Formulas for the Connes-Moscovici Hopf algebra

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Formulas for the Connes-Moscovici Hopf algebra

Frederic Menous

To cite this version:

Frederic Menous. Formulas for the Connes-Moscovici Hopf algebra. 2008. �hal-00347193�

Formulas for the Connes-Moscovici Hopf algebra

Fr´ed´eric Menous

Abstract

We give explicit formulas for the coproduct and the antipode in the Connes-Moscovici Hopf algebraHCM. To do so, we first restrict ourselves to a sub-Hopf algebra H¹_CM containing the nontrivial elements, namely those for which the coproduct and the antipode are nontrivial. There are two ways to obtain explicit formulas. On one hand, the algebraH¹_CM is isomorphic to the Fa`a di Bruno Hopf algebra of coordinates on the group of identity-tangent diffeomorphism and computations become easy using substitution automorphisms rather than diffeomorphisms. On the other hand, the algebraH¹_CMis isomorphic to a sub-Hopf algebra of the classical shuffle Hopf algebra which appears naturally in resummation theory, in the framework of formal and analytic conjugacy of vector fields. Using the very simple structure of the shuffle Hopf algebra, we derive once again explicit formulas for the coproduct and the antipode inH¹_CM.

1 Introduction.

The Connes-Moscovici Hopf algebraHCM was introduced in [5] in the context of noncommutative geometry. Because of its relation with the Lie algebra of formal vector fields, it was also proved in [5] that its subalgebra H¹_CM is iso- morphic to the Fa`a di Bruno Hopf algebra of coordinates of identity-tangent diffeomorphisms (see [5],[10]). In the past years, it appeared that this Hopf algebra was strongly related to the Hopf algebras of trees (see [2]) or graphs (see [3],[4]) underlying perturbative renormalization in quantum field theory.

Our aim is to give explicit formulas for the coproduct and the antipode in H¹_CM, since only recursive formulas seem to be known.

We remind in section 2 the definition of the Connes-Moscovici Hopf alge- bra, as well as its properties and links with the Fa`a di Bruno Hopf algebra and identity-tangent diffeomorphisms (for details, see [5],[10]). The formulas are given in section 3. We present a proof based on the isomorphism between identity-tangent diffeomorphisms and substitution automorphisms which are easier to handle in the computations. These manipulations on substitution automorphisms are very common in J. Ecalle’s work on the formal classification of differential equations, vector fields, diffeomorphism... (see [6],[7],[8],[9]). In fact, the first proof for these formulas was based on mould calculus and shuffle Hopf algebras, which we shortly describe in section 4. Sections 5 and 6 give the

outlines of the initial proof based on a Hopf morphism from H¹ ⊂ HCM in a shuffle Hopf algebra.

Contents

1 Introduction. 1

2 Connes-Moscovici and Fa`a di Bruno Hopf algebras. 3

2.1 The Connes-Moscovici Hopf algebra . . . 3

2.2 The Fa`a di Bruno Hopf algebra . . . 3

2.3 Connes-Moscovici coordinates . . . 4

3 Formulas inH¹_CM. 4 3.1 Notations . . . 4

3.2 Main formulas . . . 5

3.3 Coordinates on G2 . . . 6

3.4 Taylor expansions and substitution automorphisms . . . 7

3.5 Formulas inHFdB . . . 9

3.6 Proof of Theorems 2 and 3 . . . 10

4 Mould calculus and the shuffle Hopf algebra sh(^N∗). 12 4.1 An example of mould calculus . . . 12

4.1.1 Formal Conjugacy of equations . . . 12

4.1.2 Diffeomorphisms an substitution automorphisms . . . 13

4.1.3 Symmetral moulds and shuffle Hopf algebra . . . 14

4.2 The free group and its Hopf algebra of coordinates . . . 15

4.2.1 Lie algebra and substitution automorphisms . . . 15

4.2.2 The free group and its Hopf algebra of coordinates . . . . 15

5 Morphisms. 16 6 Initial Proofs. 18 6.1 Proof of theorem 5 . . . 18

6.2 Proof of theorem 2 . . . 20

7 Tables and conclusion. 23 7.1 The coproduct . . . 23

7.2 The antipode . . . 24

2 Connes-Moscovici and Fa` a di Bruno Hopf al- gebras.

2.1 The Connes-Moscovici Hopf algebra

The Connes-Moscovici Hopf algebraHCMdefined in [5] is the enveloping algebra of the Lie algebra which is the linear span ofY,X,δn,n≥1 with the relations, [X, Y] =X,[Y, δn] =nδn,[δn, δm] = 0,[X, δn] =δn+1 (1) for allm, n≥1. The coproduct ∆ inHCM is defined by

∆(Y) =Y⊗1+1⊗Y,∆(X) =X⊗1+1⊗X+δ1⊗Y,∆(δ1) =δ1⊗1+1⊗δ1 (2) where ∆(δn) is defined recursively, using equation 1 and the identity

∀h1, h2∈ HCM, ∆(h1h2) = ∆(h1)∆(h2) (3) The coproduct ofX and Y is given, whereas the coproduct of δn is nontriv- ial. Nonetheless, the algebra generated by{δn, n≥1} is a graded sub-Hopf algebraH¹_CM⊂ HCM where the graduation is defined by

gr(δn1. . . δn_s) =n1+. . .+ns (4) As mentioned in [5], the Hopf algebraH¹_CM is strongly linked to Fa`a di Bruno Hopf algebra.

2.2 The Fa` a di Bruno Hopf algebra

Let us consider the group of formal identity tangent diffeomorphisms : G2={f(x) =x+X

n≥1

fnxⁿ⁺¹∈^R[[x]]}

with, by convention, the productµ:G2×G2→G2 : µ(f, g) =g◦f Forn≥0, the functionals onG2 defined by

an(f) = 1

(n+ 1)!(∂xⁿ⁺¹f)(0) =fn an:G2→^R

are called de Fa`a di Bruno coordinates on the groupG2 and a0= 1 being the unit, they generates a graded unital commutative algebra

HFdB=^R[a1, . . . , an, . . .] (gr(an) =n)

Moreover, the action of these functionals on a product inG2defines a coproduct onHFdBthat turns to be a graded connected Hopf algebra (see [10] for details).

Forn≥0, the coproduct is defined by

an◦µ=m◦∆(an) (5)

wheremis the usual multiplication in^R, and the antipode reads S◦an=an◦rec

where rec(ϕ) =ϕ⁻¹ is the composition inverse ofϕ.

For example iff(x) =x+P

n≥1fnxⁿ⁺¹ andg(x) =x+P

n≥1gnxⁿ⁺¹ then ifh=µ(f, g) =g◦f and h(x) =x+P

n≥1hnxⁿ⁺¹,

a0(h) = 1 =a0(f)a0(g) → ∆a0 = a0⊗a0

a1(h) = f1+h1 → ∆a1 = a1⊗a0+a0⊗a1

a2(h) = f2+f1g1+g2 → ∆a2 = a2⊗a0+a1⊗a1+a0⊗a2

As proved in [5] and [10], there exists a Hopf isomorphism between HFdB and H¹_CM.

2.3 Connes-Moscovici coordinates

Following [5], one can define new functionals onG2 byγ0 = a0=1 (unit) and forn≥1,

γn(f) = (∂_xⁿlog(f^′))(0)

These functionals, which may be called the Connes-Moscovici coordinates on G2, freely generates the Fa`a di Bruno Hopf algebra :

HFdB=^R[a1, . . . , an, . . .] =^R[γ1, . . . , γn, . . .] gr(an) = gr(γn) =n and their coproduct is given by the formula 5. Now, see [5], [2] :

Theorem 1 The mapΘ defined byΘ(δn) =γn is a graded Hopf isomorphism betweenHFdB andH¹_CM

This means that the coproduct and the antipode inH_CM¹ can be rather com- puted inHFdB. Unfortunately, if the coproduct and the antipode is well-known for the functionalsan, using the Fa`a di Bruno formulas for the composition and the inverse of diffeomorphisms inG2, it seems that formulas for theγn cannot be easily derived. In order to do so, we will either work with substitution au- tomorphism which are easier to handle than diffeomorphisms (see section 3, or identify HFdB as a sub-Hopf algebra of a shuffle Hopf algebra and use mould calculus (see sections 4, 5, 6).

3 Formulas in H

.

3.1 Notations

In the sequel we note

N ={n= (n1, . . . , ns)∈(^N∗)^s, s≥1}

Forn= (n1, . . . , ns)∈ N,

knk=n1+. . .+ns, l(n) =s and ifn≥1,

Nn={n∈ N ; knk=n}

For a tuplen= (n1, . . . , ns)∈ N, we noten! =n1!. . . ns!. More over, Split(n) is the subset of S

t≥1N^t such that (n¹, . . . ,n^t) ∈ Split(n) if and only if the concatenation of (n¹, . . . ,n^t) is equal ton:

Split(n) ={(n¹, . . . ,n^t)∈ N^t, n¹. . .n^t=n} (6) In summation formulas, we will use the fact that

[

n∈Nn

Split(n) = [

n=(n1,...,n_s)∈Nn

Nn1×. . .× Nns (7) so that iff is a function onN andg is a function onS

t≥1N^t, forn≥1, X

n=(n1,...,n_s)∈Nn

X

m¹∈ Nn1

.. . m^s∈ Nns

f(n)g(m¹, . . . ,m^s) =

X

n∈Nn

X

m¹...m^s=n

f(km¹k, . . . ,km^sk)g(m¹, . . . ,m^s) (8)

where X

m¹...m^s=n

is the sum over Split(n).

Finally, for (n¹, . . . ,n^t)∈ N^t (t≥1), A(n¹, . . . ,n^t) = 1

l(n¹)!. . . l(n^t)!

Yt i=1

1

knⁱk+ 1 (9) and, fork≥1,

Bk(n¹, . . . ,n^t) =C_k^l(nt)

t−1Y

i=1

C_kn^l(ni+1ⁱ⁾k+...+kn^tk+k (10)

3.2 Main formulas

We will now prove the following formulas : Theorem 2 Forn≥1,

∆(δn) = δn⊗1 + 1⊗δn

+ X

(n1, . . . , n_s+1)∈ Nn

s≥1

n!

n1!. . . ns+1!αⁿ_n¹_s+1^,...,nsδn1. . . δn_s⊗δn_s+1

(11)

and, forn= (n1, . . . , ns)∈ N (l(n) =s) andm≥1, αⁿ_m=

l(n)X

t=1

C_m^t X

n¹. . .n^t=n

A(n¹, . . . ,n^t) (12)

where, for n= (n1, . . . , ns)∈ N, l(n) = 1, knk =n1+. . .+ns and with the conventionCm^t = m!

t!(m−t)! = 0if t > m.

For the antipode S: Theorem 3 Forn≥1,

S(δn) = X

n= (n1, . . . , n_s)∈ N n1+. . .+n_s=n

n!

n1!. . . ns!β^n1,...,nsδn1. . . δn_s (13)

withβⁿ¹=−1 and, ifn= (n1, . . . , ns+1)∈ N (s≥1), β^{n1,...,ns,ns+1}=

Xs t=1

X

n¹. . .n^t=n

Un^kn_s+1^1k,...,kntkA(n¹, . . . ,n^t) (14)

where, ifm= (m1, . . . , mt)∈ N/{∅}andk≥1, U_k^m=

l(m)X

i=1

(−1)ⁱ⁻¹ X

m¹. . .mⁱ=m

Bk(m¹, . . . ,mⁱ) (15)

We will now give the more recent proof of this formulas. These formulas were first conjectured and then proved using a Hopf morphism betweenH¹_CM and a shuffle Hopf algebra noted sh(^N∗). We will come back later on this morphism and the afferent proofs. Let us first look at the correspondence between FdB coordinates and the CM coordinates onG2.

3.3 Coordinates on G

Letϕ(x) =x+X

n≥1

ϕnxⁿ⁺¹. We have forn≥1 :

an(ϕ) =ϕn, γn(ϕ) = (∂_xⁿlog(ϕ^′))(0) =fn (16) Iff(x) =X

n≥1

fn

n!xⁿ, then

f(x) = log(ϕ^′(x)) ϕ(x) = Z x

0

e^f(t)dt (17)

For any sequence (un)n≥1, we note

∀n= (n1, . . . , ns)∈ N, un=un1. . . uns (18) Using equation 17, we get easily that

f(x) = X

n=(n1,...,n_s)∈N

(−1)^l(n)

l(n) (n1+ 1). . .(ns+ 1)ϕnx^knk

ϕ(x) = x+ X

n=(n1,...,n_s)∈N

1 l(n)!n!

fn

knk+ 1x^knk+1

(19)

and these formulas establish the correspondence between FdB and CM coordi- nates on G2. In order to prove theorems 2 and 3, we need to understand how these coordinates read onϕ⁻¹ and µ(ϕ, ψ) = ψ◦ϕ(ϕ, ψ∈G2). To do so, we will rather work with substitution automorphisms than with diffeomorphism.

3.4 Taylor expansions and substitution automorphisms

Definition 1 Let G˜2 be the set of linear maps from ^R[[x]] to^R[[x]] such that 1. For F ∈G˜2, the image F(x)byF of the series xis inG2.

2. For any two series AandB in^R[[x]], we have

F(A.B) =F(A).F(B) (20)

The elements of ˜G2are called substitution automorphisms and Theorem 4 G˜2 is a group for the composition and the map :

τ : G˜2 → G2

F 7→ ϕ(x) =F(x)

defines an isomorphism between the groups G˜2 and G2. Moreover, for A ∈

R[[x]],

F(A) =A◦τ(F) (21)

Proof IfF∈Ge₂, then, thanks to equation 20, fork≥0,

F(x^k) = (F(x))^k = (τ(F)(x))^k = (ϕ(x))^k (22) thus, forA(x) =P

k≥0Akx^k∈^R[[x]], F(A)(x) = FP

k≥0Akx^k

= P

k≥0AkF(x^k)

= P

k≥0Ak(ϕ(x))^k

= A◦(τ(F))(x)

(23)

This proves thatτ is injective and for anyϕ∈G2 the map F : ^R[[x]] → ^R[[x]]

A 7→ A◦ϕ

is a substitution automorphism of ˜G2 such that τ(F) = ϕ. The map τ is a bijection. Now, forF andGin ˜G2,

τ(F◦G)(x) =F(G(x)) =τ(G)◦τ(F)(x) =µ(τ(F), τ(G))(x) (24) and ifH =τ⁻¹((τ(F))⁻¹) thenF◦H =H◦F = Id. This ends the proof.

Using Taylor expansion, we also get formulas forτ⁻¹(ϕ),ϕ∈G2,

Proposition 1 Letϕ(x) =x+X

n≥1

ϕnxⁿ⁺¹∈G2 andF =τ⁻¹(ϕ), then

F = Id + X

n=(n1,...,ns)∈N

1

l(n)!ϕnx^knk+l(n)∂_x^l(n) (25) This also means thatF can be decomposed in homogeneous components :

F = Id +X

n≥1

Fn , Fn= X

n=(n1,...,n_s)∈Nn

1

l(n)!ϕnx^knk+l(n)∂x^l(n) (26) such that

∀n≥1, ∀k≥1, ∃c∈^R, Fn(x^k) =cx^n+k (27) Proof Ifϕ(x) =x+X

n≥1

ϕnxⁿ⁺¹=x+ ¯ϕ(x)∈G2, then, ifF =τ⁻¹(ϕ), then forA∈^R[[x]],

F(A)(x) = A(x+ ¯ϕ(x))

= A(x) +X

s≥1

( ¯ϕ(x))^s

s! A^(s)(x)

= A(x) +X

s≥1

X

n1≥1,...,ns≥1

1

s!ϕn1. . . ϕn_sx^n1+...+ns+sA^(s)(x)

=



Id + X

n=(n1,...,n_s)∈N

1

l(n)!ϕnx^knk+l(n)∂x^l(n)



(A(x))

The automorphismF can be seen as a differential operator acting on^R[[x]]

and from now on we note multiplicatively the action of such operators :

F.ϕ=F(ϕ) (28)

As this will be of some use later, let us give the following formula : If n= (n1, . . . , ns)∈ N andk≥1,

Fn.x^k = Fn1. . . Fn_s.x^k

= X

mⁱ∈ N_ni 1≤i≤s

ϕm¹x^n1+l(m1) l(m¹)! ∂_x^l(m1)

! . . .

ϕm^sx^ns+l(ms) l(m^s)! ∂_x^l(ms)

.x^k

= X

mⁱ∈ N_ni 1≤i≤s

Bk(m¹, . . . ,m^s)ϕm¹. . . ϕm^sx^knk+k

(29) where

Bk(m¹, . . . ,m^s) =C_k^l(ms)

s−1Y

i=1

C_km^l(mi+1ⁱ⁾k+...+km^sk+k

With these results one can already derive formulas for the FdB coordinates on G2.

3.5 Formulas in H

We recover the usual formulas : Proposition 2 We have forn≥1,

∆(an) =an⊗1 + 1⊗an+

n−1X

k=1

X

n=(n1,...,n_s)∈Nk

C_n−k+1^l(n) an⊗an−k (30) and

S(an) = X

n∈Nn

X

m¹...m^s=n

(−1)^sB1(m¹, . . . ,m^s)

!

an (31)

Proof Letϕ(x) =x+X

n≥1

ϕnxⁿ⁺¹ and ψ(x) = x+X

n≥1

ψnxⁿ⁺¹ two elements ofG2 andη=µ(ϕ, ψ) =ψ◦ϕwith

η(x) =x+X

n≥1

ηnxⁿ⁺¹ (32)

IfF,Gand H are the substitution automorphisms corresponding toϕ, ψand η, thenH =F◦G:

H = Id +X

n≥1

Hn

=



Id +X

n≥1

Fn







Id +X

n≥1

Gn





= Id +X

n≥1

Xn k=0

FkGn−k (F0=G0= Id)

(33)

But forl≥1,Gl(x) =ψlx^l+1 and then, fork≥1,

FkGl.x = X

n=(n1,...,n_s)∈Nk

1

l(n)!ϕnx^knk+l(n)∂x^l(n)(ψlx^l+1)

= X

n=(n1,...,n_s)∈Nk

ψl

1 l(n)!ϕn

(l+ 1)!

(l+ 1−l(n))!x^knk+l+1

=



 X

n=(n1,...,n_s)∈Nk

C_l+1^l(n)ϕnψl



x^k+l+1

(34)

and then, forn≥1,

ηn=ϕn+ψn+

n−1X

k=1

X

n=(n1,...,ns)∈Nk

C_l+1^l(n)ϕnψn−k (35)

If now ˜ϕ=ϕ⁻¹and ˜F =τ⁻¹( ˜ϕ), then, as ˜F F = Id we get F˜ = Id +X

s≥1

(−1)^sFn1. . . Fn_s= Id + X

n∈N

(−1)^l(n)Fn (36) but forn= (n1, . . . , ns)∈ N,

Fn(x) = X

mⁱ∈ N_ni 1≤i≤s

B1(m¹, . . . ,m^s)ϕm¹. . . ϕm^sx^knk+1 (37)

Now

˜

ϕn= X

n= (n1, . . . , n_s)∈ Nn

(−1)^s X

mⁱ∈ N_ni 1≤i≤s

B1(m¹, . . . ,m^s)ϕm¹. . . ϕm^s (38)

and this gives the attempted result.

Using the same ideas, we will finally prove theorems 2 and 3

3.6 Proof of Theorems 2 and 3

As before, letϕ(x) =x+X

n≥1

ϕnxⁿ⁺¹ andψ(x) =x+X

n≥1

ψnxⁿ⁺¹ two elements ofG2 andη=µ(ϕ, ψ) =ψ◦ϕwith

η(x) =x+X

n≥1

ηnxⁿ⁺¹ (39)

If

f(x) = log(ϕ^′(x)) = X

n≥1

fn

n!xⁿ (fn =γn(ϕ)) g(x) = log(ψ^′(x)) = X

n≥1

gn

n!xⁿ (gn=γn(ψ)) h(x) = log(η^′(x)) = X

n≥1

hn

n!xⁿ (hn=γn(η))

(40)

then h(x) = log((ψ◦ϕ)^′(x))

= log(ϕ^′(x).ψ^′(ϕ(x))

= log(ϕ^′(x)) + (logψ^′)◦ϕ(x)

= f(x) +F(g)(x)

(41)

where F is the substitution automorphism associated toϕ. We remind that F = Id +P

n≥1Fn. Because of equation 19,

Fn = X

n=(n1,...,ns)∈Nn

1

l(n)!ϕnx^knk+l(n)∂_x^l(n)

= X

n=(n1,...,ns)∈Nn

1 l(n)!

X

mⁱ∈ N_ni 1≤i≤s

A(m¹, . . . ,m^s)fm¹. . . fm^s

m¹!. . .m^s! x^knk+l(n)∂_x^l(n)

= X

n∈Nn

fn

n! X

m¹...m^s=n

A(m¹, . . . ,m^s)1

s!x^n+s∂_x^s

(42) But fork≥1,

Fn g_k k!x^k

= X

n∈Nn

fngk

n!k!

X

m¹...m^s=n

A(m¹, . . . ,m^s)Ck^sx^n+k (43) and we obtain immediately the formula for the coproduct.

Let now ˜ϕ=ϕ⁻¹and

f˜(x) = log( ˜ϕ^′(x)) =X

n≥1

f˜n

n!xⁿ ( ˜fn =γn( ˜ϕ)) (44) Since ˜ϕ◦ϕ(x) =x,

0 = log(( ˜ϕ◦ϕ)^′(x)) =f(x) +F.f˜(x) (45) thus f˜(x) =−F .f˜ (x) =−f(x)− X

n∈N

(−1)^l(n)Fn(f)(x) (46)

But, once again, fn,k(x) = X

n∈Nn

(−1)^l(n)Fn(fk

k!x^k)

= X

n∈Nn

(−1)^l(n)fk

k!

X

mⁱ∈ N_ni 1≤i≤s

Bk(m¹, . . . ,m^s)ϕm¹. . . ϕm^sx^knk+k

= X

n∈Nn

X

m¹...m^s=n

(−1)^sBk(m¹, . . . ,m^s)ϕn

fk

k!x^knk+k

= − X

n∈Nn

Uk(n)ϕn

fk

k!x^knk+k

(47) Now, replacingϕn as in equation 42,

fn,k(x) = X

n∈Nn

(−1)^l(n)Fn(fk

k!x^k)

= − X

n∈Nn

Uk(n)ϕn

fk

k!x^knk+k

= − X

n∈Nn

fnfk

n!k!

X

m¹...m^s=n

A(m¹, . . . ,m^s)Uk(km¹k, . . . ,||m^sk)x^n+k (48) Now, forl≥1,

f˜l=−fl+ Xl−1 n=1

X

n∈Nn

l!fnfl−n

n!(l−n)!

X

m¹...m^s=n

A(m¹, . . . ,m^s)Ul−n(km¹k, . . . ,||m^sk) and this gives immediately the attempted formula.

This ends the proofs for our formulas but, as we said before, the first proofs were derived from mould calculus and we will give the main ideas in the next sections.

4 Mould calculus and the shuffle Hopf algebra sh(

) .

4.1 An example of mould calculus

4.1.1 Formal Conjugacy of equations

Mould calculus, as defined by J. Ecalle (see [7],[8],[9]), appears in the study of formal or analytic conjugacy of differential equations, vector fields, diffeomor- phisms. In order to introduce it, we give here a very simple but useful example.

Letu∈G2 and the associated equation

(Eu) ∂tx=u(x) =x+X

n≥1

unxⁿ⁺¹

ForuandvinG2the equations (Eu) and (Ev) are formally conjugated if there exists an elementϕofG2such that, ifxis a solution of (Eu) theny=ϕ(x) is a solution of (Ev). This defines an equivalence relation on the set of such equations and one can easily check that there is only one class : For any equation (Eu), there exist a uniqueϕofG2 such that, ifxis a solution of (Eu) theny=ϕ(x) is a solution of

(E0) ∂ty=y The equation forϕ reads

u(x)ϕ^′(x) =ϕ(x) (49)

and, if

ϕ(x) =x+X

n≥1

ϕnxⁿ⁺¹ (50)

then

u1+ 2ϕ1 = ϕ1

u2+ 2ϕ1u1+ 3ϕ2 = ϕ2

... un+Pn−1

k=1(k+ 1)un−kϕk+ (n+ 1)ϕn = ϕn

(51)

Recursively, one can determine the valuesan(ϕ) =ϕn and thus the diffeomor- phismϕ. This does not give a direct formula for the coefficients ofϕ. Among other properties that may be useful for more sophisticated equations, we will see that the mould calculus will give explicit formulas.

Mould calculus, for this example, is based on two remarks which are detailed in the next two sections.

4.1.2 Diffeomorphisms an substitution automorphisms

As we have seen in section 3.4, to any diffeomorphismϕ∈G2 one can associate a substitution automorphismF ∈G˜2

F = Id +X

n≥1

Fn (52)

Moreover, the action of such an operator on a product of formal power series induces a coproduct

∆F=F⊗F (F(f g) = (Ff)(Fg)) (53)

which also reads

∀n≥1, ∆Fn =Fn⊗Id +

n−1

X

k=1

Fk⊗Fn−k+ Id⊗Fn (54)

4.1.3 Symmetral moulds and shuffle Hopf algebra Now, foru∈G2, the equation (Eu) reads

∂tx=



^B₀+X

n≥1

un^Bn



.x=^B.x with ^Bn=xⁿ⁺¹∂x (55) Instead of computing the conjugating map ϕ we could look for its associated substitution automorphismFin the following shape :

F = Id +X

s≥1

X

n1≥1,...,ns≥1

M^n1,...,nsBn1. . .^Bn_s (56) As we will see later, in order to get a substitution automorphism, is is sufficient to impose that for any sequencesk= (k1, . . . , ks) andl= (l1, . . . , lt),

M^kM^l=X

m

sh^k,l_m M^m (57)

where sh^k,lm is the number of shuffling of the sequencesk,lthat gives the sequence m. The set of such coefficients is called a symmetral mould. Moreover the conjugacy equation reads

BF.x=F^B0.x (58) Now we can solve the equation^BF=F^B0 by noticing that, for (n1, . . . , ns)∈ (^N∗)^s,

[^B0,^Bn1. . .^Bn_s] = (n1+. . .+ns)^Bn1. . .^Bn_s (59) and using this commutation relations, one can check that for s = 1 and a sequence (n1) we get

un1+n1Mⁿ¹= 0 (60)

and fors≥2 and a sequence (n1, . . . , ns)∈(^N∗)^s,

un1M^n2,...,ns+ (n1+. . .+ns)M^n1,...,ns = 0 (61) This defines a symmetral mould, fors≥1 and (n1, . . . , ns)∈(^N∗)^s,

M^n1,...,ns = (−1)^sun1. . . uns

(n1+. . .+ns)(n2+. . .+ns). . .(ns−1+ns)ns

(62) thus we get explicit formulas forF andϕ(x) =F.x: Forn≥1,

ϕnxⁿ⁺¹= Xn s=1

X

n1+. . .+ns=n n_i≥1

M^n1,...,nsBn1. . .^Bns.x (63)

and ϕn =

Xn s=1

X

n1+. . .+n_s=n n_i≥1

(ns+ 1)(ns−1+ns+ 1). . .(n2+. . .+ns+ 1)M^n1,...,ns (64)

We just gave the outlines of the method here. The important idea is that we only used the commutation of^B0 with the over derivations^Bn (n≥1), which means that we worked as these derivations were free of other relations. This can be interpreted in the following algebraic way.

4.2 The free group and its Hopf algebra of coordinates

4.2.1 Lie algebra and substitution automorphisms

LetA¹ the Lie algebra of formal vector fields generated by the derivations

∀n≥1, ^Bn=xⁿ⁺¹∂x (65) Its enveloping algebraU(A¹) is a graded Hopf algebra and, see [5], the Hopf algebraH¹_CM is the dual of U(A¹). Note that this dual is well-defined as the graded components ofU(A¹) are vector spaces of finite dimension. IfG(A¹)⊂ U(A¹) is the group of the group-like elements ofU(A¹), this is exactly the group of substitution automorphism describe above and it is isomorphic to the group G2

∀F ∈G(A¹),∀f ∈^R[[x]] F.f =f◦ϕ, ϕ∈G2 (66) In other terms,G(A¹) =Ge2.

4.2.2 The free group and its Hopf algebra of coordinates

Our previous mould calculus suggests to introduce, by analogy with A¹, the graded free Lie algebraA¹ generated by a set of primitive elementsXn,n≥1,

∆(Xn) =Xn⊗1 + 1⊗Xn (67)

The enveloping algebraU(A¹) is a Hopf algebra which is also called the con- catenation Hopf algebra in combinatorics (see [11]). If the unity isX_∅= 1 (∅is the empty sequence), then an elementU ofU(A¹) can be written

U = U^∅X∅+X

s≥1

X

n1,...,n_s≥1

U^n1,...,nsXn1. . . Xns

= U^∅X∅+X

s≥1

X

n1,...,ns≥1

U^n1,...,nsXn1,...,ns

= X

U^•X•

(68)

where the collection of coefficientsU^• is called a mould. The structure of the enveloping algebraU(A¹) can be described as follows : the product is given by

∀m,n∈ N, XmXn=Xmn (concatenation), (69) the coproduct is

∆(Xn) = X

n¹,n²

sh

n¹,n² n

Xn¹⊗Xn² (70)

where shⁿ_n^1,n2 is the number of shuffling of the sequences n¹,n² that givesn.

Finally, the antipodeS is defined by

S(Xn1,...,ns) = (−1)^sXns,...,n1 (71) Once again one can define the groupG(A¹) and ifF ∈G(A¹) then

F = X

n∈N ∪{∅}

FⁿXn (72)

where the mouldF^• issymmetral : F^∅= 1 and

∀n¹,n², Fⁿ¹Fⁿ² =X

n

shⁿ_n^1,n2Fⁿ (73) Moreover, ifG is the group inverse ofF, then its associated mould is given by the formulas

G^n1,...,ns= (−1)^sF^ns,...,n1

Thanks to the graduation on U(A¹), its dual H¹ is a Hopf algebra, the Hopf algebra of coordinates on G(A¹) and, if the dual basis of {Xn, n ∈ N } is {Zⁿ, n∈ N }then the product inH¹ is defined by :

∀n¹,n², Zⁿ¹Zⁿ²=X

n

shⁿn^1,n2Zⁿ (74) The coproduct is :

∆(Zⁿ) =Zⁿ⊗1 + 1⊗Zⁿ+ X

n¹n²=n

Zⁿ¹⊗Zⁿ² (75) wheren¹n²is the concatenation of the two nonempty sequencesn¹andn²and Z^∅= 1 is the unity. Finally, the antipode is given by

S(Z^n1,...,ns) = (−1)^sZ^ns,...,n1 (76) The structure ofH¹ (coproduct, antipode, ...) is fully explicit. This will be of great use since our previous mould calculus suggests that there exists a surjective morphism fromA¹ on A¹ that induces an injective morphism fromH¹_CM into H¹. In other words, H¹_CM can be identified to a sub-Hopf algebra of H¹ and, as everything is explicit inH¹, one can derive formulas for the coproduct and the antipode inH¹_CM.

5 Morphisms.

The application defined by ρ(Xn) = ^Bn = xⁿ⁺¹∂x obviously determines a morphism from A¹ (resp. U(A¹), resp. G(A¹)) on A¹ (resp. U(A¹), resp.

G(A¹)≃G2) and it is surjective : Ifϕ∈G2 and F =τ⁻¹(ϕ)∈G(A¹) = ˜G2, then, if

b(x) =x+X

n≥1

bnxⁿ⁺¹= ϕ(x)

ϕ^′(x) (77)

thenϕis the unique diffeomorphism ofG2that conjugates (Eb) to (E0) thus F = Id +X

n∈N

M^nBn=ρ X_∅+ X

n∈N

MⁿXn

!

(78) By duality, it induces a morphismρ^∗ fromH¹to H¹ by

∀γ∈ H¹, ρ^∗(γ) =γ◦ρ (79)

and, sinceρis surjective, ρ^∗ is injective : H¹_CM is isomorphic to the sub-Hopf algebraρ^∗(H¹_CM)⊂H¹. Using this injective morphism, we define

∀n≥1, Γn=ρ^∗(γn) (80) and ρ^∗(H¹_CM) is then the Hopf algebra generated by the Γn. In order to get formulas inH¹_CM, we will use the algebraρ^∗(H_CM¹ ) and express the Γn in terms of theZⁿ:

Theorem 5 Forn≥1,

Γn = n! X

n= (n1, . . . , ns)∈ Nn

Xs t=1

(−1)^t−1 t

X

n¹. . .n^t=n

Zⁿ¹. . . Z^ntSⁿ¹. . . S^nt

= n! X

n= (n1, . . . , ns)∈ Nn

QⁿZⁿ

(81) whereS^n1,...,ns=Qs

i=1(ni+ni+1+. . .+ns+ 1) =Qs

i=1(ˆni+ 1)andQ^n1,...,ns = (ns+ 1)Qs

i=2nˆi withQⁿ¹= (n1+ 1).

LetF =X_∅+P

n∈NFⁿXn∈G(A¹). IfF =ρ(F)∈G(A¹), then

Γn(F) =γn(F) =γn(ϕ) = (∂_xⁿlog(ϕ^′)(x))x=0 (82) whereϕ∈G2is defined by :

ϕ(x) =ρ(F).x=F.x=x+ X

(n1, . . . , n_s)∈ N

F^n1,...,nsBn1. . .^Bn_s.x (83)

Then

ϕ^′(x) = 1 + X

(n1, . . . , n_s)∈ N

F^n1,...,nsS^n1,...,nsx^n1+...+ns (84)

Using the logarithm and derivation, one easily gets the formula Γn(F) =n! X

n= (n1, . . . , ns)∈ Nn

Xs t=1

(−1)^t−1 t

X

n¹. . .n^t=n

Fⁿ¹. . . F^ntSⁿ¹. . . S^nt (85) We prove the second part of the formula in section 6, using the fact thatF^• is symmetral. As

Γn(F) =n! X

n= (n1, . . . , n_s)∈ Nn

FⁿQⁿ (86)

andZⁿ.F =Fⁿ, theorem 5 will be proved.

Asρ^∗(δn) = Γn ∈H¹, and, since the coproduct and the antipode are explicit inH¹, we can once again obtain the formulas given in theorems 2 and 3.

6 Initial Proofs.

6.1 Proof of theorem 5

We already proved that, forn≥1,

Γn=n! X

n= (n1, . . . , n_s)∈ Nn

Xs t=1

(−1)^t−1 t

X

n¹. . .n^t=n

Zⁿ¹. . . Z^ntSⁿ¹. . . S^nt (87) Extending the notion of shuffling, fort≥1, ifm¹, . . . ,m^t,maret+1 sequences, then sh^m_m^1,...,mt is the number of ways to obtain the sequence m by shuffling the sequencesm¹, . . . ,m^t. Then,

1

n!Γn = X

n∈Nn

l(n)X

t=1

(−1)^t−1 t

X

n¹...n^t=n

Zⁿ¹. . . Z^ntSⁿ¹. . . S^nt

= X

n∈Nn

l(n)X

t=1

(−1)^t−1 t

X

n¹...n^t=n

X

m

shⁿ_m^1,...,ntZ^m

!

Sⁿ¹. . . S^nt

= X

m∈Nn



Z^m

l(m)X

t=1

(−1)^t−1 t

X

n¹,...,n^t∈N

shⁿ_m^1,...,ntSⁿ¹. . . S^nt (88) Note that in these equations, we hadkmk=knkandl(m) =l(n). For a given sequencem∈ N, let

Q^m=

l(m)X

t=1

(−1)^t−1 t

X

n¹,...,n^t∈N

shⁿm^1,...,ntSⁿ¹. . . S^nt (89) it remains to prove that, ifm= (m1, . . . , ms) thenQ^m1,...,ms = (ms+1)Qs

i=2mˆi

withQ^m1 = (m1+ 1). We prove this formula by induction onl(m).

Ifl(m) = 1, thenm= (m1) and Q^m1= (−1)⁰

1 X

n¹∈N

shⁿm¹Sⁿ¹=S^m1=m1+ 1 (90) If l(m) = s ≥2, then let m = (m1, . . . , ms) andp = (m2, . . . , ms). For any sequence n= (n1, . . . , nk), we note m1n = (m1, n1, . . . , nk). If a shuffling of t≥1 sequencesn¹, . . . ,n^tgivesm then

• Either there exists 1≤i≤t such that nⁱ= (m1) (but then t≥2), and, omitting nⁱ = (m1), the corresponding shuffling of the t−1 remaining sequences givesp.

• Either there exists 1≤i≤t such thatnⁱ =m1n˜ⁱ ( ˜nⁱ 6=∅) (necessarily, t < l(m)) and, replacing nⁱ by ˜nⁱ, the corresponding shuffling of the t sequences givesp.

This means that : Q^m =

l(m)X

t=1

(−1)^t−1 t

X

n¹,...,n^t

shⁿ_m^1,...,ntSⁿ¹. . . S^nt

=

l(m)−1X

t=1

(−1)^t−1 t

X

n¹,...,n^t

shⁿp^1,...,nt

Xt i=1

Sⁿ¹. . . S^m1ni. . . S^nt

+

l(m)X

t=2

(−1)^t−1 t

X

n¹,...,n^t−1

shⁿ_p^1,...,nt−1 Xt−1 i=0

Sⁿ¹. . . SⁿⁱS^m1Sⁿⁱ⁺¹. . . S^nt−1 (91) but asS^m1 =m1+ 1 andS^m1ni = (m1+knⁱk+ 1)Sⁿⁱ,

Q^m =

l(m)−1X

t=1

(−1)^t−1 t

X

n¹,...,n^t

shⁿp^1,...,nt

Xt i=1

(m1+knⁱk+ 1)Sⁿ¹. . . Sⁿⁱ. . . S^nt

+

l(m)−1X

t=1

(−1)^t t+ 1

X

n¹,...,n^t

shⁿ_p^1,...,nt Xt

i=0

(m1+ 1)Sⁿ¹. . . SⁿⁱSⁿⁱ⁺¹. . . S^nt

=

l(m)−1X

t=1

(−1)^t−1

t (t(m1+ 1) +kpk) X

n¹,...,n^t

shⁿp^1,...,ntSⁿ¹. . . S^nt

+

l(m)−1X

t=1

(−1)^t

t+ 1(t+ 1)(m1+ 1) X

n¹,...,n^t

shⁿ_p^1,...,ntSⁿ¹. . . S^nt

= kpk Xl(p) t=1

(−1)^t−1 t

X

n¹,...,n^t

shⁿ_p^1,...,ntSⁿ¹. . . S^nt

= kpkQ^p

(92) And it obviously gives the right formula forQ^m.

6.2 Proof of theorem 2

Using the above formula we have

∆Γn = n! X

m∈Nn

Q^m(∆Z^m)

= n! X

m∈Nn

Q^m



Z^m⊗1 + 1⊗Z^m+ X

pq=m

Z^p⊗Z^q





= n! X

m∈Nn

Q^mZ^m

!

⊗1 + 1⊗ n! X

m∈Nn

Q^mZ^m

!

+n! X

m∈Nn

X

pq=m

Q^mZ^p⊗Z^q

= Γn⊗1 + 1⊗Γn+n! X

m∈Nn

X

pq=m

Q^mZ^p⊗Z^q

= Γn⊗1 + 1⊗Γn+ ˜∆Γn

(93)

Now ifpq=m= (m1, . . . , ms) withp,q∈ N (s≥2), then Q^m

Q^q = (ms+ 1)Qs i=2mˆi

(ms+ 1)Qs

i=l(p)+2mˆi

=

l(p)+1Y

i=2

ˆ mi=

l(p)+1Y

i=2

(ˆpi+kqk) =R_kqk^p (94) with the convention that ifi =l(p) + 1, then ˆpi = 0. As this coefficient only depends onpandkqk,

∆Γ˜ n = n! X

m∈Nn

X

pq=m

Q^mZ^p⊗Z^q

= n! X

m∈Nn

X

pq=m

(R^p_kqkZ^p)⊗Q^qZ^q

= n!

n−1X

k=1



 X

p∈Nn−k

R^p_kqkZ^p



⊗



X

q∈Nk

Q^qZ^q





=

n−1X

k=1



n!

k!

X

p∈Nk

R^p_kqkZ^p



⊗Γk

=

nX−1 k=1

P_kⁿ⊗Γk

(95)

and it remains to prove that, forn≥1 and 1≤k≤n−1,

P_kⁿ= X

(n1, . . . , n_s)∈ N n1+. . .+n_s=n−k, s≥1

n!

n1!. . . ns!k!αⁿ_k^1,...,nsΓn1. . .Γn_s (96)