Mulitgraded Dyson-Schwinger systems

(1)

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Mulitgraded Dyson-Schwinger systems

Loïc Foissy

To cite this version:

(2)

Loï Foissy

Fédération de Re her he Mathématique duNordPasde Calais FR2956

Laboratoirede MathématiquesPuresetAppliquéesJoseph Liouville

UniversitéduLittoral Cted'Opale-CentreUniversitairede la Mi-Voix

50,rueFerdinand Buisson,CS 80699, 62228Calais Cedex,Fran e

Email: foissylmpa.univ-littoral.fr

Abstra t

Westudysystemsof ombinatorialDyson-S hwingerequationswithanarbitrarynumber

N

of oupling onstants. The onsideredHopfalgebraofFeynmangraphsis

N

-graded,and

wewonderifthegradedsubalgebrageneratedbythesolutionisHopfornot.

Werstintrodu eafamilyofpre-Liealgebraswhi hwe lassify,duallyprovidingsystems

generatingaHopfsubalgebra;wealsodes ribetheasso iatedgroups,asextensionsofgroups

offormaldieomorphismsonseveralvariables.

Wethen onsidersystems omingfromFeynmangraphsofaQuantumFieldTheory. We

showthatifthenumber

N

ofindependent oupling onstantsisthenumberofintera tions

ofthe onsideredQFT,thenthegeneratedsubalgebraisHopf. ForQED,

ϕ

3

andQCD,we

alsoprovethatthisistheminimalvalueof

N

.

All theseexamplesaregeneralizationsoftherstfamilyofDyson-S hwingersystemsin

the one oupling onstant ase, alled fundamental. We also give a generalization of the

se ondfamily, alled y li .

Keywords. Dyson-S hwinger systems; Feynman graphs; pre-Lie algebras; ombinatorial

Hopfalgebras.

AMS lassi ation. 16T05,81T18,05C05.

Contents

1 Hopf algebras of de orated trees 3

1.1 Denitionand universal property . . . 3

1.2 Graduation andduality . . . 5

1.3 Completion . . . 6

2 Multigraded SDSE 6 2.1 Denition . . . 6

2.2 Simpli ation ofthehypotheses . . . 8

2.3 Operationson SDSE . . . 9

2.4 Changesofgraduation . . . 11

3 A familyof pre-Lie algebras 12 3.1 Denitionand examples . . . 12

3.2 Classi ationof deg1 pre-Liealgebras . . . 14

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4.1 Liealgebra asso iated to afundamental pre-Liealgebra . . . 18

4.2 Groupasso iated to a redu eddeg1 pre-Liealgebra . . . 20

5 SDSE asso iated to a family of Feynman graphs 23 5.1 Feynman graphs . . . 23

5.2 Graduations . . . 27

5.3 Insertions . . . 29

5.4 SDSE asso iatedto a theoryofFeynman graphs . . . 31

5.5 Minimal rankfor QCD . . . 35

6 SDSE asso iated to oloured graphs 36

Introdu tion

InaQuantumFieldTheory(shortly,QFT),theGreenfun tionsaredevelopedasaseriesinthe

oupling onstant, indexedby the set ofFeynman graphs. These series an be seenat the level

of Feynman graphs. They satisfya ertain system

(S)

of ombinatorial Dyson-S hwinger

equa-tion (briey, SDSE), whi h uses ombinatorial operators of insertion, and allows to indu tively

ompute thehomogeneous omponents of the Green fun tions, a ordingto their loop number

[1, 14, 15 , 16, 17 , 18, 19, 20 , 24, 25, 26 , 28℄. Feynman graphs areorganized as a Hopf algebra

H

F G

,gradedbythe loopnumber,andwe onsiderthe subalgebra

H

(S)

of

H

F G

generated bythe

omponentsoftheuniquesolutionof

(S)

. Anaturalquestionistoknowifthegradedsubalgebra

generated bythe Greenfun tionsisHopf ornot. Thisproblem, andrelatedquestionsabout the

nature of the obtained Hopf subalgebras, are the main obje t of study in [6, 7 , 8, 9℄. It turns

out that in the ase of QED or

ϕ

3

, whi h are QFT with only one intera tion, this subalgebra

is indeed Hopf; this is not the ase for QCD, with its four intera tions. A possibility in this

last aseisto renethe graduation, orequivalently tointrodu e more oupling onstants,whi h

makes thesubalgebra

H

(S)

generated bythe omponents of the solution bigger; we shall prove

here thatthere existsa

N

4

-graduation of the Hopf algebra of QCDFeynman graphs, su h that

H

_(S)

is aHopf subalgebra.

The aim of this text is to study SDSE giving a Hopf subalgebra when the Hopf algebra of

Feynman graphs is given a

N

-graduation, generalizing the results of [7℄ for the loop number

graduation. Re allthatifwe onsideronlyone oupling onstant,theHopfalgebraofgraphswe

onsideris

N

-graded,andwe obtainedtwo familiesofSDSE, alledfundamental and y li ,and

four operations on SDSE, allowing to obtain all SDSE giving a Hopf subalgebra. The graded

dual ofthis Hopf subalgebra isthe enveloping algebra of a pre-Lie algebra,des ribed in[9℄. In

the fundamental ase, the onstant stru tures of this pre-Lie algebra are polynomial of degree

≤ 1

. We generalize this denition to the

N

-graded ase(denition 8); these obje tsare alled

deg1 pre-Liealgebras. Their lassi ation isdone intheorem24. Asenveloping algebrasoffree

pre-Liealgebras areGrossman-LarsonHopf algebras[10 , 11℄, duallytheenveloping algebra ofa

deg1 pre-Lie algebra an be embedded in a Connes-Kreimer Hopf algebra of de orated rooted

trees [4,5℄, givinginthiswayafamilyofSDSEsu htheasso iatedsubalgebraisHopf(theorem

27). We alsodes ribethegroup asso iated to su h pre-Liealgebras; they all ontaina group of

formaldieomorphisms.

We then pro eedto SDSE oming from a QFT. We rst study all thepossible graduations

of

H

F G

whi h aredened from ombinatorial datas asso iated to Feynman graphs, su h as the

number of verti es, of internal or external half-edges or edges, or the external stru ture: we

prove that su h a

N

-graduation is asso iated to a matrix

C ∈ M

N,|V|

(Q)

, where

V

is the set

of possible verti es in the Feynman graphs of the theory (proposition 38); the rank of

C

is of

(4)

previously des ribed, ifthe rank of

C

is the ardinality of

V

. We mayask the question of the

minimalrankof

C

requiredtoobtainaHopfsubalgebra: itissmallerthan

|V|

. InQEDor

ϕ

n

,as

this ardinalityis

1

,theanswerisobviously

1

;forQCD,weproveinproposition44thatitisalso

|V| = 4

. Themainideaistoprodu eprimitiveFeynmangraphswithanarbitrarilylargenumber

of verti es of any kind, and we onje ture that for any QFT with enough primitive Feynman

graphs,the minimalrankofthegraduation isthenumberofintera tions ofthetheory. Weshall

on ludewithageneralizationofthese ondfamilyofSDSEinthe

N

-graded ase, namely y li

SDSE.

Thisarti le is organizedasfollows. Therst se tion ontains reminders onConnes-Kreimer

Hopf algebras of de orated rooted trees, their universalproperties, their graduations and their

gradedduals. Inthe se ond se tion,weintrodu e the notionof ombinatorial SDSE in

Connes-KreimerHopfalgebras; we give threeoperationsonSDSE,and alsostudy theee t of hanging

the graduation of the subalgebra

H

(S)

generated by the unique solution of su h a SDSE. We

then introdu e and lassify deg1 pre-Lie algebras in the next se tion, whi h dually give us a

rst family of

N

-graded SDSE. The group asso iated to these pre-Lie algebras are des ribed

inthe fourthse tion. Feynman graphsof agivenQFT, their Hopf-algebrai stru ture and their

SDSE areintrodu ed and studied inthe nextse tion. Thelast, independent,se tion dealswith

ageneralization of y li SDSE.

Aknowledgment. Theresear hleadingtheseresults waspartiallysupportedbytheFren h

NationalResear h Agen y underthe referen eANR-12-BS01-0017.

Notations.

1. Let

M

and

N

be nonnegative integers. We denote by

[M ]

the set of integers

{1, . . . , M }

andby

N

∗

the set ofnonzero elementsof

N

.

2. The anoni albasisof

K

N

(and of

Z

N

)is denotedby

(ǫ

1 , . . . , ǫ

N

)

.

3. Let

a, b ∈ K

. Wedenote by

F

a,b

(X)

theformalseries:

F

a,b

(X) =

∞

X

k=0

a(a − b) . . . (a − b(k − 1))

k!

X

k

₌

(

(1 + bX)

a

b

if

b 6= 0,

e

aX

if

b = 0.

Notethatfor all

a, a

′

_{, b ∈ K}

,

F

a+a

′

,b

(X) = F

a,b

(X)F

a

′

,b

(X)

.

1 Hopf algebras of de orated trees

Let us start with a few reminders on the Connes-Kreimer Hopf algebras of de orated trees

[4 , 5℄and relatedalgebrai stru tures. We onsider a nonempty set

D

,whi h we all theset of

de orations.

1.1 Denition and universal property

Denition 1 1. A tree is a nite graph, onne ted, with no loop; a rooted tree is a tree

withapointed vertex, alled the root;a rooted tree de orated by

D

isa pair

(T, d)

, where

T

isa rooted tree and

d

isa mapfrom the set

V (T )

of verti es of

T

to

D

; for all

v ∈ V (T )

,

d(v)

is alled the de oration of

v

. The set of iso lasses of rooted trees de orated by

D

is

denoted by

T

D

.

2. The algebra

H

D

of rooted trees de orated by

D

isthe free ommutative asso iative algebra

generated by

T

D

. By denition, the set

F

D

(5)

say monomialsin

T

D

, or nite disjoint unions of elements of

T

D

, is a basis of

H

D

. The produ t of

H

D

isthe disjoint union of de orated rooted forests.

Examples. We drawrootedtrees with their root at thebottom.

1. Therootedtrees de orated by

D

with

n ≤ 4

verti es are:

q

_a

, a ∈ D;

q

a

b

_{, (a, b) ∈ D}

2 _;

q

∨

a

c

b

=

q

∨

a

b

c

,

q

a

b

c

, (a, b, c) ∈ D

3 ;

q

q q

∨

a

d

c

b

=

q

q q

∨

a

c

d

b

= . . . =

q

q q

∨

a

b

c

d

,

q

∨

a

d

b

c

=

q

∨

a

b

d

c

,

q

∨

a

b

d

c

=

q

∨

a

b

c

d

,

q

a

b

c

d

, (a, b, c, d) ∈ D

4 .

2. Therootedforests de orated by

D

with

n ≤ 3

verti es are:

1;

q

_a

, a ∈ D;

q

a

b

_,

q

_a

q

_b

=

q

_b

q

_a

, (a, b) ∈ D

2 _;

q

∨

a

c

b

=

q

∨

a

b

c

,

q

a

b

c

,

q

a

b

q

_c

=

q

_c

q

a

b

_,

q

_a

q

_b

q

_c

=

q

_a

q

_c

q

_b

= . . . =

q

_c

q

_b

q

_a

, (a, b, c) ∈ D

3 _.

The algebra

H

D

an alsobedened bya universalproperty[4,27 ℄:

Proposition 2 Let

d ∈ D

. The linear endomorphism

B

d

of

H

D

sends any rooted forest

F ∈ F

D

to

B

d

(F ) ∈ T

D

obtainedingraftingthedierenttreesof

F

ona ommonrootde oratedby

d

. Thisfamilyof endomorphismssatisfy the followinguniversalproperty: if

A

isa ommutative

algebra, andfor all

d ∈ D

,

L

d

: A −→ A

isalinear endomorphism, there existsa unique algebra

morphism

φ : H

D

_{−→ A}

su h that for all

d ∈ D

,

φ ◦ B

d

= L

d

◦ φ

.

Example. If

a, b, c, d ∈ D

,

B

a

(

q

b

q

c

d

_{) =}

q

∨

a

c

b

d

.

This universalproperty an be usedto dene theConnes-Kreimer oprodu t of

H

D

:

Proposition 3 1. There existsa unique oprodu t on

H

D

su hthat forall

d ∈ D

,for all

x ∈ H

D

:

∆ ◦ B

d

(x) = B

d

(x) ⊗ 1 + (Id ⊗ B

d

) ◦ ∆(x).

With this oprodu t,

H

D

be omes a Hopfalgebra. Its ounitis the map:

ε :

H

D

_{−→ K}

F ∈ F

D

−→ δ

F,1

.

2. Let

A

be a ommutative Hopf algebra, and for all

d ∈ D

, let

L

d

: A −→ A

a linear

endomorphism su h that for all

x ∈ A

:

∆ ◦ L

d

(x) = L

d

(x) ⊗ 1 + (Id ⊗ L

d

) ◦ ∆(x).

The unique algebra morphism

φ : H

D

_{−→ A}

su h that for all

d ∈ D

,

φ ◦ B

d

= L

d

◦ φ

is a

Hopfalgebra morphism.

This oprodu tadmitsa ombinatorial des riptionintermsofadmissible uts. Forexample,

if

a, b, c, d ∈ D

:

∆

q

∨

d

c

b

a

=

q

∨

d

c

b

a

⊗ 1 + 1 ⊗

q

∨

d

c

b

a

+

q

b

a

_⊗

q

d

c

₊

q

_a

⊗

q

∨

d

c

b

+

q

_c

⊗

q

d

b

a

+

q

b

a

q

_c

⊗

q

_d

+

q

_a

q

_c

⊗

q

d

b

_.

(6)

Proposition 4 Let

a = (a

d

)

d∈D

be a family of elements of

K

. We denote by

φ

a

the unique

Hopf algebra endomorphism of

H

D

su h that for all

d ∈ D

,

φ ◦ B

d

= a

d

B

d

◦ φ

. For any forest

F ∈ F

D

, denoting by

V (F )

the set of verti es of

F

:

φ

a

(F ) =





Y

v∈V (F )

a

_d(v)



 F.

Consequently, if for all

d ∈ D

,

a

d

6= 0

,

φ

a

isan automorphism.

Proof. We onsidertheendomorphism

ϕ

dened by:

∀F ∈ F

D

, ϕ(F ) =





Y

v∈V (F )

a

_d(v)



 F.

Let

F, F

1 , F

2 ∈ F

D

. As

V (F

1 F

2 ) = V (F

1 ) ⊔ V (F

2 )

,

ϕ(F

1 F

2 ) = ϕ(F

1 )ϕ(F

2 )

,

ϕ

is an algebra

endomorphism. As

V (B

d

(F )) = V (F ) ⊔ {root(F )}

,

ϕ(B

d

(F )) = a

d

B

d

(ϕ(F ))

. Consequently,

ϕ ◦ B

d

= a

d

B

d

◦ ϕ

. By uni ity inthe universalproperty,

ϕ = φ

a

.

1.2 Graduation and duality

Denition 5 1. A

N

-gradedsetisapair

(D, deg)

,where

D

isasetand

deg : D −→ N

N

isa map. For all

α ∈ N

N

, we put

D

α

= deg

−1

_(α)

. We shall say that the

N

-graded

D

is

onne ted if

D

0 = ∅

and if for all

α ∈ N

N

,

deg

−1

_(α)

isnite. 2. Let

D

be a

N

-graded onne ted set. For all forest

F ∈ F

D

, we put:

deg(F ) =

X

v∈V (F )

deg(d(v)).

Thisindu es a onne ted

N

-graduation of the Hopfalgebra

H

D

,with:

∀α ∈ N

N

, (H

D

)

α

= V ect(F ∈ F

D

| deg(F ) = α).

Moreover, for this graduation,

B

d

is homogeneous of degree

deg(d)

for all

d ∈ D

.

If

D

is a

N

-graded onne ted set, then, as

H

D

is a graded onne ted Hopf algebra, its

graded dual

(H

D

₎

∗

is also a Hopf algebra [13 , 23 ℄. Asa ve tor spa e, it an be identied with

H

D

,bythehelpof thesymmetri pairingdened by:

∀F, G ∈ F

D

, hF, Gi = s

F

δ

F,G

,

where

s

F

is thenumberofsymmetries of

F

. The oprodu t

∆

′

of

(H

D

₎

∗

isgiven by:

∀T

1 , . . . , T

k

∈ T

D

, ∆

′

(T

1 . . . T

k

) =

X

I⊆[k]

Y

i∈I

T

i

!

⊗

Y

i /

∈I

T

i

!

.

Itsprodu t

⋆

isgivenbygraftings: thisistheGrossman-Larsonprodu t[10 ,11,12 ℄. Forexample:

q

a

b

_⋆

q

c

d

₌

q

a

b

q

c

d

₊

q

∨

c

d

a

b

+

q

c

d

a

b

.

(7)

By the Cartier-Quillen-Milnor-Moore's theorem,

(H

D

₎

∗

is the enveloping algebra of a Lie

algebra

g

D

. By onstru tion of the oprodu t

∆

′

, the set

T

D

is a basis of

g

D

; by denition of

theGrossman-Larson produ t,for all

T, T

′

_{∈ T}

D

:

[T, T

′

] =

X

v

′

_{∈V (T}

′

₎

grafting of

T

on

v

′

₋

X

v∈V (T )

grafting of

T

′

on

v.

We dene a produ t

∗

on

g

D

by:

T ∗ T

′

=

X

v

′

_{∈V (T}

′

₎

grafting of

T

on

v

′

_.

For any

x, y ∈ g

D

,

[x, y] = x ∗ y − y ∗ x

. For example:

q

_c

∗

q

a

b

₌

q

∨

a

b

c

+

q

a

b

c

,

q

a

b

_∗

q

_c

=

q

c

a

b

.

Thisprodu tisnot asso iative,but ispre-Lie:

Denition 6 A (left) pre-Lie algebra is a pair

(V, ∗)

, where

V

is a ve tor spa e and

∗

is a

bilinear produ t on

V

, su h that for all

x, y, z ∈ V

:

(x ∗ y) ∗ z − x ∗ (y ∗ z) = (y ∗ x) ∗ z − y ∗ (x ∗ z).

If

(V, ∗)

ispre-Lie, the bra ket dened by

[x, y] = x ∗ y − y ∗ x

isa Liebra ket.

Moreover, Chapoton and Livernet proved, using the theory of operads, that

g

D

is a free

pre-Liealgebra [2, 3℄:

Theorem 7 Let

A

be a pre-Lie algebra and let

a

d

∈ A

for all

d ∈ D

. There exists a unique

pre-Lie algebra morphism

φ : g

D

_{−→ A}

su h that

φ(

q

_d

) = a

d

for all

d ∈ D

. In other words,

g

D

is,as a pre-Lie algebra, freely generated by the elements

q

d

,

d ∈ D

.

1.3 Completion

We graduate

H

D

bythe number ofverti es of forests,that isto saywe onsiderthe graduation

indu edbythemap

deg : D −→ N

,sending every element of

D

to

1

. Thisgraduation indu es a

distan e

d

on

H

D

,dened by:

d(f, g) = 2

−val(f −g)

.

The metri spa e

H

D

is not omplete: its ompletion is denoted by

H

d

D

. As a ve tor spa e, it

is the spa e of ommutative formal series in

T

D

. The produ t of

H

D

, being homogeneous of

degree

0

,is ontinuous, so anbeextendedto

H

d

D

: thisgivestheusual produ tofformalseries.

Similary, for any

d ∈ D

,

B

d

, being homogeneous of degree

1

, is ontinuous so an be extended

to amap

B

d

: d

H

D

_{−→ d}

_H

D

.

2 Multigraded SDSE

2.1 Denition

Denition 8 Let

D = D

1 ⊔. . .⊔D

M

beapartitionedset. Let

(f

d

)

d∈D

beafamilyofelements

of

Khhx

1 , . . . , x

M

ii

. ThesystemofDyson-S hwingerequations(briey,SDSE)asso iatedtothese

elements is:

∀i ∈ [M ], X

i

=

X

d∈D

i

B

d

(f

d

(X

1 , . . . , X

M

)),

where

X = (X

1 , . . . , X

M

)

belongs to

H

d

D

M

.

(8)

By onvenien e,wegenerallyindexthefamilyofunknows by

[M ]

,butitisof oursepossible

to indexthem by anynite set.

Proposition 9 Let

(S)

be a SDSE. Ithas a unique solution.

Proof. If

X = (X

1 , . . . , X

M

)

isasolutionof

(S)

,thenforall

i

,

X

i

isainnitespanoftrees,

so belongs to the augmentation ideal

H

D

+

. Hen e, it is enough to prove that

(S)

has a unique

solutionin

H

d

D

+

M

. Letus onsiderthefollowing map:

Θ :











d

H

D

M

_{−→ d}

_H

D

M

(X

1 , . . . , X

M

) −→





X

d∈D

i

B

i

(f

d

(X

1 , . . . , X

M

))





i∈[M ]

.

As

B

d

ishomogeneous ofdegree

1

forall

d

,we obtainthatfor all

f, g ∈ d

H

D

M

:

d(Θ(f ), Θ(f )) ≤

1

2 d(f, g).

So

Θ

is a ontra ting map. As

H

d

D

M

is omplete,

Θ

has a unique xed point

(X

1 , . . . , X

M

)

,

whi his theuniquesolution of

(S)

.

Remarks.

1. Asthe

D

i

aredisjoint, the nonzero

X

i

aresum of trees with roots de orated by elements

of

D

i

,soarealgebrai allyindependent.

2. If

X

i

= 0

,we an delete the

i

-thequationof

(S)

andrepla e

f

d

by

(f

d

)

|x

i

=0

forall

d ∈ D

,

without hanging

H

(S)

.

We now assume that all the

X

i

are nonzero(and, as a onsequen e,are algebrai ally

independent).

Denition 10 Let

D

be a onne ted

N

-graded set, indu inga onne ted

N

-graduation of

the Hopfalgebra

H

D

. Let

(S)

be a SDSEon

D

.

1. The unique solution of

S

is denoted by

X = (X

1 , . . . , X

M

)

, and the homogeneous

ompo-nentsof

X

i

are denoted by

X

i

(α)

,

i ∈ [M ]

,

α ∈ N

N

.

2. The subalgebra of

H

D

generated by the

X

i

(α)

's isdenoted by

H

(S)

.

3. We shall say that

(S)

is Hopfif

H

(S)

isa Hopf subalgebra of

H

D

.

Notethat

H

(S)

dependsonthe hoi e ofthe graduation.

Example. Here is an example of SDSE. Le us x

k ≥ 1

and

d

0 , . . . , d

k

∈ N

. For any

α = (α

0 , . . . , α

k

) ∈ [N ]

k+1

,we put:

deg(α) = d

0 ǫ

α

0 + . . . + d

k

ǫ

α

k

∈ Z

N

_.

Theset ofde orations is:

(9)

TheHopf algebra

H

D

inherits a onne ted

N

-graduation. We onsider theSDSE:

(S)

F dB

: ∀i ∈ [N ], X

i

=

X

α∈[N ]

k

B

(i,α)

(1 + X

α

1 )

d

1 _{. . . (1 + X}

α

k

)

d

k

_{(1 + X}

i

)

d

0 +1

.

(1)

Inparti ular, if

(d

0 , . . . , d

k

) = (0, 1, . . . , 1)

,this gives:

∀i ∈ [N ], X

i

=

X

α∈[N ]

k

B

_(i,α)

((1 + X

α

1 ) . . . (1 + X

α

k

)) .

Taking

k = 2

, the omponents of

X

are a ommutative version of the elements of Denition

20 in[6 ℄, whi h generate a Hopf algebra isomorphi to the free Faà di Bruno Hopf algebra on

N

variables. We shall prove that it is indeed a Hopf SDSE, related to the Faà di Bruno Hopf

algebra on

N

variables.

2.2 Simpli ation of the hypotheses

Lemma 11 Let

(S)

be a HopfSDSE, andlet

d ∈ D

. If

f

d

(0, . . . , 0) = 0

, then

f

d

= 0

.

Proof. Let

i ∈ [M ]

, su h that

d ∈ D

i

. As

f

d

(0, . . . , 0) = 0

,

q

d

does not appear in

X

i

,

and

q

d

never appears in any element of

H

(S)

. Let us assume that

f

d

6= 0

. As the

X

j

are

algebrai ally independent,

f

d

(X

1 , . . . , X

N

) 6= 0

, and there exists a linear form

g

on

H

d

D

, su h

that

g(f

d

(X

1 , . . . , X

N

)) = 1

. Then

(g ⊗ Id) ◦ ∆(X

i

)

is an element of

H

(S)

, where the term

g(f

d

(X

1 , . . . , X

N

))

q

_d

=

q

_d

appears: this is a ontradi tion. So

f

d

= 0

.

Consequently,if

H

(S)

isHopfand

f

d

0 (0, . . . , 0) = 0

for a ertain

d

0 ∈ D

i

,we an rewritethe

i

-thequationof

(S)

inthefollowing way:

X

i

=

X

d∈D

i

\{d

0 }

B

d

(f

d

(X

1 , . . . , X

M

)).

We now assume that for all

d

,

f

d

(0, . . . , 0) 6= 0

.

Lemma 12 We onsider the two SDSE:

(S) : ∀i ∈ [M ], X

i

=

X

d∈D

i

B

d

(f

d

(X

1 , . . . , X

M

)),

(S

′

) : ∀i ∈ [M ], Y

i

=

X

d∈D

i

B

d

f

d

(Y

1 , . . . , Y

M

)

f

d

(0, . . . , 0)

.

For all

d ∈ D

, we put

a

d

= f

d

(0, . . . , 0)

. Let

φ

a

be the Hopf algebra isomorphism dened in

proposition 4. Thenfor all

i ∈ [M ]

,

X

i

= φ

a

(Y

i

)

;

H

(S)

= φ

a

(H

(S

′

)

and

(S)

isHopf, ifand only

if,

(S

′

₎

isHopf. Proof. We put:

g

d

(x

1 , . . . , x

M

) =

f

d

(x

1 , . . . , x

M

)

f

d

(0, . . . , 0)

.

As

φ

a

◦ B

d

= f

d

(0, . . . , 0)B

d

◦ φ

a

for all

d

,we obtain:

φ

a

(Y

i

) =

X

d∈D

i

φ

a

◦ B

d

(g

d

(Y

1 , . . . , Y

M

))

=

X

d∈D

i

f

d

(0, . . . , 0)B

d

◦ φ

a

(g

d

(Y

1 , . . . , Y

M

))

=

X

d∈D

i

f

d

(0, . . . , 0)B

d

(g

d

(φ

a

(Y

1 ), . . . , φ

a

(Y

M

)))

=

X

d∈D

i

B

d

(f

d

(φ

a

(Y

1 ), . . . , φ

a

(Y

M

))).

(10)

So

(φ

a

(Y

1 ), . . . , φ

a

(Y

M

))

is theunique solution of

(S)

.

We now assume that

f

d

(0, . . . , 0) = 1

for all

d ∈ D

.

Lemma 13 Let

(S)

be a Hopf SDSE,

d

1 , d

2

be two elements in the same

D

i

, of the same

degree. Then

f

d

1 = f

d

2

.

Proof. Let us denoteby

α

the ommon degreeof

d

1

and

d

2

. The homogeneous omponent

of

X

i

of degree

α

hasthe form

q

d

1 +

q

_d

2 + . . .

; onsequently, ifwe onsider thelinearforms:

f

1 :

H

D

_{−→ K}

F ∈ F

D

−→ δ

F,

q

_d

1 ,

f

2 :

H

D

_{−→ K}

F ∈ F

D

−→ δ

F,

q

_d

2 ,

thentherestri tion of

f

1

and

f

2

to

H

(S)

areequal. As

H

(S)

isHopf:

f

d

1 (X

1 , . . . , X

M

) = (Id ⊗ f

1 ) ◦ ∆(X

i

) = (Id ⊗ f

2 ) ◦ ∆(X

i

) = f

d

2 (X

1 , . . . , X

M

).

So

f

d

1 = f

d

2

.

Notethat, ifthe SDSE is Hopf,we an writeit undertheform:

∀i ∈ [M ], X

i

=

X

α∈N

N

∗





X

i∈D

i

,deg(i)=α

B

i





|

{z

}

=B

i,α

(f

α

(X

1 , . . . , X

M

)) =

X

α∈N

N

∗

B

i,α

(f

α

(X

1 , . . . , X

M

)).

2.3 Operations on SDSE Denition 14 Let

D = D

1 ⊔. . .⊔D

M

bea

N

-graded onne ted partitionedset. We onsider

the SDSEgiven by:

(S) : ∀i ∈ [M ], X

i

=

X

d∈D

i

B

d

(f

d

(X

1 , . . . , X

M

)).

1. (Change of variables) Let

a = (a

1 , . . . , a

M

)

be a family of nonzero s alars. The SDSE

obtained from

(S)

by the hange of variables asso iated to these oe ients is:

(S)

a

: ∀i ∈ [M ], Y

i

=

X

d∈D

i

B

d

(f

d

(a

1 Y

1 , . . . , a

M

Y

M

)).

2. (Restri tion) Let

I ⊆ [M ]

. The restri tion of

(S)

to

I

isthe SDSE given by:

(S)

|I

: ∀i ∈ I, X

i

=

X

d∈D

i

B

d

(g

d

(X

j

, j ∈ I)),

where for all

d ∈ I

,

g

d

= f

d|x

j

=0

forall

j /

∈I

∈ K[[X

j

, j ∈ I]]

.

Proposition 15 1. Let

(S)

be a SDSEandlet

(S)

a

be another SDSE,obtained from

(S)

by a hange of variables. We dene the oe ients

a

d

,

d ∈ D

,by:

a

d

= a

i

if

d ∈ D

i

.

Let

φ

a

be the Hopf algebra isomorphism dened in proposition 4. The unique solution of

(S)

a

is:

1 a

1 φ

a

(X

1 ), . . . ,

1 a

M

φ

a

(X

M

)

.

Hen e,

H

(S)

(11)

2. Let

I ⊆ M

. We dene the oe ients

a

d

,

d ∈ D

, by:

a

d

=







1

if

d ∈

G

i∈I

D

i

,

0

otherwise

.

Let

φ

a

be the Hopfalgebra morphismdenedin proposition 4. The uniquesolutionof

(S)

|I

is:

(φ

a

(X

i

))

i∈I

.

Hen e,

H

(S)

|I

= φ

a

(H

(S)

)

and,if

(S)

is a HopfSDSE, then

(S)

|I

isalso a HopfSDSE.

Proof. 1. For all

i ∈ [M ]

,we put

Y

i

=

1 a

i

φ

a

(X

i

)

. Then:

Y

i

=

1 a

i

X

d∈D

i

φ

a

◦ B

d

(f

d

(X

1 , . . . , X

M

))

=

X

d∈D

i

B

d

◦ φ

a

(f

d

(X

1 , . . . , X

M

))

=

X

d∈D

i

B

d

(f

d

(φ

a

(X

1 ), . . . , φ

a

(X

M

)))

=

X

d∈D

i

B

d

(f

d

(a

1 Y

1 , . . . , a

N

Y

M

)).

So

Y = (Y

1 , . . . , Y

M

)

is thesolutionof

(S)

a

.

2. Proved ina similarway,noting that

φ

a

(X

i

) = Y

i

ifi

∈ I

and

0

otherwise.

Denition 16 (Con atenation) Let

(S)

and

(S

′

₎

be two SDSE,respe tively asso iated to

partitioned

N

-graded sets

D = D

1 ⊔ . . . ⊔ D

M

and

D

′

_{= D}

′

1 ⊔ . . . ⊔ D

M

′

, andto formal series

(f

d

)

d∈[M ]

and

(f

′

d

)

d∈[M

′

_]

. The on atenation of

(S)

and

(S

′

₎

is the system asso iated to the

N

-graded partitioned set

D ⊔ D

′

_{= D}

1 ⊔ . . . ⊔ D

M

⊔ D

1 ′

⊔ . . . ⊔ D

M

′

given by:

(S) ⊔ (S

′

) :











if

1 ≤ i ≤ M, , X

i

=

X

d∈D

i

B

d

(f

d

(X

1 , . . . , X

M

)),

if

M + 1 ≤ i ≤ M + M

′

_{, X}

i

=

X

d∈D

′

i−M

B

d

(f

d

′

(X

M +1

, . . . , X

M +M

′

)).

Proposition 17 Let

(S)

and

(S

′

₎

be two SDSE.Then

(S) ⊔ (S

′

₎

isHopfif, andonlyif,

(S)

and

(S

′

₎

are Hopf.

Proof.

=⇒

. Let us assumethat

(S) ⊔ (S

′

₎

isHopf. Then

(S) ⊔ (S

′

₎

|[M ]

= (S)

and, upto a reindexation,

(S) ⊔ (S

′

₎

|[M +M

′

_{]\[M ]}

= (S

′

)

. By proposition15,

(S)

and

(S

′

₎

areHopf.

⇐=

. Letusassumethat

(S)

and

(S

′

₎

areHopf. Then

H

(S)⊔(S

′

₎

isisomorphi to

H

(S)

⊗H

(S

′

₎

⊆

H

D

_{⊗ H}

D

′

⊆ H

D⊔D

′

. As

H

(S)

and

H

(S

′

₎

are Hopf subalgebras of

H

D

and

H

D

′

,

H

(S)

⊗ H

(S

′

₎

is a Hopfsubalgebra of

H

D⊔D

′

,so

(S) ⊔ (S

′

₎

is Hopf.

Remark. Asin[7℄,itispossibleto dene anoperation ofdilatation for multigradedSDSE.

(12)

Let

D

bea

N

-graded onne tedset. Let

C ∈ M

N

′

,N

(Q)

. Weassumethefollowing hypothesis:

if

α ∈ N

N

satises

D

α

6= (0)

, then

Cα ∈ N

N

′

∗

. We give

D

a

N

′

-graduation by:

D

_β

′

=

G

α∈N

N

_,Cα=β

D

α

.

Thisdenesanother onne tedgraduationof

D

. Consequently,

H

D

inheritsase ondgraduation:

H

D

_(β)

′

=

M

α,Cα=β

H

D

_(α)

.

Let

(S)

bea SDSE on

D

. The solution

X

of

(S)

an be de omposedinto two ways:

X

i

=

X

α∈N

N

X

i

(α) =

X

β∈N

N ′

X

_i

′

(β).

Hen e, we obtaintwo subalgebras,denoted by

H

(S)

and

H

′

(S)

.

Lemma 18 Under the pre eding hypotheses:

1.

H

′

(S)

⊆ H

(S)

; if

Ker(C) = (0)

, this is an equality.

2. If

H

′

(S)

is Hopf,then

H

(S)

isHopf.

Proof. Let

β ∈ N

N

′

. Then:

X

_i

′

(β) =

X

Cα=β

X

i

(α).

Hen e,

H

′

(S)

⊆ H

(S)

. Let us assumethat

Ker(C) = (0)

. Let

α ∈ N

N

. We put

β = Cα

. As

C

is inje tive,

X

′

i

(β) = X

i

(α)

,so

X

i

(α) ∈ H

′

(S)

,and nally

H

(S)

= H

(S)

′

.

Letus assume that

H

′

(S)

isHopf. We denoteby

π

α

the anoni alproje tion on

H

D

_(α)

. For all

β ∈ N

N

′

:

π

α

(X

i

′

(β)) =

(

X

i

(α)

if

Cβ = α,

0

otherwise

.

Moreover, forall

x, y ∈ H

D

:

π

α

(xy) =

X

α

′

_+α

′′

_=α

π

α

′

(x)π

_α

′′

(y).

Thisimpliesthatfor all

α ∈ N

N

∗

,

π

α

H

′

(S)

⊆ H

(S)

. For

β = Cα

:

∆(X

i

(α)) = ∆ ◦ π

α

(X

i

′

(β))

=

X

α

′

_+α

′′

_=α

(π

α

′

⊗ π

_α

′′

) ◦ ∆(X

_i

(β))

∈

X

α

′

_+α

′′

_=α

π

α

′

H

′

_(S)

⊗ π

α

′′

H

_(S)

′

∈ H

(S)

⊗ H

(S)

.

So

H

(S)

isa Hopfsubalgebra of

H

D

.

Weshalloftenrestri t ourselvesto matri es

C

whose rankis

N

′

. One naturalquestion is to

ndthesmallest

N

su h thatthere existsa

N

(13)

If

(S)

isa Hopf SDSE, as

X

i

is an innitespanof trees with roots de orated by

D

i

. Moreover,

in

H

(S)

, any linear span of rooted trees with roots de orated by

D

i

is a linear span of

X

i

(α)

;

hen e, we anwrite the oprodu tof

X

i

underthe form:

∆(X

i

) = X

i

⊗ 1 +

X

α∈N

N

∗

P

i,α

(X

1 , . . . , X

n

) ⊗ X

i

(α).

So

H

(S)

isa ommutative ombinatorial Hopfalgebra inthesense of[21 ℄. Hen e, its dualisthe

envelopingofalgebra ofapre-Liealgebra

g

(S)

. Itisgeneratedbytheelements

f

i

(α)

,dualto the

nonzero

X

i

(α)

;for all

i, j ∈ [M ]

,for all

α, β ∈ N

N

∗

,there existsa s alar

λ

i,j

(α, β)

,su h that:

f

j

(β) ∗ f

i

(α) = λ

i,j

(α, β)f

i

(α + β),

where

∗

isthepre-Lieprodu tof

g

(S)

. When

N = 1

,ifthesystemisfundamental, weproved in

[9℄ thatthese oe ients arepolynomialof degree

≤ 1

. Wehere generalize this asefor any

N

.

3.1 Denition and examples

Denition 19 Let

(g, ∗)

be a pre-Lie algebra. We shall say that it is deg1 if there exists a

basis

(f

i

(α))

i∈[M ],α∈N

N

∗

of

g

,and

A

(i,j)

_{∈ K}

N

,

b

(i,j)

_{∈ K}

, su h that for all

i, j ∈ [M ]

,

α, β ∈ N

N

∗

:

f

j

(β) ∗ f

i

(α) = (A

(i,j)

· α + b

(i,j)

)f

i

(α + β),

where we denote by

·

the usual inner produ t of

K

N

. The elements

A

(i,j)

and

b

(i,j)

willbe alled

the stru ture oe ients of

g

.

Example. We take

M = N

. The pre-Lie produ t of the

N

-dimensional Faà di Bruno Lie

algebra isgiven by:

f

j

(β) ∗ f

i

(α) = (α

j

+ δ

i,j

)f

i

(α + β).

Here,

A

(i,j)

_{= ǫ}

j

, and

b

(i,j)

_{= δ}

i,j

.

Let

(g, ∗)

beadeg1pre-Liealgebraofstru ture oe ients

A

(i,j)

and

b

(i,j)

. Let

λ

i

∈ K − {0}

for all

i ∈ [M ]

. We put

g

i

(α) = λ

i

f

i

(α)

for all

i ∈ [M ]

,

α ∈ N

N

∗

. Then:

g

j

(β) ∗ g

i

(α) = (λ

j

A

(i,j)

· α + λ

j

b

(i,j)

)g

i

(α + β).

So thedeg1 pre-Liealgebra withstru ture oe ients

A

(i,j)

and

b

(i,j)

isisomorphi to thedeg1

pre-Lie algebra withstru ture oe ients

λ

j

A

(i,j)

and

λ

j

b

(i,j)

: we shallsay thatthese two

pre-Lie algebras are equivalent. Our aim in this se tion is to nd all deg1 pre-Lie algebras, up to

equivalen e.

Lemma 20 Let

g

be a ve tor spa e with a basis

(f

i

(α))

i∈[M ],α∈N

N

∗

, elements

A

(i,j)

_{∈ K}

N

,

b

(i,j)

∈ K

, for

i, j ∈ [M ]

. We dene a produ t

∗

on

g

by:

f

j

(β) ∗ f

i

(α) = (A

(i,j)

· α + b

(i,j)

)f

i

(α + β).

Then

(g, ∗)

isa pre-Lie algebra if, andonly if,for all

i, j, k ∈ [M ]

:

(A

(i,j)

= 0

and

b

(i,j)

_{= 0)}

or

(A

(i,j)

_{= A}

(i,k)

_),

(2)

A

(i,j)

b

(j,k)

= A

(i,k)

b

(k,j)

,

(3)

b

(i,j)

b

(j,k)

= b

(i,k)

b

(k,j)

.

(4)

(14)

Proof. Let

α, β, γ ∈ N

N

∗

,

i, j, k ∈ [M ]

. Then:

(f

k

(γ) ∗ f

j

(β)) ∗ f

i

(α) − f

k

(γ) ∗ (f

j

(β) ∗ f

i

(α))

= (A

(i,j)

· α + b

(i,j)

)(A

(j,k)

· β + b

(j,k)

)f

i

(α + β + γ)

− (A

(i,j)

· α + b

(i,j)

)(A

(i,k)

· (α + β) + b

(i,k)

)f

i

(α + β + γ)

= (A

(i,j)

· α + b

(i,j)

)((A

(j,k)

− A

(i,k)

) · β − A

(i,k)

· α + b

(j,k)

− b

(i,k)

)f

i

(α + β + γ).

Consequently:

(g, ∗)

is pre-Lie

⇐⇒ ∀i, j, k ∈ [M ], ∀α ∈ N

N

_∗

,











(A

(i,j)

· α + b

(i,j)

_)(A

(j,k)

_{− A}

(i,k)

_{) = 0,}

(A

(i,k)

· α + b

(i,k)

)(A

(k,j)

− A

(i,j)

) = 0,

(A

(i,j)

_{· α + b}

(i,j)

_)(b

(j,k)

_{− b}

(i,k)

_{− A}

(i,k)

_{· α) = (A}

(i,k)

_{· α + b}

(i,k)

_)(b

(k,j)

_{− b}

(i,j)

_{− A}

(i,j)

_{· α),}

⇐⇒ ∀i, j, k ∈ [M ],











A

(i,j)

_{= 0}

or

A

(j,k)

_{= A}

(i,k)

_,

b

(i,j)

= 0

or

A

(j,k)

_{= A}

(i,k)

_,

A

(i,j)

(b

(j,k)

− b

(i,k)

) − b

(i,j)

A

(i,k)

= A

(i,k)

(b

(k,j)

− b

(i,j)

) − b

(i,k)

A

(i,j)

,

b

(i,j)

)(b

(j,k)

− b

(i,k)

_{) = b}

(i,k)

_(b

(k,j)

_{− b}

(i,j)

_),

whi his equivalent to onditions (2)-(4).

Proposition 21 Let

[M ] = I

0 ⊔ . . . ⊔ I

k

be a partition of

[M ]

, su hthat

I

1 , . . . , I

k

6= ∅

(note

that

I

0

may be empty),

A

1 , . . . , A

k

∈ K

N

,

b

1 , . . . , b

p

∈ K

, and

b

(i)

p

∈ K

for all

i ∈ I

0

and

p ∈ [k]

.

We dene a deg1 pre-Lie algebra by:

A

(i,j)

=

(

A

q

if

j ∈ I

q

, q ≥ 1,

0

if

j ∈ I

0 .

b

(i,j)

=











δ

p,q

b

q

if

j ∈ I

q

, q ≥ 1, i ∈ I

p

, p ≥ 1,

0

if

j ∈ I

0 ,

b

(i)

q

if

j ∈ I

q

, q ≥ 1, i ∈ I

0 .

This pre-Lie algebra will be alled the fundamental deg1 pre-Lie algebra of parameters

I =

(I

0 , . . . , I

k

)

,

A = (A

1 , . . . , A

k

) ∈ M

N,k

(K)

,

b = (b

1 , . . . , b

k

) ∈ K

k

and

b

(i,j)

.

Proof. Dire tveri ationsprovethatthesestru ture oe ientssatisfy onditions(2)-(4).

Remarks.

1. For example,the Faà diBruno pre-Liealgebra ofdimension

N

is fundamental, with

I

j

=

{j}

for all

j ∈ [M ]

,

I

0 = ∅

,

A = I

N

and

b = (1, . . . , 1)

.

2. Thepre-Lieprodu tofsu hapre-Liealgebraisgiveninthefollowingway: if

i ∈ I

p

,

j ∈ I

q

,

α, β ∈ N

N

_∗

,

f

j

(β) ∗ f

i

(α) =











(A

q

· α + δ

p,q

b

q

)f

i

(α + β)

if

p, q 6= 0,

(A

q

· α + b

(i)

q

)f

i

(α + β)

if

p = 0, q 6= 0,

0

if

q = 0.

(15)

Let

g

be adeg1 pre-Lie algebra. We atta hto it anorientedgraph

G(g)

,dened asfollows:

•

Theverti es of

G(g)

aretheelements of

[M ]

.

•

There existsan orientededgefrom

i

to

j

if,and only if,

b

(i,j)

_{6= 0}

.

Weshallwrite

i −→ j

ifthere isan oriented edgefrom

i

to

j

in

G(g)

.

Lemma 22 Let

g

be a fundamental deg1 pre-Lie algebra and let

i −→ j −→ k

in

G(g)

.

Then,in

G(g)

:

j

::

jj

**

k

yy

i

@@

✂

]]❁❁

❁❁

_❁❁

Proof. By ondition (4), if

i −→ j −→ k

, then

b

(i,j)

_b

(j,k)

_{= b}

(i,k)

_b

(k,j)

_{6= 0}

, so

i −→ k

and

k −→ j

. Withthe same argument, as

k −→ j −→ k

,

k −→ k

. As

j −→ k −→ j

,

j −→ j

.

Proposition 23 Let

g

be a fundamental deg1 pre-Lie algebra. The graph

G(g)

has the

fol-lowing stru ture:

1. The set of verti es

[M ]

admitsa partition

[M ] = I

0 ⊔ . . . ⊔ I

k

.

2. For all

1 ≤ p ≤ k

, the omplete subgraph of

G(g)

whose verti es are the elements of

I

p

is,

either omplete, either an isolated vertex.

3. For all

i ∈ I

0

, there exists

D(i) ⊆ [k]

, su h that for all

j ∈ [M ]

,

i −→ j

if, and only if,

j ∈

G

p∈D(i)

I

p

.

4. If

i ∈ I

0

, there isno vertex

j

su h that

j −→ i

.

Proof. Firststep. Let

i

0 ∈ [M ]

. For all

p ≥ 1

,we denoteby

J

p

thesets of verti es

j ∈ [M ]

,

su h that there exists

i

1 , . . . , i

p−1

∈ [M ]

,

i

0 −→ i

1 −→ . . . −→ i

p−1

−→ j

. We put

J =

[

p≥1

J

p

and we onsider a onne ted omponent

K

of thesubgraph of

G(g)

of verti es

J

. Let us prove

that

K

is either omplete, or is an isolated vertex. First, observe that if

j −→ k

in

K

, by

denitionof

J

,there exists

j

p−1

,su hthat

j

p−1

−→ j −→ k

. Bylemma22,

{j, k}

is a omplete

subgraph of

K

.

If

K

hasno edge,asitis onne ted,itisan isolatedvertex; letus assumeithasatleastone

edge

j −→ k

. By the pre edingobservation,

{j, k}

isa omplete subgraph of

K

,so

K

ontains

ompletesubgraphs. Let

L

beamaximal ompletesubgraphof

K

. If

L ( K

,as

K

is onne ted,

there exists

k ∈ K \ L

,

l ∈ L

,su h that

k −→ l

or

l −→ k

. We already observed that

{k, l}

is

omplete inboth ases. Let

l

′

_{∈ L}

. As

L

is omplete,then

k −→ l −→ l

′

and

l

′

_{−→ l −→ k}

: by lemma 22,

k −→ l

′

,and

l

′

_{−→ k}

:

L ⊔ {k}

is omplete, whi h ontradi ts the maximalityof

L

. So

K = L

is omplete.

Se ond step. We denote by

I

0

theset of verti es

i

su h thatthere is no

j

with

j −→ i

. Let

K

be a onne ted omponent of the subgraph of verti es

[M ] \ I

0

. If

k ∈ K

, then

k /

∈ I

0

, so

there exists

j ∈ I

, su h that

j −→ k

. By the rst step,

K

is an isolated vertexor is omplete.

We denoteby

I

1 ⊔ . . . ⊔ I

k

the de ompositionof

[M ] \ I

0

in onne ted omponents. Let

i

0 ∈ I

0

,

and

j

su hthat

i

0 −→ j

. Then

j /

∈ I

0

,sothere exists

p ≥ 1

,

j ∈ I

p

. If

I

p

is an isolated vertex,

then

i

0 −→ j

′

for any

j

′

_{∈ I}

p

. If

I

p

is omplete,forany

j

′

_{∈ I}

p

,then

i

0 −→ j −→ j

′

,so

i

0 −→ j

′

by lemma 22. Denotingby

D(i

0 )

theset of

p

su h that there exists

j ∈ I

p

with

i

0 −→ j

, then

(16)

Theorem 24 Let

g

be a deg1 pre-Lie algebra. Up to an equivalen e, it isthe dire t sum of

fundamental deg1 pre-Lie algebras.

Proof. First ase. We assume rst that

G(g)

is omplete. Let us hoose

i

0 ∈ I

. For all

j

,

b

(i

0 ,j)

_{6= 0}

: up to an equivalen e,we assume that

b

(i

0 ,j)

_{= 1}

for all

j

. Condition (4), with

i = i

0

be omes: for all

j, k

,

b

(j,k)

_{= b}

(k,j)

. Still by ondition (4), as

b

(j,k)

_{= b}

(k,j)

_{6= 0}

, for all

i, j, k

,

b

(i,j)

= b

(i,k)

. Hen e, for all

i, j

:

b

(i,j)

= b

(i,i

0 )

_{= b}

(i

0 ,i)

_{= 1.}

Condition (2)be omes: for all

i, j, k

,

A

(j,k)

_{= A}

(i,k)

. We denote by

A

(k)

the unique ve tor su h

that

A

(i,k)

_{= A}

(k)

for all

i

. Condition (3) be omes: for all

j, k

,

A

(k)

_{= A}

(j)

. So there exists

a unique ve tor

A

, su h that for all

i, j

,

A

(i,j)

_{= A}

. Finally,

g

is a fundamental deg1 pre-Lie

algebra,with

[M ] = I

1

.

Se ond ase. We assume that

G(g)

is onne ted. We use the notations of proposition 23.

If there is an edge from

i

to

j

, by ondition (2), for all

k

,

A

(j,k)

_{= A}

(i,k)

. By onne tivity,

thereexistsve tors

A

(k)

,su h thatfor all

i, j, k

,

A

(i,k)

_{= A}

(j,k)

_{= A}

(k)

. We onsider thepre-Lie

subalgebra

g

p

of

g

generated by the elements

f

i

(α)

,

i ∈ I

p

,

α ∈ N

N

∗

. They are deg1 pre-Lie

algebras; if

p ≥ 1

and

I

p

is not a single element, then the graph asso iated to

g

p

is omplete.

By therst step, up to an equivalen e, we an assume that

A

(k)

is onstant on

I

p

: thereexists

a ve tor

A

p

su h that

A

(k)

_{= A}

p

for all

k ∈ I

p

,

p ≥ 1

. Moreover, there existsa s alar

b

p

, su h

that

b

(i,j)

_{= b}

p

for all

i, j ∈ I

p

,if

p ≥ 1

.

Let

j ∈ I

0

. By onne tivityof

G(g)

,and bydenitionof

I

0

,thereexists

k

su h that

j −→ k

,

so

b

(j,k)

_{6= 0}

and

b

(k,j)

_{= 0}

. By ondition (3),

A

(i,j)

_{= 0}

for all

i

,so

A

(j)

_{= 0}

if

j ∈ I

0

.

By denition of the graph, if

i ∈ I

p

,

j ∈ I

q

,

p, q ≥ 1

and

p 6= q

, then

b

(i,j)

_{= 0}

. If

j ∈ I

0

,

then

b

(i,j)

_{= 0}

for all

i

. Let

i ∈ I

0

,

j, k ∈ I

p

,

p ≥ 1

. If

j = k

, then

b

(i,j)

_{= b}

(i,k)

. If

j 6= k

,then

I

p

is omplete and

j −→ k

in

G(g)

:

b

(j,k)

_{= b}

(j,k)

_{6= 0}

. By ondition (4),

b

(i,k)

_{= b}

(i,j)

. So there exists

b

(i)

p

,su hthat

b

(i,j)

_{= b}

(i)

p

for all

j ∈ I

p

. Finally,thestru ture oe ients aregiveninthe

following arrays:

A

(i,j)

:

i \ j I

0 I

1 . . .

I

k

I

0

0 A

1 . . . A

k

I

1

0

. . . . . . . . .

0

. . . . . .

I

k

0 A

1 . . . A

k

b

(i,j)

:

i \ j I

0 I

1 . . .

I

k

I

0

0 b

(i)

1 . . . b

(i)

k

I

1

0 b

1 . . .

0

. . .

0

. . . . . . . . .

I

k

0

0 . . .

b

k

Sothis isa fundamental deg1 pre-Liealgebra.

General ase. Let

G

1 , . . . , G

l

be the onne ted omponents of

G(g)

. By the se ond step,

up to an equivalen e of

g

, the pre-Lie subalgebra of

g

orresponding to these subgraphs are

fundamental deg1 pre-Liealgebras.

Firstsub ase. Letusassumethatthereexists

i ∈ G

p

,

j ∈ G

q

,with

p 6= q

,su hthat

A

(i,j)

_{6= 0}

.

By ondition (2),for all

k

,

A

(j,k)

_{= A}

(i,k)

. By onne tivityof

G

p

and

G

q

,we dedu ethatfor all

i

′

∈ G

p

,

j

′

_{∈ G}

q

,for all

k

,

A

(i

′

_,k)

= A

(j

′

,k)

.

Se ond sub ase. Let us assume thatfor all

i ∈ G

p

,

j ∈ G

q

,

A

(i,j)

_{= 0}

. As

b

(i,j)

_{= 0}

,for all

α, β ∈ N

N

_∗

,forall

i ∈ G

p

,

j ∈ G

q

,

f

j

(β) ∗ f

i

(α) = 0

.

Wedene an equivalen e relation

∼

on

[M ]

inthefollowing way:

i ∼ j

iffor all

k

,

A

(i,k)

₌

A

(j,k)

. Therstsub aseimpliesthattheequivalen e lassesaredisjoint unionof

G

p

: we denote them by

H

1 , . . . , H

n

. The se ond step gives that the orresponding subalgebras

g

1 , . . . g

n

are