CHEVALLEY COHOMOLOGY FOR KONTSEVICH'S GRAPHS

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CHEVALLEY COHOMOLOGY FOR KONTSEVICH’S GRAPHS

Didier Arnal, Angela Gammella, Mohsen Masmoudi

To cite this version:

Didier Arnal, Angela Gammella, Mohsen Masmoudi. CHEVALLEY COHOMOLOGY FOR KONT-

SEVICH’S GRAPHS. Pacific Journal of Mathematics, 2005, 218, pp.201-239. �hal-00092521�

CHEVALLEY COHOMOLOGY FOR KONTSEVICH’S GRAPHS

DIDIER ARNAL, ANGELA GAMMELLA, AND MOHSEN MASMOUDI

Abstract. We introduce the Chevalley cohomology for the graded Lie algebra Tpoly(R^d) of polyvector fields on R^d. This cohomology occurs naturally in the problem of construction and classification of formalities on the space R^d. Consi- dering only graph formalities,i.e. formalities defined with the help of graphs like in the original construction of Kontsevich in [K1, K2], we define, as in [AM] for the Hochschild cohomology, the Chevalley cohomology directly on spaces of graphs.

More precisely, observing first a noteworthy property for the Kontsevich’s explicit formality onR^d, we restrict ourselves to graph formalities with that property. With this restriction, we obtain some simple expressions of the Chevalley coboundary operator, especially, we can write this cohomology directly on the space of purely aerial, non-oriented graphs. We give finally examples and applications.

1. Introduction

In this article, we study formalities on the space R

. A formality is a formal non linear mapping F between two formal graded manifolds T

( R

)[1] and D

( R

)[1], intertwinning their natural vector fields Q and Q

.

Here, T

( R

)[1] (resp. D

( R

)[1]) denotes the space of polyvector fields (resp.

polydifferential operators) on R

graded by |α| = degree(α) = k − 2 if α is a k-vector field (resp. |D| = m −2 if D is an m-differential operator) -[1] stands for this choice of translation on degrees- and viewed as a formal manifold (see [K1, K2]). The monomial functions α

.α

. . . . .α

on T

( R

) are elements of the space S

(T

( R

)[1]) of symmetric n-polyvector fields on T

( R

)[1] (that means α

.α

= (−1)

α

.α

).

The manifold T

( R

)[1] is equipped with the formal bilinear vector field Q = Q

, defined with the help of the Schouten bracket [ , ]

:

Q

(α

.α

) = (−1)

[α

, α

]

. Similarly, D

( R

)[1] is equipped with the formal vector field

Q

= Q

+ Q

, defined by

Q

(D

) = −d

D

, Q

(D

.D

) = (−1)

[D

, D

]

.

Date: 08/02/04.

1

Here, d

denotes the usual Hochschild coboundary operator: if D is a m-differential operator,

d

D(f

, . . . , f

) = f

D(f

, . . . , f

) − D(f

f

, . . . , f

) + · · · + + (−1)

D(f

, . . . , f

)f

and [ , ]

is the Gerstenhaber bracket.

A formality F is then given by a sequence of mappings F

: F

: S

T

( R

)[1]

→ D

( R

)[1], homogeneous with degree 0 and such that the ‘formality equation’:

d

(F

)(α

. . . . .α

) = 1 2

X

ItJ={1,...,n},|I|6=0,|J|6=0

ε

(I, J)Q

F

(α

).F

(α

)

− 1 2

X

k6=`

ε

(k`, 1 . . . k` . . . n)F b

Q

(α

.α

).α

. . . . . α [

α

. . . . .α

holds. Here, if I = {i

< · · · < i

}, the notation α

means α

. . . . .α

.

We shall impose moreover that F

is the canonical mapping F

from T

( R

) to D

( R

) defined by

F

(ξ

∧ . . . ∧ ξ

)(f

, . . . , f

) = 1 n!

X

σ∈Sn

ε(σ)

n

Y

i=1

ξ

(f

), for any vector fields ξ

and any functions f

.

Let us now choose a coordinate system (x

) on R

. In [K2], M. Kontsevich built explicitly a formality U for R

, using families of graphs drawn on configuration spaces.

A graph Γ has aerial and terrestrial vertices. The aerial vertices are labelled p

, . . . , p

and elements of the Poincar´ e half plane

H = {z ∈ C , =(z) > 0}.

The terrestrial vertices q

< · · · < q

are on the real line. The edges of Γ are arrows starting from an aerial vertex, ending to a terrestrial or an aerial vertex; there are no arrow of the form p ~

p

and no multiple arrow. If we fix a total ordering O on the edges of Γ, we get an oriented graph (Γ, O). We say that O is compatible if, for all i, the arrows starting from p

are before those starting from p

and denote by GO

the set of oriented graphs (Γ, O) with n labelled aerial vertices, m labelled terrestrial vertices and O compatible.

Let us consider such an oriented graph (Γ, O) in GO

. Let us also suppose there

are k

edges starting from the vertex p

(1 ≤ i ≤ n). In [K2], Kontsevich defines a

natural operator B

assigning a m-differential operator B

(α

⊗ · · · ⊗ α

) to a

n-uple (α

, . . . , α

) of polyvector fields α

. This operator vanishes except if, for each

i, α

belongs to T

( R

) (α

is a k

-polyvector field). Let us first consider all the

multi-indexes (t

, . . . , t

) with |k| = P

k

and 1 ≤ t

≤ d for all 1 ≤ r ≤ |k|. We denote by end(a) the set of edges arriving on the vertex a and, if these edges are e

, . . . , e

, by ∂

the operator

∂

= ∂

∂x

1

. . . ∂x

. Then, we denote by star(p

) the ordered set e

< · · · < e

ki

of edges starting from p

and, if α

is a k

-vector field, by α

the following component of α

:

α

= α

. Finally, if, for each i, α

is a k

-vector field,

B

(α

⊗ · · · ⊗ α

)(f

, . . . , f

) = X

1≤t1,...,t_|k|≤d n

Y

i=1

∂

α

m

Y

j=1

∂

f

.

B

will be called the graph operator associated with (Γ, O).

The explicit formality U of Kontsevich can now be written as a sum U = X

n

U

with

U

= X

m≥0

X

(Γ,O)∈GO_n,m

w

B

,

where the coefficient w

, the weight of (Γ, O) is an integral on a compactified configuration space. To be precise, for 2n + m − 2 ≥ 0, let Conf (n, m) be the space of n distinct points p

in H and m distinct points q

on the real line ∂H. Let us act on Conf (n, m) by the group G of transformations z 7→ az + b (a > 0, b real). Consider the quotient space

C

= Conf (n, m)/G.

In [K2], Kontsevich associates with each oriented graph (Γ, O) the following form ω

on C

:

ω

= 1 k!

n

^

i=1

dΦ

1

∧ . . . ∧ dΦ

ki

here {e

< e

< · · · < e

} denotes the ordered set star(p

) formed by the k

edges starting from p

, k! := k

! . . . k

! and, if e

= p ~

a,

Φ

`

= Φ

= 1 2π Arg

a − p

a − p ¯

.

The weight w

is then defined as the value of the integral ω

on the connected

component C

of C

for which q

< · · · < q

.

In this work, we are looking for graph formalities, i.e. formalities on the space R

of the form F = P

n

F

where the F

are homogeneous mapping (with degree 0) of the form

F

= X

m≥0

X

(Γ,O)∈GOn,m

c

B

,

with real coefficients c

. We shall use the notation F

= B

where γ

is the linear combination

γ

= X

m≥0

X

(Γ,O)∈GOn,m

c

(Γ, O).

Let us assume now we found F

, . . . , F

(with F

= F

= U

) such that the formality equation holds up to order n − 1. Then, the next term F

, if it exists, is a solution of an equation:

d

◦ F

= E

, that is

d

(F

(α

. . . . .α

)) = E

(α

. . . . .α

) = E

(α

),

where E

(α

) is a Hochschild cocycle. It is well-known that the Hochschild cohomology is localized in T

( R

)[1]. More precisely, the total skewsymmetrization

a

◦ E

(α

) of E

(α

) is a polydifferential operator of order 1, . . . , 1, that is the image by F

of a polyvector field. Moreover, there exists an operator A

such that

E

(α

) = (

◦ E

+ d

◦ A

)(α

).

Put now

ϕ

= F

◦

◦ E

, that is

ϕ

(α

) = F

E

(α

)

! ,

then ϕ

: S

(T

( R

)[1] → T

( R

)[1] is homogeneous with degree |ϕ

| = 1.

In section 2, we define the Chevalley coboundary operator ∂ on T

( R

). Then, we show that the mapping ϕ

described above is a Chevalley cocycle, and, if it is a coboundary i.e. ϕ

= ∂φ

, we can add to F

a Hochschild coboundary so that

a

(E

) vanishes and thus find a F

for which the formality equation holds up to order n.

In Section 3, we etablish a remarkable property for the Kontsevich’s weights. For any graph Γ (with k

edges starting from p

), denote by ∆ the purely aerial graph obtained by cutting the legs p ~

q

and the feet q

of Γ and by `

the number of aerial edges starting from p

. We prove that:

a

X

(Γ,O)∈GO⁽¹⁾_n,m

w

B

= X

(∆,O∆)∈GO⁽⁰⁾_n

w

1 m!

X

GO⁽¹⁾n,m3(Γ,O)⊃(∆,O_∆)

`!

k! ε(Γ)B

.

Here GO

denotes the subspace of GO

formed by the oriented graphs having exactly one leg for each foot, GO

is the set of purely aerial oriented graphs (∆, O

) with n aerial vertices and O

compatible and ε(Γ) is an explicit sign depending only on Γ.

This property suggests us to study what we call K-graph formalities. A K-graph formality up to order n is a graph formality F at order n−1 such that ϕ

= F

◦

◦E

has the form ϕ

= X

(∆,O∆)∈GO⁽⁰⁾_n

c

C

with real coefficients c

and where

C

= X

m≥0

1 m!

X

GOn,m⁽¹⁾ 3(Γ,O)⊃(∆,O_∆)

`!

k! ε(Γ)B

.

In Section 4, we first give some simple expressions of our Chevalley coboundary operator. Then, we restrict ourselves to K-graph formalities and study the Chevalley cohomology related to the question of building such formalities.

In Section 5, we show that the coboundary operator ∂ can be written directly on the aerial part of the graphs.

We devote the last section to explicit computations and applications. In particular, we prove the triviality of the cohomology for small value of n and give the restriction of the cohomology for linear formalities.

2. Chevalley cohomology and formalities

In this section, we first define a graded Chevalley cohomology in a general algebraic setting i.e. for cochains C : S

(

[1]) →

[1] where

is a graded Lie algebra and

M

a graded

-module. In fact two Chevalley coboundary operators are naturally associated with the formality equation for R

. The first one, say ∂

, is obtained by endowing D

( R

) with a T

( R

)-graded module structure, cochains are mappings C : S

T

( R

)[1]

→ D

( R

)[1]. The other one, ∂, is obtained by considering T

( R

) as a graded module over itself, cochains are mapping C : S

T

( R

)[1]

→ T

( R

)[1]. Using both ∂ and ∂

, we show that the obstructions to formalities can be interpreted as cocycles for ∂.

2.1. Chevalley cohomology.

Let (

, [ , ]) be a graded Lie algebra and

a graded module over

. For reasons

of homogeneity, we prefer to work with

[1] and

[1]. Thus, we replace [ , ] and the

action of

on

respectively by [ , ]

and [ , ]

, defined for homogeneous α, β in

[1]

and m in

[1] with degree |α| (resp. |β|, |m|) by

[α, β]

= (−1)

[α, β]

[α, m]

= (−1)

α.m.

The space C

(

,

) of n-cochains consists of mappings C from S

(

[1]) to

[1]. The Chevalley coboundary ∂C of a n-cochain C, homogeneous with degree |C|, is the n + 1-cochain defined as

∂C (α

. . . . .α

) =

n+1

X

i=1

(−1)

ε

(i, 1 . . .ˆ ı . . . n + 1)[α

, C(α

. . . . α ˆ

. . . .α

)]

− 1 2

X

i6=j

ε

(ij, 1.. ı . . . , n b + 1)(−1)

C([α

, α

]

.α

. . . α d

α

. . . .α

).

Here the α

are homogeneous elements of

, |α

| denotes the degree of α

in

[1] and for any permutation σ of {1, . . . , n}, ε

(σ) is the sign of σ in the graded sense. We shall denote by C

(

,

) the subspace of C

(

,

) formed by the n-cochains of degree q and by H

(

,

) the corresponding cohomology group. Note that ∂ sends C

(

,

) into C

(

,

).

Extending usual techniques to the graded case, it is possible to prove Lemma 2.1. (See [Ga] for an explicit computation)

The operator ∂ is a cohomology operator i.e. ∂

= ∂ ◦ ∂ = 0.

Let us now return to the graded Lie algebras (T

( R

), [ , ]

) and (D

( R

), [ , ]

) where [ , ]

is the Schouten bracket and [ , ]

the Gerstenhaber bracket. Let us precise first our conventions for these spaces and brackets.

Let α be a k-vector field and {e

} the canonical basis of R

. We put:

α = X

j1,...,jk

α

e

⊗ · · · ⊗ e

= X

j1<j2<···<j_k

α

e

∧ . . . ∧ e

= 1 k!

X

j1...jk

α

e

∧ . . . ∧ e

.

For any k

-vector field α

and k

-vector field α

(the degree of α

is k

− 1 in T

( R

)), we define first a polyvector field α

• α

with its components

α

• α

= 1 k

!k

!

X

σ∈S_k

1+k2−1

ε(σ)

k1

X

`=1

(−1)

d

X

s=1

α

∂

α

.

Now, [α

, α

]

can be written as

[α

, α

]

= (−1)

α

• α

− (−1)

α

• α

.

Note that this choice for the Schouten bracket is denoted [ , ]

in [AMM] and [MT].

On the other hand, for any m

-differential operator D

and any m

-differential operator D

(the degree of D

is m

− 1 in D

( R

)), we may write [D

, D

]

in the form

[D

, D

]

= D

◦ D

− (−1)

D

◦ D

where

D

◦ D

(f

, . . . , f

) =

m1

X

j=1

(−1)

D

(f

, . . . , f

, D

(f

, . . . , f

), f

, . . . , f

).

Recall the canonical mapping F

from T

( R

) into D

( R

): each k-vector field α can be viewed as a k-differential operator F

(α) of order 1,. . . ,1:

F

(α)

(f

, . . . , f

) =< α, df

∧ . . . ∧ df

>= 1

k! α

∂

f

. . . ∂

f

. Now we consider the following action of T

( R

):

α.D =

◦ [F

(α), D]

∀α ∈ T

( R

), ∀D ∈ D

( R

)

where

denotes the usual skewsymmetrization of differential operators and [ , ]

the Gerstenhaber bracket. This action defines a T

( R

)-graded module structure on D

( R

). Indeed, one can prove

Proposition 2.2.

The following equalities

i)

◦ [D

, D

]

=

◦ [D

,

◦ D

]

ii) F

([α

, α

]

) =

◦ [F

(α

), F

(α

)]

iii)

◦ [F

([α

, α

]

), D]

=

◦ [F

(α

),

◦ [F

(α

), D]

]

−

− (−1)

◦ [F

(α

),

◦ [F

(α

), D]

]

hold for any D

, D

, D in D

( R

), any k

-vector field α

and k

-vector field α

in T

( R

). Especially, item (iii) means

[α

, α

]

.D = α

.(α

.D) − (−1)

α

.(α

.D)

and thus D

( R

) is a T

( R

)-module.

Let us now endow D

( R

) with the T

( R

)-graded structure described above.

If C : V

T

( R

)

= S

T

( R

)[1]

→ D

( R

)[1] is a mapping homogeneous with degree |C|, we can define its Chevalley coboundary ∂

C. The latter can be written using the vector fields Q and Q

, associated respectively with T

( R

) and D

( R

):

∂

C(α

. . . . .α

) =

n+1

X

i=1

(−1)

ε

(i, 1 . . . ˆ ı . . . n + 1)

◦ Q

F

(α

).C(α

. . . . α ˆ

. . . .α

)

− 1 2

X

i6=j

ε

(ij, 1 . . . ı . . . n b + 1)(−1)

C

Q

(α

.α

).α

. . . α d

α

. . . .α

.

To simplify the writing, we will sometimes write α

instead of F

(α

).

On the other hand, considering T

( R

) as a graded module over itself, one can define the Chevalley cohomology for T

( R

) . If C : S

T

( R

)[1]

→ T

( R

)[1]

is an n-cochain, homogeneous with degree |C|, its coboundary ∂C is:

∂C(α

. . . . .α

) =

n+1

X

i=1

(−1)

ε

(i, 1 . . .ˆ ı . . . n + 1)Q

α

.C(α

. . . . α ˆ

. . . .α

)

− 1 2

X

i6=j

ε

(ij, 1 . . . ı . . . n b + 1)(−1)

C

Q

(α

.α

).α

. . . α d

α

. . . .α

.

Remark 2.1.

For any ϕ : S

T

( R

)[1]

→ T

( R

)[1], we have:

∂

(F

◦ ϕ) = F

◦ ∂ϕ.

2.2. Obstruction to formalities.

The two Chevalley coboundary operators ∂ and ∂

enable us to reformulate the for- mality equation. Indeed, suppose we want to construct a formality F from T

( R

) to D

( R

). We need thus to solve recursively the formality equation (see [K, AMM]

for notations)

d

(F

)(α

. . . . .α

) = 1 2

X

ItJ={1,...,n},|I|≥1,|J|≥1

ε

(I, J )Q

F

(α

).F

(α

)

− 1 2

X

k6=`

ε

(k`, 1 . . . k` . . . n)F b

Q

(α

.α

).α

. . . . . α [

α

∂ . . . .α

,

where d

is the Hochschild coboundary operator.

Let us impose the first component F

to be F

. Assume there exist mappings F

, . . . , F

, homogeneous with degree 0, and satisfying the formality equation up to order n − 1. Denote by E

the right hand side of the equation at the order n.

Then E

is a Hochschild cocycle: d

E

= 0 (see [AM] for instance). Thus E

=

◦ E

+ d

C,

where

◦ E

is a differential operator of order 1,. . . , 1 and E

is a coboundary if and only if

◦ E

= 0. But:

a

◦ E

(α

. . . . .α

) = ∂

F

(α

. . . . .α

) +

R

(α

. . . . .α

), where

R

(α

. . . . .α

) = 1 2

X

ItJ={1,...,n},|I|≥2,|J|≥2

ε

(I, J )Q

F

(α

).F

(α

) .

It follows directly from this expression that the degree of R

and

◦ R

are both 1, |R

| = |

◦ R

| = 1. Moreover,

Theorem 2.3.

The skewsymmetrization

◦ E

of E

can be identified through the inverse mapping of F

with a ∂-cocycle. If this cocycle is exact, we can find F

and F

, homogeneous with degree 0, such that F

, . . . , F

, F

, F

satisfy the formality equation up to order n.

Proof: The proof proceed in three steps.

1. Let us first see that

◦ R

is a cocycle for ∂

:

∂

R

(α

. . . . .α

) =

=

n+1

X

i=1

(−1)

ε

(i, 1 . . . ˆ ı . . . n + 1)

Q

α

.

R

(α

. . . . . α b

. . . . .α

) + 1

2 X

i6=j

ε

(ij, 1 . . . ı . . . n b + 1)

R

Q

(α

.α

).α

. . . . . α d

α

. . . . .α

= 1 2

n+1

X

i=1

(−1)

ε

(i, 1 . . . ˆ ı . . . n + 1) X

ItJ={1...ˆı...n+1},|I|≥2,|J|≥2

ε

(I, J )

a

Q

α

.

Q

(F

(α

).F

(α

))

+ 1 4

X

i6=j

ε

(ij, 1 . . . ı . . . n b + 1) X

ItJ={0,1...bı...n+1},|I|≥2,|J|≥2

ε

(I, J)

a

Q

F

(α

).F

(α

)

= 1

2 (I) + 1 4 (II),

here α

:= Q

(α

.α

), ε

:= ε

and ε

:= ε

. The first term is:

(I) =

n+1

X

i=1

X

ItJ={1...ˆı...n+1}

|I|≥2,|J|≥2

(−1)

ε

(i, I, J)

Q

a

Q

(F

(α

).F

(α

)).α

.

By Proposition 2.2,

Q

satisfies the graded Jacobi identity, thus:

(I) = −

n+1

X

i=1

X

ItJ={1...ˆı...n+1}

|I|≥2,|J|≥2

(−1)

ε

(i, I, J )

Q

a

Q

(F

(α

).α

).F

(α

)

− X

ItJ={1...ˆı...n+1}

|I|≥2,|J|≥2

ε

(i, I, J)

Q

a

Q

(α

.F

(α

)).F

(α

)

= − 2 X

ItJ={1...ˆı...n+1}

|I|≥2,|J|≥2

ε

(i, I, J)

Q

a

Q

(α

.F

(α

)).F

(α

) .

Similarly, the second term is:

(II) = X

i6=j

ε

(ij, 1 . . . ı . . . n b + 1) X

ItJ={0,1...ı...n+1}b

|I|≥2,|J|≥2

ε

(I, J)

Q

F

(α

).F

(α

)

= X

i6=j

X

I=I1t{0}

I1tJ={1...ı...n+1}b

ε

(ij, I

, J)

Q

F

(Q

(α

.α

).α

).F

(α

)

+ X

i6=j

X

J=J1t{0}

ItJ₁={1...ı...n+1}b

ε

(ij, 1 . . . ı . . . n b + 1)ε

(I, {0}, J

)

a

Q

F

(α

).F

(Q

(α

.α

).α

)

= X

i6=j

h X

I=I1t{0}

I1tJ={1...bı...n+1}

ε

(ij, I

, J)

Q

F

(Q

(α

.α

).α

).F

(α

) +

+ X

J=J1t{0}

ItJ1={1...bı...n+1}

ε

(ij, I, J

)(−1)

Q

F

(α

).F

(Q

(α

.α

).α

) i

= 2 X

i6=j

X

I=I1t{0}

I1tJ={1...bı...n+1}

ε

(ij, I

, J)

Q

F

(Q

(α

.α

).α

).F

(α

) .

Putting (I) and (II) together, we get:

∂

(

R

)(α

. . . . .α

) = 1

2 (I) + 1 4 (II)

= X

I⁰tJ={1...n+1}

|J|≥2,|I⁰|≥3

ε

(I

, J) h

− X

i∈I⁰(I⁰=It{i})

ε

(i, I)

Q

a

Q

(α

.F

(α

)).F

(α

)

+ 1 2

X

i6=j∈I⁰,I⁰=I1t{ij}

I=I1t{0}

ε

1

(ij, I

)

Q

F

(Q

(α

.α

).α

).F

(α

) i

.

Now, Proposition 2.2 and the definition of ∂

yield:

(∗) ∂

(

R

)(α

. . . . .α

) = − X

I⁰tJ={1...n+1}

|J|≥2, |I⁰|≥3

ε

(I, J )

Q

∂

F

(α

).F

(α

) .

On the other hand, since the formality equation holds up to order n − 1, we have:

∂

F

+

R

=

(E

) =

(d

(F

)) = 0 ∀p ≤ n − 1.

But |I

| ≤ n − 1 for all I

in the expression (∗), thus:

−∂

F

(α

) =

R

(α

) = 1 2

X

StT=I⁰,|S|≥2,|T|≥2

ε

(S, T )

Q

(F

(α

).F

(α

)).

Finally, (∗) becomes:

∂

(

R

)(α

. . . . .α

) =

= 1 2

X

StTtJ={1...n+1}

|S|≥2,|T|≥2,|J|≥2

ε

(S ∪ T, J )ε

(S, T )

Q

a

Q

(F

(α

).F

(α

)).F

(α

)

= 1 2

X

StTtJ={1...n+1}

|S|≥2,|T|≥2,|J|≥2

ε

(S, T, J )

Q

a

Q

(F

(α

).F

(α

)).F

(α

)

:= S

.

But thanks to the Jacobi identity, S

vanishes. Hence ∂

(

R

) = 0 and ∂

(

E

) = 0.

2. Let us put

ϕ

= F

◦

◦ E

. Since

∂

(

◦ E

) = ∂

F

(ϕ

) = F

(∂ϕ

) = 0,

ϕ

is a cocycle for ∂ .

3. Let us assume that ϕ

= ∂φ

where φ

: S

T

( R

)[1]

→ T

( R

)[1].

Of course, d

F

(φ

) = 0. Therefore, the mappings F

= F

, . . . , F

= F

, F

= F

− F

◦ φ

satisfy the formality equation up to order n − 1. Moreover, the Hochschild cocycle E

corresponding to these F

satisfies

a

◦ E

=

◦ E

− ∂

(F

◦ φ

) =

◦ E

− F

(∂φ

) = 0.

We are now able to find F

such that E

= d

F

. This ends the proof.

3. Skewsymmetrization

The aim of this section is the proof of a noteworthy property of the Kontsevich’s weights and the definition of K-graph formalities.

3.1. Skewsymmetrization and 1-graphs.

Let us first consider a m-differential operator D on R

, vanishing on constants. We can decompose D as

D = D

+ D

,

where D

has order 1 in each of its arguments and D

has order larger than 1 for at least one of its arguments. The skewsymmetrization

(D) of D i.e.

a

(D)(f

, . . . , f

) = 1 m!

X

σ∈Sm

ε(σ)D(f

, . . . , f

) satisfies

a

(D) =

(D

) +

(D

) and therefore

a

(D)

=

(D

).

We assume D to be defined with the help of graphs:

D

= X

Γ

c

B

(α

⊗ · · · ⊗ α

)

where the c

are real. To compute a(D)

, we need only to consider D

1,...,αn)

= X

Γ∈G⁽¹⁾

c

B

(α

⊗ · · · ⊗ α

),

where G

denotes the family of 1-graphs i.e. graphs having exactly one leg for each foot.

However, as in [K2], to define B

, we need to choose a total ordering O on the set

E(Γ) of edges of Γ. To be exact, we first choose a labelling on the aerial vertices of Γ,

say p

, . . . , p

. Then we put away the arrows starting from p

, from p

. . . and finally

from p

. We get a total ordering of E(Γ) which is compatible with the ordering