Independance of hyperlogarithms over function fields through algebraic combinatorics on words.

M. Deneufchâtel, G. H. E. Duchamp, Hoang Ngoc Minh and Allan I. Solomon ^{∗}

Contents

Abstract

Group-like series ...

1 Introduction

2 Series with variable coefficients.

2.1 Motivations.

In this section, we implement an abstract setting which is intended to apply on function spaces. As a motivation, let us illustrate this by an example. Let V be a connected and simply connected analytic variety of dimension one (for example, the doubly cut plane ℂ − (] − ∞, 0[∪]1, +∞[), or the universal covering of ℂ − {0, 1}), H = C^{ω}(V; ℂ) be the space of all analytic fonctions on V. This space is a differential algebra with the derivative \frac{d}{dz}. One extends at once this derivative (then denoted by d) to H⟨⟨X⟩⟩ by

d(S) = ∑_{w∈X^{∗}} \frac{d}{dz} (⟨S|w⟩)w  (1)

it is easy to check (proof in the general case below) that d is a derivation of the algebra H⟨⟨X⟩⟩. Differential equations of the type

d(S) = MS  (2)

where M = ∑_{x∈X} u_{x}(z)x were widely considered in the domains of (à faire automatique, Drinfel’d, Weshung, etc...) and provide, through integrators build by iterated integrals, spaces of special functions. An immediate application of theorem (à faire ??) below provides the result that, for any solution of (??) (with ⟨S|1⟩ = 1), the family of functions (⟨S|w⟩)_{w∈X^{∗}} is free over the field of rational functions on V .

∗LIPN - UMR 7030 CNRS - Université Paris 13 F-93430 Villetaneuse, France

2.2 General setting

Let M, be a locally finite monoid [?] and H⟨⟨M⟩⟩, be the large algebra [?] of M with coefficients in H). Let M be a locally finite monoid [?] and V be a set (where the variable z stands) and H ⊂ C^{V}, an algebra of fonctions. Every series S ∈ H⟨⟨M⟩⟩ can be written

S := ∑_{m∈M}⟨S|m⟩m  (3)

(as the family (⟨S|m⟩m)_{m∈M} is summable). Thus, one can consider the series of H⟨⟨M⟩⟩ as functions on V (with values in ℂ⟨⟨M⟩⟩) and specialize them by

S(z_{0}) := ∑_{m∈M} (⟨S|m⟩)|_{z=z_{0}}m .  (4)

Moreover, if d is a derivation in H, its extension to H⟨⟨M⟩⟩ “coefficient by coefficient” given as

d(S) := ∑_{m∈M} d(⟨S|m⟩)m  (5)

is a derivation of H⟨⟨M⟩⟩. Let us show, on an example, how the proof of theorem à faire ?? below works on an example. The data are

1. X = {x_{0}, x_{1}}

2. V is a connected and simply connected subset of ℂ

3. H = C^{ω}(V, ℂ) which does not contain \{0, 1\}, eendowed with the derivation \frac{d}{dz}

4. M = \frac{x_{0}}{z} + \frac{x_{1}}{1−z}

5. Statement : If S is any solution of

d(S) = MS ; ⟨S|1⟩ = 1  (6)

then the functions (⟨S|w⟩)_{w∈X^{∗}} are linearly independant over the field of rational functions i.e. if

f_{i} = \frac{p_{i}}{q_{i}} , i = 1 ⋯ N ; p_{i}, q_{i} ∈ ℂ[z]  (7)

and w_{i} ∈ X^{∗} are such that

∑_{i=1}^{N} f_{i}(z)⟨S|w_{i}⟩ = 0  (8)

on some open (non void) set (which does not contain the opes of the f_{i}) then

(∀i ∈ [1 ⋯ N])(f_{i} ≡ 0) .  (9)

6. An example for the recursion Order X by x_{0} < x_{1} and use ≺_{glex}, the graded lexicographic ordering on X^{∗}. For each non-trivial relation (Rel) (if there are such)

∑_{i=1}^{N} f_{i}(z)⟨S|w_{i}⟩ = 0  (10)

we consider the leading monomial lead(Rel) = sup{w_{i}|f_{i} ≢ 0} . If there were nontrivial relatons, we could take one with the least possible leading monomial and the relation itself. Assumption 1 Suppose that the set of its monomials be {x_{0}, x_{1}, x_{1}x_{0}, x_{0}^{2}x_{1}} with lead(Rel) = x_{0}^{2}x_{1}. One has

f_{x_{0}}(z)⟨S|x_{0}⟩ + f_{x_{1}}(z)⟨S|x_{1}⟩ + f_{x_{1}x_{0}}(z)⟨S|x_{1}x_{0}⟩ + f_{x_{0}^{2}x_{1}}(z)⟨S|x_{0}^{2}x_{1}⟩ = 0 . (11)

which is defined on U intersection of the domains U_{0} = dom(f_{w_{i}}). At the cost of restricting the relation to U_{1} = U_{0} \ O_{f_{x^{2}_{0}x_{1}}} one has also

g_{x_{0}}(z)⟨S|x_{0}⟩ + g_{x_{1}}(z)⟨S|x_{1}⟩ + g_{x_{1}x_{0}}(z)⟨S|x_{1}x_{0}⟩ + ⟨S|x_{0}^{2}x_{1}⟩ = 0  (12)

with g_{i} = _{\frac{f_{i}}{f_{x^{2}_{0}x_{1}}}} . Differentiating (??), we get

g′_{x_{0}}(z)⟨S|x_{0}⟩ + g_{x_{0}}(z)⟨S′|x_{0}⟩ + g′_{x_{1}}(z)⟨S|x_{1}⟩ + g_{x_{1}}(z)⟨S′|x_{1}⟩+
g′_{x_{1}x_{0}}(z)⟨S|x_{1}x_{0}⟩ + +g_{x_{1}x_{0}}(z)⟨S′|x_{1}x_{0}⟩ + ⟨S′|x_{0}^{2}x_{1}⟩ = 0 .  (13)

As S′ = MS, we have

⟨S′|x_{0}u⟩ = ⟨MS|x_{0}u⟩ = \frac{1}{z}⟨S|u⟩ and

⟨S′|x_{1}u⟩ = ⟨MS|x_{1}u⟩ = \frac{1}{1 − z} ⟨S|u⟩ .  (14)

With this in hand (??) becomes

g′_{x_{0}}(z)⟨S|x_{0}⟩ + g_{x_{0}}(z)\frac{1}{z}⟨S|1⟩ + g′_{x_{1}}(z)⟨S|x_{1}⟩ + g_{x_{1}}(z)\frac{1}{1 − z} ⟨S|1⟩+

g′_{x_{1}x_{0}}(z)⟨S|x_{1}x_{0}⟩ + +g_{x_{1}x_{0}}(z)\frac{1}{1 − z} ⟨S|x_{0}⟩ + \frac{1}{z}⟨S|x_{0}x_{1}⟩ = 0 .  (15)

which is of rank strictly less than (??) and then should be trivial. A contradiction as \frac{1}{z} is zero in no non-void open subset ; “Assumption 1” must be false and we are done. □

2.3 Non commutative differential equations (abstract setting).

The ground field k is supposed commutative and of characteristic zero. We suppose given a commutative differential k-algebra (A, d) that is a k-algebra (associative and commutative with unit) A endowed with an element d ∈ 𝔇𝔢𝔯(A). We will suppose that the ring of constants ker(d) is exactly k.

An alphabet X being given, one can at once extend the derivation d to the algebra A⟨⟨X⟩⟩, as in (??) by

d(S) = ∑_{w∈X^{∗}} d(⟨S|w⟩)w .  (16)

Theorem 2.1 Let (A, d) be a k-commutative associative differential algebra with unit (ch(k) = 0) and C be a differential subfield of A (i.e. d(C) ⊂ C). We suppose that S ∈ A⟨⟨X⟩⟩ is a solution of the differential equation

d(S) = MS ; ⟨S|1⟩ = 1  (17)

where the multiplier M = ∑_{x∈X} u_{x}x ∈ C⟨⟨X⟩⟩ is an homogeneus series (a polynomial in case X is finite) of degree 1. The following condition are equivalent :

i) The family (⟨S|w⟩)_{w∈X^{∗}} of coefficients of S is free over C. ii) The family of coefficients (⟨S|y⟩)_{y∈X∪{1_{X^{∗}}}} is free over C. iii) The family (u_{x})_{x∈X} is such that, for f ∈ C and α_{x} ∈ k

d(f) = ∑_{x∈X} α_{x}u_{x} ⟹ (∀x ∈ X)(α_{x} = 0) .  (18)

iv) The family (u_{x})_{x∈X} is free over k and

d(C) ∩ span_{k}((u_{x})_{x∈X}) = {0} .  (19)

Proof — (i)⟹(ii) Obvious. (ii)⟹(iii) Suppose that the family (⟨S|y⟩)_{y∈X∪{1_{X^{∗}}}} (coefficients taken at letters and the emty word) of coefficients of S is free over C and let us consider a relation as eq. (??)

d(f) = ∑_{x∈X} α_{x}u_{x} .  (20)

We form the polynomial P = −f1_{X^{∗}} + ∑_{x∈X} α_{x}x. One has d(P) = −d(f)1_{X^{∗}} and

d(⟨S|P ⟩) = ⟨d(S)|P ⟩+⟨S|d(P)⟩ = ⟨MS|P ⟩−d(f)⟨S|1_{X^{∗}}⟩ = (∑_{x∈X} α_{x}u_{x})−d(f) = 0 (21)

and, then ⟨S|P ⟩ must be a constant, say λ ∈ k. For Q = P − λ.1_{X^{∗}}, we have

supp(Q) ⊂ X ∪ {1_{X^{∗}}} and ⟨S|Q⟩ = ⟨S|P ⟩ − λ⟨S|1_{X^{∗}}⟩ = ⟨S|P ⟩ − λ = 0 .

This implies that Q = 0 and, as Q = −(f + λ)1_{X^{∗}} + ∑_{x∈X} α_{x}x, one has, in particular, all the α_{x} = 0. (iii)⟺(iv) Obvious, (iv) being a geometric reformulation of (iii). (iii)⟺(i) Let K be the kernel of P ↦ ⟨S|P ⟩ (a linear form C⟨X⟩ → C) i.e.

K = {P ∈ C⟨X⟩|⟨S|P ⟩ = 0} .  (22)

If K = {0}, we are done. Otherwise, let us adop the following strategy. First, we order X by some well-ordering < ([?] III.2.1) and X^{∗} by the graded lexicographic ordering ≺ defined by

u ≺ v ⟺ |u| < |v| or (u = pxs_{1} , v = pys_{2} and x < y)  (23)

it is easy to check that ≺ is also a well-ordering relation. For each nonzero polynomial P, we note lead(P) its leading monomial i.e. the greatest element of its support supp(P) (for ≺). Now, as R = K − {0} is not empty, let w_{0} be the minimmal element of lead(R) and choose a P ∈ R such that lead(P) = w_{0}. We write

P = fw_{0} + ∑_{u≺w_{0}}⟨P |u⟩u ; f ∈ C − {0} .  (24)

the polynomial Q = \frac{1}{f}P is also in R with the same leading monomial, but the leading coefficient is now 1 and Q reads

Q = w_{0} + ∑_{u≺w_{0}}⟨Q|u⟩u .  (25)

Differentiating ⟨S|Q⟩ = 0, one gets

0 = ⟨d(S)|Q⟩ + ⟨S|d(Q)⟩ = ⟨MS|Q⟩ + ⟨S|d(Q)⟩ =
⟨S|M^{†}Q⟩ + ⟨S|d(Q)⟩ = ⟨S|M^{†}Q + d(Q)⟩  (26)

with

M^{†}Q + d(Q) = ∑_{x∈X} u_{x}(x^{†}Q) + ∑_{u≺w_{0}} d(⟨Q|u⟩)u ∈ C⟨X⟩ .  (27)

It is impossible that M^{†}Q + d(Q) ∈ R because it would be of leading monomial strictly less than w_{0}, hence M^{†}Q + d(Q) = 0. This is equivalent to the recursion

d(⟨Q|u⟩) = − ∑_{x∈X} u_{x}⟨Q|xu⟩ ; for x ∈ X , v ∈ X^{∗}  (28)

From this last relation, we derive that ⟨Q|w⟩ ∈ k for every w of length deg(Q) and, because ⟨S|1⟩ = 1, one must have deg(Q) > 0. Then, write w_{0} = x_{0}v and compute the coefficient at v

d(⟨Q|v⟩) = − ∑_{x∈X} u_{x}⟨Q|xv⟩ = ∑_{x∈X} α_{x}u_{x}  (29)

with coefficients α_{x} = −⟨Q|xv⟩ ∈ k as |xv| = deg(Q) for all x ∈ X. Condition PI implies that all coefficients ⟨Q|xu⟩ are zero, in particular, as ⟨Q|x_{0}u⟩ = 1, we get a contradiction. This proves that K = {0}. □

3 Applications of the main theorem.

3.1 Independance of the polylogarithms.

Let V be a connected and simply connected analytic variety of dimension 1 (for example, the doubly cut plane ℂ − (] − ∞, 0] ∪ [1, +∞[), or the universal covering of ℂ − {0, 1}), A = C^{ω}(V; ℂ) be the space of analytic fonctions on V endowed with the derivative d = \frac{d}{dz}. Let X = {x_{0}, x_{1}} and X^{∗} be the free monoid on X. It is locally finite [?] and we note A⟨⟨X⟩⟩ its large algebra [?] (with coefficients in A).

4 Réserve à ordonner

We will use three types of differential equations.

a) Left-sided equation

\frac{d}{dz} S(z) = M(z)S(z)  (30)

b) Right-sided equation

\frac{d}{dz} S(z) = S(z)M(z)  (31)

c) Two-sided equation

\frac{d}{dz} S(z) = M_{1}(z)S(z) + S(z)M_{2}(z)  (32)

with M, M_{i} ∈ H_{≥1}⟨⟨M⟩⟩. One first give the resolution of equations of type (??) as their properties specialize, with M_{2} = 0 (resp. M_{1} = 0) to the type (??) (resp. (??)).

Theorem 4.1 With the preceding assumptions.

i) Equation (??) has solutions all of the form

S = (H_{z_{0}}^{z})^{∗}S_{0}  (33)

where H_{z_{0}}^{z} is the operator

G ↦ ∫_{z_{0}}^{z} (M_{1}(s)G(s) + G(s)M_{2}(s))ds  (34)

and S_{0} = S(z_{0}) is a constant series. ii) Two solutions which coincide at a point do coincide everywhere. iii) Let Δ be a closable comultiplication with constant coefficients and suppose that M_{1}, M_{2} in (??) are primitive elements for Δ i.e.

Δ(M_{i}) = M_{i} ⊗ 1 + 1 ⊗ M_{i} ; i = 1, 2 .

Then if S, a solution of (??), is group-like at a point of V, it is group-like everywhere in V. iv) The constant term of S is contant on V (and is that of S_{0}), in particular, if a solution is invertible at a point, it is so everywhere (these solutions will be called regular).

v) Let S be a regular solution of an equation of type (??) with primitive multiplier. Let F be a filter on V (neighbourhoods of 0, of 1, of infinity etc.) and one supposes that S is asymtotically equivalent to G (w.r.t. F i.e. lim_{F}(G^{−1}S) = 1). Then S is group-like.

4.0.1 Proof of theorem (??)

i) The integrator H = H_{z_{0}}^{z} ∈ End(H⟨⟨M⟩⟩) satisfies

H^{n}(H⟨⟨M⟩⟩) ⊂ H_{≥n}⟨⟨M⟩⟩

for all n ∈ ℕ. This implies that (H^{n})_{n≥0} is summable for the 𝔐-adic topology of End^{filtr}(H_{≥1}⟨⟨M⟩⟩) given by the ideal 𝔐 of operators ϕ such that ϕ(H⟨⟨M⟩⟩) ⊂ H_{≥1}⟨⟨M⟩⟩. Let H^{∗} = ∑_{n≥0} H^{n} be its sum. For S_{0} a constant series, one has

\frac{d}{dz} (H^{∗}[S_{0}]) = \frac{d}{dz} (I + HH^{∗})[S_{0}] =

\frac{d}{dz} (S_{0}) + \frac{d}{dz} (HH^{∗})[S_{0}] = M_{1}H^{∗}[S_{0}] + H^{∗}[S_{0}]M_{2}  (35)

Conversely, if S is a solution of (??), then S_{0} = (I − H)[S] is a constant series as

\frac{d}{dz} (S_{0}) = \frac{d}{dz} (I − H)[S] = \frac{d}{dz} S − \frac{d}{dz} (H)[S] = M_{1}S + SM_{2} − (M_{1}S + SM_{2}) = 0 .

Moreover S = (I − H)^{−1}[S_{0}] = H^{∗}[S_{0}] and then S_{0} = S(α). ii) If two solutions S_{1}, S_{2} coincide at z_{1} ∈ V, one can construct the operator H with formula (??) and α = z_{1}. One has then S_{1} = H^{∗}(S_{1}(z_{1})) = H^{∗}(S_{2}(z_{1})) = S_{2}. iii) A comultiplication with constant coefficients permute with the derivation operator hence, if S is a solution of (??), one has

\frac{d}{dz} (Δ(S)) = Δ(\frac{d}{dz} (S)) = Δ(M_{1}S + SM_{2}) =

Δ(M_{1})Δ(S) + Δ(S)Δ(M_{2})  (36)

this proves that Δ(S) satifies a two sided differential equation. On the other hand, S ⊗ S satisfies

\frac{d}{dz} (S ⊗ S) = \frac{d}{dz} (S) ⊗ S + S ⊗ \frac{d}{dz} (S) =

(M_{1}S + SM_{2}) ⊗ S + S ⊗ (M_{1}S + SM_{2}) =

(M_{1} ⊗ 1 + 1 ⊗ M_{1})(S ⊗ S) + (S ⊗ S)(M_{1} ⊗ 1 + 1 ⊗ M_{1})  (37)

which proves that, if M_{1} and M_{2} are primitive, Δ(S) and S⊗S satisfy the same differential equation. By virtue of (ii), if S is group-like at a point, it is so everywhere. In particular (and this will be used in (v)), if S is an invertible solution of an equation of type (??), with M_{1} and M_{2} primitive, then, for z_{0} ∈ V, S(z)S(z_{0})^{−1} is group-like.

iv) Indeed

⟨S|1_{X^{∗}}⟩ = ⟨H^{∗}[S_{0}]|1_{X^{∗}}⟩ = ⟨(I + HH^{∗})[S_{0}]|1_{X^{∗}}⟩ =
⟨S_{0}|1_{X^{∗}}⟩ + ⟨HH^{∗}[S_{0}]|1_{X^{∗}}⟩ = ⟨S_{0}|1_{X^{∗}}⟩ .  (38)

v) For z_{0} ∈ V, one defines

R(z, z_{0}) = S(z)S(z_{0})^{−1}G(z_{0})G(z)^{−1} = S(z)(G(z_{0})^{−1}S(z_{0}))^{−1}G(z)^{−1} .

R = R(z, z_{0}) is the product of two group-like series (S(z)S(z_{0})^{−1} and G(z_{0})G(z)^{−1}). Thus, z being fixed, one has

(S(z) ⊗ S(z))(G(z)^{−1} ⊗ G(z)^{−1}) = (S(z)G(z)^{−1}) ⊗ (S(z)G(z)^{−1}) =
lim_{z_{0}:F}R(z, z_{0}) ⊗ R(z, z_{0}) = lim_{z_{0}:F}Δ(R(z, z_{0})) =

Δ(lim_{z_{0}:F}(R(z, z_{0})) = Δ(S(z)G(z)^{−1}) = Δ(S(z))Δ(G(z)^{−1}) =
Δ(S(z))(G(z)^{−1} ⊗ G(z)^{−1})  (39)

and, finally Δ(S) = S ⊗ S. □

Remark 4.2 The proof of the theorem provides an integrator

H(G) = ∑_{w∈M} ( ∑_{uv=w} ∫_{z_{0}}^{z} (⟨M_{1}|u⟩(s)⟨G|v⟩(s) + ⟨G|u⟩(s)⟨M_{2}|v⟩(s))ds)w

but any similar operator H such that \frac{d}{dz}(H(G)) = M_{1}G+GM_{2} would do. In particular, one can construct operators with varied lower integration bounds. For example, the operator

H(G) = ∑_{w∈M} ( ∑_{uv=w} ( ∫_{a(u)}^{z}⟨M_{1}|u⟩(s)⟨G|v⟩(s) + ∫_{b(v)}^{z}⟨G|u⟩(s)⟨M_{2}|v⟩(s))ds)w (40)

is fairly admissible. We will see in Paragraph (??) an application of such a principle.

5 Coordinates of group-like elements.

5.1 Through the looking glass: passing from right to left.

Let S ∈ H⟨⟨X⟩⟩, we call F(S) the ℂ-vector space generated by the coefficients of S, one has

F(S) = {⟨S|P ⟩}_{P ∈ℂ⟨X⟩} .  (41)

We will use the following increasing filtrations

F_{≤α}(S) = {⟨S|P ⟩}_{P ∈ℂ_{≤α}⟨X⟩} .  (42)

or

F_{≤n}(S) = {⟨S|P ⟩}_{P ∈ℂ_{≤n}⟨X⟩} .  (43)

Proposition 5.1 We have the following properties :

i) If T ∈ ℂ⟨⟨X⟩⟩ then F(ST) ⊂ F(S) and one has equality if T is invertible. ii) If S is group-like, then F(S) is a unital sub-algebra of H, which is filtered w.r.t. (??) and (??) i.e.

F_{≤α}(S)F_{≤β}(S) ⊂ F_{≤α+β}(S)  (44)

Proof — (i) The space F(ST) is spanned by the

⟨ST |w⟩ = ∑_{uv=w}⟨S|u⟩⟨T |v⟩ ∈ F(S)

if T is invertible one has F(S) = F(ST T^{−1}) ⊂ F(ST) which proves the equality. ii) If S is group-like, one has

⟨S|u⟩⟨S|v⟩ = ⟨S ⊗ S|u ⊗ v⟩ = ⟨Δ(S)|u ⊗ v⟩ = ⟨S|u v⟩  (45)

In the case when all functions ⟨S|w⟩ are ℂ-linearly independant, one has a correspondence between the Differential Galois group of a differential equation of type (??) (acting on the right) and the group of automorphisms of F(S) compatible with the preceding filtration (they turn out to be unipotent).

Proposition 5.2 Let S be a group-like series. The following conditions are equivalent:

i) For every x ∈ X, ker_{ℂ}(S) ⊂ ker_{ℂ}(Sx). ii) For every x ∈ X, there is a derivation δ_{x} ∈ 𝔇𝔢𝔯(F(S)) such that

δ_{x}(S) = Sx  (46)

iii) For every x ∈ X, there is a one-paramater group of automorphisms ϕ_{x}^{t} ∈ Aut(F(S)); t ∈ ℝ such that

ϕ_{x}^{t}(S) = Se^{tx}  (47)

iv) For every C ∈ Lie_{C}⟨⟨X⟩⟩, there is δ ∈ 𝔇𝔢𝔯(F(S)) such that

δ(S) = SC  (48)

v) For every C ∈ Lie_{C}⟨⟨X⟩⟩, there is ϕ ∈ Aut(F(S)) such that

ϕ(S) = Se^{C}  (49)

vi) The functions (⟨S|w⟩)_{w∈X^{∗}} are ℂ-linearly independant.

Proof — i) ⟹ ii) From the inclusion, we derive that, for all x ∈ X there exists a ℂ-linear mapping ϕ ∈ End(F(S)) such that for all w ∈ M, ϕ(⟨S|w⟩) = ⟨Sx|w⟩. It must be a derivation of F(S) as

ϕ(⟨S|u⟩⟨S|v⟩) = ϕ(⟨S|u v⟩) = ⟨Sx|u v⟩ = ⟨S|(u v)x^{−1}⟩ =
⟨S|(ux^{−1} v) + (u vx^{−1})⟩ = ⟨S|(ux^{−1} v)⟩⟨S|(u vx^{−1})⟩ =
⟨Sx|u⟩⟨S|v⟩ + ⟨S|u⟩⟨Sx|v⟩ = ϕ(⟨S|u⟩)⟨S|v⟩ + ⟨S|u⟩ϕ(⟨S|v⟩)  (50)

from the fact that (⟨S|w⟩)_{w∈X^{∗}} spans F(S). ii) ⟹ iv) As (⟨S|w⟩)_{w∈X^{∗}} spans F(S), the derivation ϕ is uniquely defined. Let us note it δ_{x} and notice that, doing so, we have constructed a mapping Φ : X → 𝔇𝔢𝔯(F(S)) (which is Lie algebra. Therefore, there is a unique extension of this mapping as a morphism Lie_{ℂ}⟨X⟩ → 𝔇𝔢𝔯(F(S)). This correspondence, which we will note P → δ(P) is (uniquely) recursively defined by

δ(x) = δ_{x} ; δ([P, Q]) = [δ(P), δ(Q)] .  (51)

For C = ∑_{n≥0} C_{n} ∈ Lie_{ℂ}⟨⟨X⟩⟩ with C_{n} ∈ Lie_{ℂ}⟨X⟩_{n}, we remark that the sequence ⟨S ∑_{0≤n≤N} C_{n}|w⟩ is stable (for large N). Set δ_{≤N} := δ(∑_{0≤n≤N} C_{n}). We see that δ_{≤N} is stable (for large N) on every F_{α} and we note δ(C) its limit. It is clear that this limit is a derivation and that it corresponds to C. iv) ⟹ v) For evey C = ∑_{n≥0} C_{n} ∈ Lie_{ℂ}⟨⟨X⟩⟩, the exponential e^{C} defines a mapping ϕ ∈ End(F(S)) as indeed e^{δ_{≤N}} is stationnary. It is easily checked that this mapping is an automorphism of algebra of F(S).

v) ⟹ iii) For C_{i} ∈ Lie_{ℂ}⟨⟨X⟩⟩; ı = 1, 2 which commute we have

Se^{C_{1}}e^{C_{2}} = ϕ_{C_{1}}(S)e^{C_{2}} = ϕ_{C_{1}}(Se^{C_{2}}) = ϕ_{C_{1}}ϕ_{C_{2}}(S)  (52)

this proves the existence, for a C ∈ Lie_{ℂ}⟨⟨X⟩⟩ of a one-parameter (rational) group ϕ_{C}^{t} in Aut(F(S)) such that Se^{tC} = ϕ_{C}^{t}(S). This one-parameter (rational) group can be extended to ℝ as continuity is easily checked by taking the scalar products ⟨ϕ_{C}^{t}(S)|w⟩ = ⟨Se^{tC}|w⟩ and it suffices to specialize the result to C = x. iii) ⟹ ii) By stationary limits one has

⟨Sx|w⟩ = lim_{t→0} \frac{1}{t}(⟨Se^{tx}|w⟩ − ⟨S|w⟩) = lim_{t→0} \frac{1}{t}(⟨ϕ_{x}^{t}(S)|w⟩ − ⟨S|w⟩)  (53)

v) ⟹ i) Let x ∈ X, t ∈ ℝ, we take C = tx and ϕ_{t} ∈ Aut(F(S)) s.t. ϕ_{t}(S) = Se^{tx}. It there is P ∈ ℂ⟨X⟩ such that ⟨S|P ⟩ = 0 one has

0 = ⟨S|P ⟩ = ϕ_{t}(⟨S|P ⟩) = ⟨ϕ_{t}(S)|P ⟩ = ⟨Se^{tx}|P ⟩ = ∑_{n=0}^{deg(P)} \frac{t^{n}}{n!}⟨Sx^{n}|P ⟩  (54)

and then, for all z ∈ V, the polynomial

∑_{n=0}^{deg(P)} \frac{t^{n}}{n!}⟨S(z)x^{n}|P ⟩  (55)

is identically zero over ℝ hence so are all of its coeefficients in particular ⟨S(z)x|P ⟩ for all z ∈ V. This proves the claim.

i) ⟹ vi) Let P ∈ ker_{ℂ}(S) if P ≠ 0 take it of minimal degree with this property. For all x ∈ X, one has P ∈ ker_{ℂ}(Sx) which means ⟨Sx|P ⟩ = 0 and then P x^{†} = 0 as deg(P x^{†}) = deg(P) − 1. The reconstruction lemma implies that

P = ⟨P |1⟩ + ∑_{x∈X}(P x^{†})x = ⟨P |1⟩  (56)

Then, one has 0 = ⟨S|P ⟩ = ⟨S|1⟩⟨P |1⟩ = ⟨P |1⟩ which shows that ker_{ℂ}(S) = {0}. This is equivalent to the statement (vi). vi) ⟹ i) Is obvious as ker_{ℂ}(S) = {0}. □

It is possible to enlarge somehow the range of proposition (??) to coefficients that are analytic functions f : dom(f) → ℂ.

Definition 5.3 We call here differential field of germs w.r.t. a filter basis 𝔹 of open connected subsets of V, a map C defined on 𝔹 such that for every U ∈ 𝔹, C[U] is a subring of C^{ω}(U, ℂ) and

1. C is compatible with restrictions i.e. if U, V ∈ 𝔹 and V ⊂ U, one has

res_{V U}(C[U]) ⊂ C[V]

2. if f ∈ C[U] \ {0} then there exists V ∈ 𝔹 s.t. V ⊂ U − O_{f} and f^{−1} (defined on V) is in C[V] .

For any U ∈ 𝔹, we note C[U] the ring of functions in C defined on U and restricted to this set.

There are important cases when the conditions (??) are satified as shows the following theorem.

Theorem 5.4 Let V be a simply connected non-void open subset of ℂ − {a_{0}, ⋯ a_{n}} (\{a_{0}, ⋯ a_{n}\} are distinct points), M = ∑_{i=0}^{n} \frac{λ_{i}x_{i}}{z−a_{i}} be a multiplier on X = \{x_{0}, ⋯ x_{n}\} with all λ_{i} ≠ 0 and S be any regular solution of

\frac{d}{dz} S = MS .  (57)

Then, let C be a differential field of functions defined on V which do not contain linear combinations of logarithms on any domain but which contains z and the constants (as, for example the rational functions). If U is a non-void domain of C and P ∈ C[U]⟨X⟩, one has

⟨S|P ⟩ = 0 ⟹ P = 0  (58)

Proof — Let U ∈ 𝔹. For every non-zero Q ∈ C[U]⟨X⟩, we note lead(Q) the greatest word in the support of Q for the graded lexicographic ordering ≺ (we have endowed X with any linear ordering) and call Q monic if the leading coefficient ⟨Q|lead(Q)⟩ is 1. A monic polynomial then reads

Q = w + ∑_{u≺w}⟨Q|u⟩u .  (59)

Suppose now that it is possible to find U and P ∈ C[U]⟨X⟩ (not necessarily monic) such that ⟨S|P ⟩ = 0, we choose P with lead(P) minimal for ≺.

Then

P = f(z)w + ∑_{u≺w}⟨P |u⟩u  (60)

with f ≢ 0. Thus U_{1} = U \ O_{f} ∈ 𝔹 and Q = \frac{1}{f(z)}P ∈ C[U_{1}]⟨X⟩ is monic and satisfies

⟨S|Q⟩ = 0 .  (61)

Differentiating eq. (??), we get

0 = ⟨S′|Q⟩ + ⟨S|Q′⟩ = ⟨MS|Q⟩ + ⟨S|Q′⟩ = ⟨S|Q′ + M^{†}Q⟩ .  (62)

Remark that one has

Q′ + M^{†}Q ∈ C[U_{1}]⟨X⟩  (63)

If Q′+M^{†}Q ≠ 0, one has lead(Q′+M^{†}Q) ≺ lead(Q) and this is not possible because of the minimality hypothesis of lead(Q) = lead(P). Hence, one must have R = Q′ + M^{†}Q = 0. With |w| = n, write now

Q = Q_{n} + ∑_{|u|<n}⟨Q|u⟩u .  (64)

where Q_{n} = ∑_{|u|=n}⟨Q|u⟩u is the dominant homogeneous component of Q. For every |u| = n we have

(⟨Q|u⟩)′ = −⟨M^{†}Q|u⟩ = −⟨Q|Mu⟩ = 0  (65)

thus all the coefficients of Q_{n} are constant.

If n = 0, Q ≠ 0 is constant which is impossible by eq. (??) and because S is regular. If n > 0, for any word |v| = n − 1, we have

(⟨Q|v⟩)′ = −⟨M^{†}Q|v⟩ = −⟨Q|Mv⟩ = − ∑_{i=0}^{n} \frac{λ_{i}}{z − a_{i}} ⟨Q|x_{i}v⟩ = − ∑_{i=0}^{n} \frac{λ_{i}}{z − a_{i}} ⟨Q_{n}|x_{i}v⟩ (66)

Because all x_{i}v are of length n. Then

⟨Q|v⟩ = − ∑_{i=0}^{n}⟨Q_{n}|x_{i}v⟩ ∫_{α}^{z} \frac{λ_{i}}{s − a_{i}} ds + const  (67)

But all the functions ∫_{α}^{z} \frac{λ_{i}}{s−a_{i}} ds are linearly independant over ℂ and not all the scalars ⟨Q_{n}|x_{i}v⟩ are zero (write w = x_{k}v and choose v such). This contradicts the fact that Q ∈ C[U_{1}]⟨X⟩ as C does contain no linear combination of logarithms. □

Remark 5.5 i) If a series satifies the equivalent conditions of the theorem (??), then every Se^{C} so does. ii) Series as the one of polylogarithms and all the exponential solutions of equation

\frac{d}{dz} (S) = (\frac{x_{0}}{z} + \frac{x_{1}}{1 − z})S  (68)

satisfy conditions of the theorem (??) as shows theorem (??). iii) One could ask oneself what happens when these conditions are not satisfied. In fact the set of Lie series C ∈ Lie_{ℂ}⟨⟨𝕏⟩⟩ such that it exists a ϕ ∈ End(F(S)) (then a derivation) s.t. SC = ϕ(S) is a closed Lie subalgebra of Lie_{ℂ}⟨⟨𝕏⟩⟩ which we will note Lie_{S}. For example

• for X = {x_{0}, x_{1}} and S = e^{zx_{0}} one has x_{0} ∈ Lie_{S} ; x_{1} ∉ Lie_{S}

• for X = {x_{0}, x_{1}} and S = e^{z(x_{0}+x_{1})}, one has x_{0}, x_{1} ∉ Lie_{S} but (x_{0} + x_{1}) ∈ Lie_{S}.

5.2 Polylogarithms and related functions

Here X is still the finite alphabet {x_{0}, x_{1}} equipped with the order x_{0} < x_{1} and let C be the ring ℂ[z, z^{−1}, (1 − z)^{−1}].

The iterated integral over ω_{0}, ω_{1} associated to w = x_{i_{1}} ⋯ x_{i_{k}} over X and along the integration path z_{0} ⇝ z is the following multiple integral defined by

∫_{z_{0}⇝z} ω_{i_{1}} ⋯ ω_{i_{k}} = ∫_{z_{0}}^{z} ω_{i_{1}}(t_{1}) ∫_{z_{0}}^{t_{1}} ω_{i_{2}}(t_{2}) … ∫_{z_{0}}^{t_{r−2}} ω_{i_{r}}(t_{r−1}) ∫_{z_{0}}^{t_{r−1}} ω_{i_{r}}(t_{r}),  (69)

where t_{1} ⋯ t_{r−1} is a subdivision of the path z_{0} ⇝ z. In a shortened notation, we denote this integral by α_{z_{0}}^{z}(w) and^{1} α_{z_{0}}^{z}(1_{X^{∗}}) = 1. One can check that the polylogarithm Li_{s_{1},...,s_{r}} is also the value of the iterated integral over ω_{0}, ω_{1} and along the integration path 0 ⇝ z [?, ?] :

Li_{w}(z) = α_{0}^{z}(x_{0}^{s_{1}−1}x_{1} … x_{0}^{s_{r}−1}x_{1}).  (70)

1Here, 1_{X^{∗}} stands for the empty word over X.

The definition of polylogarithms is extended over the words w ∈ X^{∗} by putting Li_{x_{0}}(z) := log z. The {Li_{w}}_{w∈X^{∗}} are C-linearly independent [?, ?]. In order to, define L = ∑_{w∈X^{∗}} Li_{w}w, one also can use an integrator with variable lower integration bounds as one described by (??) with M_{2} = 0, a(u) = 1 for u ∈ x_{0}^{∗} and a(u) = 0 for u ∈ X^{∗}x_{1}X^{∗}. Indeed, L is group-like but, to show this one cannot use Thm ?? (iii) because the lower bounds of the integrals are different. So one first shows that

lim_{z→0} exp(−x_{0} log z)L(z) = lim_{z→0}L(z) exp(−x_{0} log z) = 1  (71)

which can be done as follows. One first remarks that, in case w contains at least ons x_{1} (i.e. |w|_{x_{1}} ≥ 1) and for every k

lim_{z→0} log(z)^{k}⟨L(z)|w⟩ = 0  (72)

then, setting L^{+}(z) = ∑_{|w|_{x_{1}}≥1}⟨L(z)|w⟩w, one has

lim_{z→0} exp(−x_{0} log z)L^{+}(z) = L^{+}(z) exp(−x_{0} log z) = 0  (73)

and as

L(z) = L^{+}(z) + ∑_{w∈(x_{0})^{∗}}⟨L(z)|w⟩w = L^{+}(z) + exp(x_{0} log z)  (74)

the result follows.

The following functions

∀w ∈ X^{∗}, P_{w}(z) = (1 − z)^{−1}Li_{w}(z),  (75)

are also C-linearly independent, as C is an integral domain, by the following lemma easy to check

Lemma 5.6 Let A be an integral domain and M an A-module. If (x_{i})_{i∈I} is a linearly independant family and b ≠ 0 in A, then (bx_{i})_{i∈I} is linearly independant.

Since, for any w ∈ Y^{∗}, P_{w} is the ordinary generating function of the sequence {H_{w}(N)}_{N≥0} :

P_{w}(z) = ∑_{N≥0} H_{w}(N) z^{N}  (76)

then, as a consequence of the classical isomorphism between convergent Taylor series and their associated sums, the harmonic sums {H_{w}}_{w∈Y^{∗}} are ℂ-linearly independent. Firstly, ker P = {0} and ker H = {0}, and secondly, P is a morphism transporting the stuffle to the Hadamard product :

P_{u}(z) ⊙ P_{v}(z) = ∑_{N≥0} H_{u}(N)H_{v}(N)z^{N} = ∑_{N≥0} H_{u v}(N)z^{N} = P_{u v}(z).  (77)

6 Conclusion

To sum up what has been done in this paper? we can state that the deformed algebra LDIAG(q_{c}, q_{s}), which originates from a special quantum field theory [?], is free and its law can be constructed from very general procedures: it is a shifted twisted law. Before shifting, one can observe that the law is, in fact, dual to a comultiplication on a free algebra. This comultiplication is a perturbation, with q_{s} (the superposition parameter) of the shuffle comultiplication on this free algebra. The parameter q_{s} is obtained by addition of a perturbating factor which is just dual to a (diagonally) deformed law of a semigroup whereas the crossing parameter q_{c} is obtained by extending to the tensor structure (i.e. to words) a colour factor of an algebra.

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