x 0 ), µ ₂ (x 1 )}, η ₂ ) (t ² x ₀ x ₁ ) ^∗ ↔ ( ν ₁ , { µ ₁ (x 0 ), µ ₁ (x 1 )}, η ₁ ).

22

see [41].

23

In fact (we will see it) a unital A-algebra for conc and

⊔⊔

.

24

The first constructions are already treated in [21,40], only the last one is new.

ν 1 = 1 0

, µ 1 (x 0 ) =



 0 t 0 0



 , µ 1 (x 1 ) =



 0 0 t 0



, η 1 =



 1 0



 ,

ν ₂ = 1 0

, µ ₂ (x ₀ ) =



 0 it 0 0



 , µ ₂ (x ₁ ) =



 0 0 it 0



, η ₂ =



 1 0



 .

(−t ² x ₀ x ₁ ) ^∗

^⊔⊔

(t ² x ₀ x ₁ ) ^∗ ↔ ( ν , { µ (x 0 ), µ (x 1 )}, η )

= ( ν ₁ ⊗ ν ₂ , { µ ₁ (x 0 ) ⊗ I _n

₂

+ I _n

₁

⊗ µ ₂ (x 0 ), µ 1 (x 1 ) ⊗ I n

₂

+ I n

₁

⊗ µ 2 (x 1 ), η 1 ⊗ η 2 ).

(1, I) start

(2, I)

(2, II)

(1, II) x ₀ , it

x ₀ , t

x ₀ , t

x ₁ , itx 1 ,t

x ₁ , it x ₁ , t

x ₀ , it

ν =

1 0 0 0 ,

µ (x 0 ) =



 

 

 



0 0 t 0 0 0 0 t 0 0 0 0 0 0 0 0



 

 

 

 +



 

 

 



0 it 0 0 0 0 0 0 0 0 0 it 0 0 0 0



 

 

 



=



 

 

 



0 it t 0 0 0 0 t 0 0 0 it 0 0 0 0



 

 

 

 ,

µ (x 1 ) =



 

 

 



0 0 0 0 0 0 0 0 t 0 0 0 0 t 0 0



 

 

 

 +



 

 

 



0 0 0 0 it 0 0 0 0 0 0 0 0 0 it 0



 

 

 



=



 

 

 



0 0 0 0 it 0 0 0 t 0 0 0 0 t it 0



 

 

 

 ,

η =



 

 

 

 1 0 0 0



 

 

 

 .

With the notations of Definition 2.3.(iii) and from Theorem 2.4, it follows that

Proposition 2.7 Suppose A to be a field K. We have (a) Assertions of Theorem 2.4 are equivalent to

(iv) There exists a finite double family of series (G i , D _i ) i∈F such that ²⁵

∆ conc S = ∑

i∈F

G i ⊗ D i

(b) For S ∈ H ^◦

⊔⊔

( X ), since A is a field then the previous identity is equivalent to

∀P , Q ∈ H

_⊔⊔

( X ), hS | PQi = ∑

i∈I

hG i | PihD i | Qi.

Therefore, (K ^rat hh X ii,

^⊔⊔

, 1 X

^∗

, ∆ conc , e) (resp. (K ^rat hhY ii, , 1 X

^∗

, ∆ conc , e)) is the Sweedler’s dual of H

_⊔⊔

( X ) (resp. H (Y )).

Now, let us characterize characters of (AhX i, conc, 1 _X

^∗

).

Proposition 2.8 (Kleene stars of the plane) Let R ∈ A ^rat hh X ii, hR | 1 X

^∗

i = 1 A . The following assertions are equivalent

(i) hR | •i realizes a character ²⁶ of (AhX i, conc, 1 X

^∗

).

(ii) There is a family of coefficients (c x ) _x∈ X such that R = (∑ _x∈ X c x x) ^∗ . (iii) The series R admits a linear representation of dimension one ²⁷ . Moreover, we have ²⁸

( α ₀ x ₀ + α ₁ x ₁ ) ^∗

^⊔⊔

( β ₀ x ₀ + β ₁ x ₁ ) ^∗ = (( α ₀ + β ₀ )x 0 + ( α ₁ + β ₁ )x 1 ) ^∗

s≥1 ∑

a _s y _s

∗

s≥1 ∑

b _s y _s ∗

=

s≥1 ∑

(a s + b _s )y s + ∑

r,s≥1

a _s b _r y _s+r ∗

, where, for any i = 0, 1 and s ≥ 1, α i , β i , a _s , b _s ∈ C.

Example 2.9 [Identity (−t ² y ₂ ) ^∗ (t ² y ₂ ) ^∗ = (−4t ⁴ y ₄ ) ^∗ , [34,35]]

start 1 start 2 start 3

y ₂ , −t ² y ₂ , t ² y ₄ , −t ⁴

(−t ² y ₂ ) ^∗ ↔ ( ν 2 , µ 2 (y 2 ), η 2 )

= (1, −t ² , 1),

(t ² y ₂ ) ^∗ ↔ ( ν 1 , µ 1 (y 2 ), η 1 )

= (1, t ² , 1),

(−t ⁴ y ₄ ) ^∗ ↔ ( ν , µ (y 4 ), η )

= (1, −t ⁴ , 1).

25

See [39] for a way to obtain this finite double family of series (G

i

, D

_i

)

i∈F

.

26

For A = K being a field, this can be rephrased as “R is a group like element of K

^rat

hh X ii”.

27

The dimension is here (as in [1]) the size of the matrices.

28

In particular, (a

s

y

_s

)

^∗

(a

r

y

_r

)

^∗

= (a

s

y

_s

+ a

_r

y

_r

+ a

_s

a

_r

y

_s+r

)

^∗

and (a

s

y

_s

)

^∗

(−a

s

y

_s

)

^∗

= (−a

²_s

y

_2s

)

^∗

.

3 Triangularity, solvability and rationality

3.1 Syntactically exchangeable rational series

Now, we have to study a special set of series in order to work with the rational series of this class: a series S ∈ Ahh X ii is called syntactically exchangeable if and only if it is constant on multi-homogeneous classes, i.e.

(∀u, v ∈ X ^∗ )([(∀x ∈ X )(|u| x = |v| x )] ⇒ hS | ui = hS | vi). (37) A series S ∈ Ahh X ii is syntactically exchangeable iff it is of the following form

S = ∑

α ∈N

^(X)

,supp(α)={x

₁

,...,x

_k

}

s _α , x ^α(x ₁

¹

⁾

^⊔⊔

. . .

^⊔⊔

x ^α(x _k

^k

⁾ . (38)

The set of these series, a shuffle subalgebra of AhhX ii, will be denoted A ^synt _exc hh X ii . When A is a field, the rational and exchangeable series are exactly those who admit a representation with commuting matrices (at least the minimal one is such, see Theorem 3.2 below). We will take this as a definition as, even for rings, this property implies syntactic exchangeability.

Definition 3.1 A series S ∈ A ^rat hh X ii will be called rationally exchangeable if it admits a representation ( ν , µ , η ) such that { µ (x)} _x∈ X is a set of commuting matrices, the set of these series, a shuffle subalgebra of AhhX ii, will be denoted A ^rat _exc hh X ii.

Theorem 3.2 (See [24,39]) Let A ^synt _exc hh X ii denote the set of (syntactically) ex- changeable series. Then

(i) In all cases, one has A ^rat _exc hh X ii ⊂ A ^rat hh X ii ∩ A ^synt _exc hh X ii. The equality holds when A is a field and

A ^rat _exc hhXii = A ^rat hhx 0 ii

^⊔⊔

A ^rat hhx 1 ii =

^⊔⊔

x∈X

A ^rat hhxii, A ^rat _exc hhY ii ∩ A ^rat _fin hhY ii = ^[

k≥0

A ^rat hhy 1 ii

^⊔⊔

. . .

^⊔⊔

A ^rat hhy k ii ( A ^rat _exc hhY ii,

where A ^rat _fin hhY ii = ∪ F⊂

f inite

Y A ^rat hhFii, the algebra of series over finite subal- phabets ²⁹ .

(ii) (Kronecker’s theorem [1,51]) One has A ^rat hhxii = {P(1 − xQ) ⁻¹ } _P,Q∈A[x] (for x ∈ X ) and if A = K is an algebraically closed field of characteristic zero one

29

The last inclusion is strict as shows the example of the following identity [6]

(ty

1

+ t

²

y

₂

+ . . .)

^∗

= lim

k→+∞

(ty

1

+ . . .+ t

^k

y

_k

)

^∗

= lim

k→+∞

(ty

1

)

^∗⊔⊔

. . .

⊔⊔

(t

^k

y

_k

)

^∗

=

⊔⊔

k≥1

(t

^k

y

_k

)

^∗

which lives in A

^rat_exc

hhY ii but not in A

^rat_exc

hhY ii ∩ A

^rat_fin

hhY ii.

also has K ^rat hhxii = span _K {(ax) ^∗

^⊔⊔

Khxi|a ∈ K}.

(iii) The rational series (∑ _x∈ X α x x) ^∗ are conc-characters and any conc- character is of this form.

(iv) Let us suppose that A is without zero divisors and let ( ϕ i ) i∈I be a family within A d X which is Z-linearly independent then, the family L yn( X ) ⊎ { ϕ _i ^∗ } i∈I is algebraically free over A within (A ^rat hh X ii,

^⊔⊔

, 1 X

^∗

).

(v) In particular, if A is a ring without zero divisors {x ^∗ } x∈ X (resp. {y ^∗ } y∈Y ) are algebraically independent over (Ah X i,

^⊔⊔

, 1 X

^∗

) (resp. (AhY i, , 1 _Y

^∗

)) within (A ^rat hh X ii,

^⊔⊔

, 1 X

^∗

) (resp. (A ^rat hhY ii, , 1 Y

^∗

)).

Proof.

(i) The inclusion is obvious in view of (38). For the equality, it suffices to prove that, when A is a field, every rational and exchangeable series admits a repre- sentation with commuting matrices. This is true of any minimal representation as shows the computation of shifts (see [20,24,39]).

Now, if X is finite, as all matrices commute, we have

w∈ ∑ X

^∗

µ (w)w =

x∈ ∑ X

µ (x)x ∗

=

^⊔⊔

x∈ X

( µ (x)x) ^∗

and the result comes from the fact that R is a linear combination of matrix elements. As regards the second equality, inclusion ⊃ is straightforward. We remark that the union ^S _k≥1 A ^rat hhy 1 ii

^⊔⊔

. . .

^⊔⊔

A ^rat hhy k ii is directed as these alge- bras are nested in one another. With this in view, the reverse inclusion comes from the fact that every S ∈ A ^rat _fin hhY ii is a series over a finite alphabet and the result follows from the first equality.

(ii) Let A = {P(1 − xQ) ⁻¹ } _P,Q∈A[x] . Since P(1 − xQ) ⁻¹ = P(xQ) ^∗ then it is obvi- ous that A ⊂ A ^rat hhxii. Next, it is easy to check that A contains Ahxi(= A[x]) and it is closed by +, conc as, for instance,

(1 − xQ ₁ )(1 − xQ ₂ ) = (1 − x(Q ₁ + Q ₂ − xQ ₁ Q ₂ )).

We also have to prove that A is closed for ∗. For this to be applied to P(1 − xQ) ⁻¹ , we must suppose that P(0) = 0 (as, indeed, hP(1 − xQ) ⁻¹ | 1 x

^∗

i = P(0)) and, in this case, P = xP ₁ . Now

P 1 − xQ

∗

=

1 − P

1 − xQ −1

= 1 − xQ

1 − x(Q + P ₁ ) ∈ A .

(iii) Let S = (∑ _x∈ X α x x) ^∗ and note that S = 1 + (∑ _x∈ X α x , x)S. Then hS | 1 X

^∗

i =

1 A and, if w = xu, we have hS | xui = α x hS | ui, then by recurrence on the

length, hS | x ₁ . . . x _k i = ∏ ^k _i=1 α x

i

which shows that S is a conc-character. For

the converse, we have Schützenberger’s reconstruction lemma which says

that, for every series S S = hS | 1 X

^∗

i.1 A + ∑

x∈ X

x.x ⁻¹ S

but, if S is a conc-character, hS | 1 X

^∗

i = 1 and x ⁻¹ S = hS | xiS, then the previous expression reads

S = 1 A +

x∈ ∑ X

hS | xix

S

this last equality being equivalent to S = (∑ _x∈ X hS | xi.x) ^∗ proving the claim.

(iv) As (Ah X i,

^⊔⊔

, 1 X

^∗

) and (AhY i, , 1 Y

^∗

) are enveloping algebras, this prop- erty is an application of the fact that, on an enveloping U , the characters are linearly independent w.r.t. to the convolution algebra U _∞ ^∗ (see the gen- eral construction and proof in [25] or [27]). Here, this convolution algebra ( U _∞ ^∗ ) contains the polynomials (is equal in case of finite X ). Now, consider a monomial

( ϕ _i ^∗

₁

)

^⊔⊔

^α

¹

. . .( ϕ _i ^∗

_n

)

^⊔⊔

^α

ⁿ

= _n

k=1 ∑

α _i

_k

ϕ _i

_k

∗

The Z-linear independence of the monomials in ( ϕ i ) _i∈I implies that all these monomials are linearly independent over Ah X i which proves algebraic inde- pendence of the family ( ϕ _i ) i∈I .

To end with, the fact that L yn( X ) ⊎ { ϕ _i ^∗ } i∈I is algebraically free comes from Radford theorem (Ah X i,

^⊔⊔

, 1 X

^∗

) ≃ A[ L yn( X )] and the transitivity of polynomial algebras (see [3] ch III.2 Proposition 8).

(v) Comes directly as an application of the preceding point.

✷ Remark 3.3 (Point (ii) of Theorem 3.2 above) Kronecker’s theorem which can be rephrased in terms of stars as A ^rat hhxii = {P(xQ) ^∗ } _{P ,Q∈A[x]} holds for every ring and is therefore characteristic free, unlike the shuffle version requiring algebraic closure and denominators.

3.2 Exchangeable rational series and their linear representations

As examples, one can consider the following forms (F ₀ ), (F ₁ ) and (F ₂ ) of rational series in A ^rat hhX ii [33,39]:

(F 0 ) E ₁ x _i

₁

. . . E _j x _i

_j

E _j+1 , where x _i

₁

, . . .,x i

j

∈ X , E ₁ , . . ., E _j ∈ A ^rat hhx 0 ii,

(F ₁ ) E ₁ x i

₁

. . . E j x i

j

E _j+1 , where x i

₁

, . . .,x i

j

∈ X , E ₁ , . . ., E j ∈ A ^rat hhx ₁ ii,

(F 2 ) E ₁ x _i

₁

. . . E _j x _i

_j

E _j+1 , where x _i

₁

, . . . , x _i

_j

∈ X , E ₁ , . . . , E _j ∈ A ^rat _exc hhX ii.

Using linear representations, we also have

Theorem 3.4 (Triangular sub bialgebras of (A ^rat hh X ii,

^⊔⊔

, 1 _X

^∗

, ∆ conc , e), [39]) Let ρ = ( ν , µ , η ) a representation of R ∈ A ^rat hh X ii. Then

(i) If the matrices { µ (x)} _x∈ X commute between themselves and if the alphabet is finite, every rational exchangeable series decomposes as

R =

n i=1 ∑

⊔⊔

x∈ X

R ⁽ⁱ⁾ _x with R ⁽ⁱ⁾ _x ∈ A ^rat hhxii.

(ii) If L consists of upper-triangular matrices then R ∈ A ^rat _exc hh X ii

^⊔⊔

Ah X i.

(iii) For any x ∈ X , letting M(x) := µ (x)x and then extending, in the obvious way, this representation to Ahh X ii by M(S) = ∑ _w∈ X

^∗

hS | wi µ (w)w, we have

R = ν M( X ^∗ ) η . Moreover, we have

(a) If { µ (x)} x∈ X are upper-triangular then M( X ) = D( X )+N( X ), where D( X ) and N( X ) are diagonal and strictly upper-triangular letter ma- trices, respectively, such that ³⁰

M( X ^∗ ) = ((D( X ^∗ )N( X )) ^∗ D( X ^∗ )).

(b) We get ³¹ (for X = X )

M((x ₀ + x ₁ ) ^∗ ) = (M(x ^∗ ₁ )M(x ₀ )) ^∗ M(x ^∗ ₁ ) = (M(x ^∗ ₀ )M(x ₁ )) ^∗ M(x ^∗ ₀ ) and the modules generated by the families (F 0 ), (F 1 ) and (F 2 ) are closed by conc,

^⊔⊔

(and coproducts if A = K is a field). From this, it follows that R is a linear combination of expressions in the form (F 0 ) (resp. (F 1 )) if M(x ^∗ ₁ )M(x 0 ) (resp. M(x ^∗ ₀ )M(x 1 )) is strictly upper-triangular.

(c) If A is a Q-algebra then M( X ^∗ ) =

ց l∈ L ∏ yn X

e ^S

^l

^µ(P

^l

⁾ .

Remark 3.5 (i) The point (i) of Theorem 3.4 is no longer true for an infinite alphabet as shows the example of the series S = ∑ _k≥1 y _k in A ^rat hhY ii.

(ii) On a general ring it can happen that R is exchangeable, ρ minimal and never- theless L is noncommutative, as shows the case of A = Q[x,t ]/t ³ Q[x, t] and

X = {a, b}, µ (a) = t



 1 0 x 1



 , µ (b) = t



 1 x 0 1



 , ν = 1 1

, η =



 1 1



 .

30

by Lazard factorization [43,50].

31

idem.

With these data, R = 2 + (xt + 2t)(a + b) + (x ² t ² + 2xt ² + 2t ² )(ab + ba) which is an exchangeable polynomial but

µ (a) µ (b) =



 t ² xt ² xt ² x ² t ² + t ²



 , µ (b) µ (a) =



 x ² t ² + t ² xt ² xt ² t ²





Now the representation is minimal because if it were of dimension 1, ¹ ₂ R would be a conc-character, which is not the case. Otherwise, if it were of dimension 0, R would be zero.

In order to establish Theorem 3.7 below, we will use the following

Lemma 3.6 Let ( ν , τ , η ) a representation of S of dimension r such that, for all x ∈ X , ( τ (x) − c(x)I r ) is strictly upper triangular, then S ∈ K _exc ^rat hh X ii

^⊔⊔

Kh X i.

Proof. Let (e i ) 1≤i≤r be the canonical basis of K ^1×r . We construct the representa- tions ρ ₁ = ( ν , (x 7→ τ (x) − c(x)I r ), η ), ρ ₂ = (e ₁ , (x 7→ c(x)I r ), e ^∗ ₁ ) of S ₁ and S ₂ and remark that S ₁

^⊔⊔

S ₂ admits the representation

ρ 3 = ( ν ⊗ e ₁ , (( τ (x) − c(x)I r ) ⊗ I _r + I _r ⊗ c(x)I r ) _x∈ X , η ⊗ e ^∗ ₁ )

as I _r ⊗ c(x)I r = c(x)I r ⊗ I _r , ρ 3 is, in fact, ( ν ⊗ e ₁ , ( τ (x) ⊗ I _r ) _x∈ X , η ⊗ e ^∗ ₁ ) which represents S, the result now comes from the fact that S ₁ ∈ Kh X i and S ₂ =

(∑ _x∈ X c(x)x) ^∗ ∈ K _exc ^rat hh X ii. ✷

We first begin by properties essentially true over algebraically closed fields.

Theorem 3.7 (Triangular sub bialgebras of (K ^rat hh X ii,

^⊔⊔

, 1 X

^∗

, ∆ conc , e), [39]) We suppose that K is an algebraically closed field and that ρ = ( ν , µ , η ) is a linear representation of R ∈ K ^rat hh X ii of minimal dimension n, we note L = L ( µ ) ⊂ K ^n×n the Lie algebra generated by the matrices ( µ (x)) _x∈ X . Then

(i) L is commutative iff R ∈ K _exc ^rat hh X ii, (ii) L is nilpotent iff R ∈ K _exc ^rat hh X ii

^⊔⊔

Kh X i,

(iii) L is solvable iff R is a linear combination of expressions in the form (F 2 ).

Moreover, denoting K _nil ^rat hh X ii (resp. K _sol ^rat hh X ii), the set of rational series such that L ( µ ) is nilpotent (resp. solvable), we get a tower of sub Hopf algebras of the Sweedler’s dual, K _nil ^rat hh X ii ⊂ K _sol ^rat hh X ii ⊂ H ^◦

⊔⊔

( X ).

Proof.

(i) Let us remark that, for x, y ∈ X , p, s ∈ X ^∗ , we have hR | pxysi = hR | pyxsi

which is due to the commutation of matrices. Conversely, since ρ is minimal

then there is P _i , Q _i ∈ Kh X i, i = 1...n such that (see [1,20,48])

∀u ∈ X ^∗ , µ (u) = (hP i ⊲ R ⊳ Q _i | ui) 1≤i,j≤n = (hR | Q _i uP _i i) 1≤i, j≤n . Now, for x, y ∈ X , we have

µ (xy) = (hR | Q _i xyP _i i) 1≤i, j≤n ∗

= (hR | Q _i yxP _i i) 1≤i, j≤n = µ (yx) equality = ^∗ being due to exchangeability.

(ii) Let us consider K ⁿ as the space of the representation of L given by µ . Let K ⁿ = ^{L m} _j=1 V j be a decomposition of K ⁿ into indecomposable L -modules (see [16], Theorem 1.3.19 where it is done for ch(K) = 0, or [5] Chapter VII

§1 Propopsition 9 for arbitrary characteristic), we know that each V j is a L - module and that the action of L is triangularizable with constant diagonals inside each sector V _j . Thus, it is an invertible matrix P ∈ GL(n, K) such that

∀x ∈ X , P µ (x)P ⁻¹ = blockdiag(T 1 , T ₂ . . . , T _k ) =



 

 

 



T ₁ 0 0 . . . 0

0 T ₂ 0 . . . 0 ... ... ... ... ...

0 0 . . . 0 T _k



 

 

 

 where the T j are upper triangular matrices with scalar diagonal i.e. is of the form T _j (x) = λ (x)I + N(x) where N(x) is strictly upper-triangular ³² . Set d _j to be the dimension of T _j (so that n = ∑ ^m _j=1 d _j ), partitioning ν P ⁻¹ = ν ^′ (resp.

P η = η ^′ ) with these dimensions we get blocks so that each ( ν ^′ _j , T _j , η ^′ _j ) is the representation of a series R _j and R = ∑ ^m _j=1 R _j . It suffices then to prove that, for all j, R j ∈ K _exc ^rat hh X ii

^⊔⊔

Kh X i. This is a consequence of Lemma 3.6.

Conversely, if ρ i = ( ν i , τ i , η i ), i = 1, 2, are two representations then [ τ 1 (x) ⊗ I _r + I _r ⊗ τ ₂ (x), τ ₁ (y) ⊗ I _r + I _r ⊗ τ ₂ (y)] = [ τ ₁ (x), τ ₁ (y)] ⊗ I _r + I _r ⊗ [ τ ₂ (x), τ ₂ (y)]

and a similar formula holds for m-fold brackets (Dynkin combs), so that if L ( τ i )’s are nilpotent, the Lie algebra L ( τ 1 ⊗ I _r + I _r ⊗ τ 2 ) is also nilpotent.

The point here comes from the fact that series in K _exc ^rat hh X ii as well as in Kh X i admit nilpotent representations, so, let ( α , τ , β ) such a representation and ( α ^′ , τ ^′ , β ^′ ) its minimal quotient (obtained by minimization, see [1]), then L ( τ ^′ ) is nilpotent as a quotient of L ( τ ). Now two minimal representations being isomorphic, L ( µ ) is isomorphic to L ( τ ) and then it is nilpotent.

(iii) As L is solvable and K algebraically closed, using Lie’s theorem, we can find a conjugate form of ρ = ( ν , µ , η ) such that the matrices µ (x) are upper- triangular. Since this form also represents R, letting D( X ) (resp. N( X )) be

32

Even, as K is infinite, there is a global linear form on L , λ

lin

such that, for all g ∈ L , PgP

⁻¹

−

λ

lin

(g)I is strictly upper-triangular.

the diagonal (rep. strictly upper-triangular) letter matrice such that M( X ) = D( X ) + N( X ) then

R = ν M( X ^∗ ) η = ν (D( X ^∗ )N( X )) ^∗ D( X ^∗ ) η .

Since D( X ^∗ )N( X ) being nilpotent of order n then (D( X ^∗ )N( X )) ^∗ =

∑ ⁿ _j=0 (D( X ^∗ )N( X )) ^j . Hence, letting S be the vector space generated by forms of type (F 2 ) which is closed by concatenation, we have D( X ^∗ )N( X ) ∈ S ^n×n and then (D( X ^∗ )N( X )) ^∗ ∈ S ^n×n . Finally, R = ν M( X ^∗ ) η ∈ S which is the claim.

Conversely, as sums and quotients of solvable representations are solvable is suffices to show that a single form of type F ₂ admits a solvable representa- tion and end by quotient and isomorphism as in (ii). From Proposition (2.5), we get the fact that, if R _i admit solvable representations so does R ₁ R ₂ , then the claim follows from the fact that, firstly, single letters admit solvable (even nilpotent) representations and secondly series of

^⊔⊔

{K ^rat hhxii} _x∈ X admit solv- able representations. Finally, we choose (or construct) a solvable representa- tion of R, call it ( α , τ , β ) and ( α ^′ , τ ^′ , β ^′ ) its minimal quotient, then L ( τ ^′ ) is solvable as a quotient of L ( τ ). Now two minimal representations being isomorphic, L ( µ ) is isomorphic to L ( τ ), hence solvable.

Moreover and ff.] Comes from the computation of the coproduct by inser- tion of identity ∑ ⁿ _i=1 e ^∗ _i e i .

✷ Remark 3.8 For an example of series S with solvable representation but such that S ∈ / K _exc ^rat hh X ii

^⊔⊔

Kh X i. One can take X = {a, b} and S = a ^∗ b(−a) ^∗ .

To end this section (of combinatorial framework), for a need of the proof of Theorem 4.8 below, let us extend the pairing (16) as a partially defined map

Dom(h?k?i) −→ A, (39)

T ⊗ S −→ hT kSi := ∑

w∈ X

^∗

hT | wihS | wi. (40)

where Dom(h?k?i) ⊂ Ahh X ii ⊗ Ahh X ii.

Here, the family ∑ _w∈ X

^∗

hT | wihS | wi is summable, for some topology on A.

Its sum is denoted by hT kSi and the set of these series S is denoted by Dom _word (T ).

This proof will also use the following lemma as a consequence of Theorem 2.4 Lemma 3.9 For any ring A without zero divisors, let R ∈ A ^rat hh X ii of linear rep- resentation ( ν , µ , η ) of dimension n. Then any family {R ⊳ P _i |P i ∈ Ah X i} i=1...m>n

is linearly dependent, i.e. there are { α i } i=1...m in A, not all zero, such that

∑ ^m _I=1 α _i (R ⊳ P _i ) = 0.

4 Towards a noncommutative Picard-Vessiot theory

Let ( A , d) be a commutative associative differential ring (ker(d) = k being a field), C ₀ be a differential subring of A (d( C ₀ ) ⊂ C ₀ ) which is an integral domain con- taining the field of constants and C{{(g _i ) i∈I }} be the differential subalgebra of A generated by (g i ) i∈I , i.e. the k-algebra generated by g i ’s and their derivatives [49].

4.1 Noncommutative differential equations

Let us consider the following differential equation, with homogeneous series of degree 1 as multiplier (a polynomial in the case of finite alphabet).

dS = MS; hS | 1i = 1, where M = ∑

x∈ X

u _x x ∈ C ₀ hh X ii (41) Example 4.1 [Drinfel’d equation] X = {x ₀ , x ₁ } and Ω = C \ (] − ∞, 0] ∪ [1, +∞[).

(KZ 3 ) dS = (x 0 u _x

₀

+ x ₁ u _x

₁

)S with u _x

₀

(z) = z ⁻¹ , u _x

₁

(z) = (1 − z) ⁻¹ .

This equation was introduced in [17,18] and a complete study was presented in [39]

(solutions via polylogarithms and their special values, polyzetas).

Example 4.2 Y = {y i } i≥1 and Ω = {z ∈ C | | z |< 1}.

dS =

i≥1 ∑

y _i u _y

_i

S with u _y

_i

(z) = ∂ ℓ i (z).

where, denoting γ the Euler’s constant and ζ the Riemann zeta function, ℓ 1 (z) := γ z − ∑

k≥2

ζ (k) (−z) ^k

k and for r ≥ 2, ℓ r (z) := − ∑

k≥1

ζ (kr) (−z ^r ) ^k k .

This equation was introduced in [9] to study the independence of a family of eule- rian functions.

Let us also recall the following useful result for proving Theorem 4.8 bellow.

Proposition 4.3 ([34,35,37]) Let S ∈ A hh X ii be solution of (41). Then S satis- fies the differential equations d ^l S = Q _l S, for l ≥ 0, where Q _l ∈ C{{(u _i ) i≥0 }}h X i satisfying the recursion Q ₀ = 1 and Q _l = Q _l−1 M + dQ _l−1 .

More explicitly, Q _l can be computed as follows (suming over words w = x _i

₁

. . . x _i

_l

and derivation multi-indices r = (r 1 , . . . , r _l ) of degree deg r =| w |= l and of weight wgt r = l + r ₁ + . . . + r _l )

Q _l = ∑

wgtr=l w∈Xdegr

deg r

∏ l=1

∑ ^l _j=1 r _j + j − 1 r _l

τ r (w) and

 



τ r (w) = τ r

₁

(x i

₁

) . . . τ r

l

(x i

l

) =

( ∂ z ^r

¹

u x

i1

)x i

₁

. . . ( ∂ z ^r

^l

u x

_il

)x i

_l

.

Theorem 4.4 Suppose that the C-commutative ring A is without zero divisors and equipped with a differential operator ∂ such that C = ker ∂ .

Let S ∈ A hh X ii be a group-like solution of (41), in the following form S = 1 X

^∗

+ ∑

w∈ X

^∗

X

hS | wiw = 1 X

^∗

+ ∑

w∈ X

^∗

X

hS | S _w iP w =

ց l∈ L ∏ yn X

e ^hS|S

^l

^iP

^l

. Then

(i) If H ∈ A hh X ii is another group-like solution of (41) then there exists C ∈ L ie A hh X ii such that S = He ^C (and conversely).

(ii) The following assertions are equivalent

(a) {hS | wi} w∈ X

^∗

is C ₀ -linearly independent, (b) {hS | li} _l∈ L yn X is C ₀ -algebraically independent, (c) {hS | xi} _x∈ X is C ₀ -algebraically independent, (d) {hS | xi} _x∈ X ∪{1

X∗

} is C ₀ -linearly independent,

(e) The family {u x } _x∈ X is such that, for f ∈ Frac( C ₀ ) and (c x ) _x∈ X ∈ C ^{( X )} ,

x∈ ∑ X

c _x u _x = ∂ f = ⇒ (∀x ∈ X )(c x = 0).

(f) The family (u x ) _x∈ X is free over C and ∂ Frac( C ₀ ) ∩span _C {u x } _x∈ X = {0}.

Proof. [Sketch] The first item has been treated in [35]. The second is a group-like version of the abstract form of Theorem 1 of [15]. It goes as follows

• due to the fact that A is without zero divisors, we have the following embeddings C ₀ ⊂ Frac( C ₀ ) ⊂ Frac(A), Frac(A) is a differential field, and its derivation can still be denoted by ∂ as it induces the previous one on A ,

• the same holds for A hh X ii ⊂ Frac(A)hh X ii and d

• therefore, equation (41) can be transported in Frac(A)hh X ii and M satisfies the same condition as previously.

• Equivalence between a-d comes from the fact that C ₀ is without zero divisors and then, by denominator chasing, linear independances w.r.t C ₀ and Frac( C ₀ ) are equivalent. In particular, supposing condition d, the family {hS | xi} _x∈ X ∪{1

X∗

}

(basic triangle) is Frac( C ₀ )-linearly independent which imply, by the Theorem 1 of [15], condition e,

• still by Theorem 1 of [15], e is equivalent to f, implying that {hS | wi} _w∈ X

^∗

is Frac( C ₀ )-linearly independent which induces C ₀ -linear independence (i.e. a).

✷

Now, let us go back to notations of Section 1 and equip the differential rings of