---
abstract: 'Let $M$ be a smooth manifold, ${{\mathcal{S}}}$ the space of polynomial on fibers functions on $T^*M$ (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, ${\mathrm{Vect}}(M)$, of vector fields on $M$ with coefficients in the space of linear differential operators on ${{\mathcal{S}}}$. This cohomology space is closely related to the ${\mathrm{Vect}}(M)$-modules, ${{\mathcal{D}}}_\l(M)$, of linear differential operators on the space of tensor densities on $M$ of degree $\l$.'
author:
- 'P.B.A. Lecomte [^1]'
- 'V.Yu. Ovsienko [^2]'
title: Cohomology of the vector fields Lie algebra and modules of differential operators on a smooth manifold
---

ł

.1truein \[section\] \[thm\][Lemma]{}\[thm\][Corollary]{}\[thm\][Proposition]{}\[thm\][Example]{}\[thm\][Remark]{}\[thm\][Definition]{}

Introduction and the Main Theorem
=================================

Let $M$ be a smooth manifold and ${\mathrm{Vect}}(M)$ the Lie algebra of vector fields on $M$.

The main purpose of this article is to study the cohomology of ${\mathrm{Vect}}(M)$ with coefficients in the space of linear differential operators acting on tensor fields. This cohomology is, actually, a natural generalization of the Gelfand-Fuchs cohomology (i.e., of ${\mathrm{Vect}}(M)$-cohomology with coefficients in the modules of tensor fields on $M$).

The problem of computation of such cohomology spaces naturally arises if one considers *deformations* of the ${\mathrm{Vect}}(M)$-module structure on the space of tensor fields.

The general theory of deformations of Lie algebra modules is due to Nijenhuis and Richardson [@NR; @Ric]. Let ${\mathfrak{g}}$ be a Lie algebra and $V$ a ${\mathfrak{g}}$-module, then the problem of deformation of the ${\mathfrak{g}}$-module structure on $V$ is related to the cohomology spaces: ${\mathrm{H}}^1({\mathfrak{g}};{\mathrm{End}}(V))$ and ${\mathrm{H}}^2({\mathfrak{g}};{\mathrm{End}}(V))$. More precisely, the first cohomology space classifies *infinitesimal* deformation, while the second one contains the obstructions to integrability of a given infinitesimal deformation.

The origin of our investigation is related to the space of scalar linear differential operators on $M$ viewed as a module over ${\mathrm{Vect}}(M)$. It is quite clear a-priori that this module should be considered as a deformation of the corresponding module of *symbols* (i.e., of polynomial on fibers functions on $T^*M$). We are, therefore, led to study the first cohomology of ${\mathrm{Vect}}(M)$ with coefficients in the ${\mathrm{Vect}}(M)$-module of operators on the space of symbols.

Differential operators on symmetric contravariant tensor fields {#SkSpace}
---------------------------------------------------------------

Consider the space, ${{\mathcal{S}}}(M)$ (or ${{\mathcal{S}}}$ for short), of symmetric contravariant tensor fields on $M$ (i.e., ${{\mathcal{S}}}=\Gamma(STM)$). As a ${\mathrm{Vect}}(M)$-module it is isomorphic to the space of smooth functions on $T^*M$ polynomial on the fibers. Therefore, ${{\mathcal{S}}}$ is a Poisson algebra with a natural graduation given by the decomposition $$\label{decomp}
{{\mathcal{S}}}=\bigoplus_{k=0}^\infty{{\mathcal{S}}}_k,$$ where ${{\mathcal{S}}}_k$ is the space of $k$-th order tensor fields. Obviously, ${{\mathcal{S}}}_0$ is isomorphic to $C^\infty(M)$ and ${{\mathcal{S}}}_1$ to ${\mathrm{Vect}}(M)$. The Poisson bracket on ${{\mathcal{S}}}$ is usually called the (symmetric) Schouten bracket (see e.g. [@Fuc]).

The action of $X\in{\mathrm{Vect}}(M)$ on ${{\mathcal{S}}}$ is given by the Hamiltonian vector field $$\label{Hamilton}
L_X=
\frac{\partial{}X}{\partial\xi_i}\,
\frac{\partial}{\partial{}x^i}
-
\frac{\partial{}X}{\partial{}x^i}\,
\frac{\partial}{\partial\xi_i}\,,$$ where $(x,\xi)$ are local coordinates on $T^*M$ (we identified $X$ with the first-order polynomial $X=X^i\xi_i$; the summation over repeated indices is understood).

Let us introduce the space, ${{\mathcal{D}}}({{\mathcal{S}}})$, of all linear differential operators on ${{\mathcal{S}}}$. This space is a ${\mathrm{Vect}}(M)$-module with a filtration $$\label{filtr1}
{{\mathcal{D}}}^0({{\mathcal{S}}})
\subset
{{\mathcal{D}}}^1({{\mathcal{S}}})
\subset\cdots\subset
{{\mathcal{D}}}^r({{\mathcal{S}}})
\subset\cdots,$$ where ${{\mathcal{D}}}^r({{\mathcal{S}}})$ is the space of $r$-th order differential operators.

In this article we compute the first cohomology space $$\label{DefCohom}
{\mathrm{H}}^1({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}})).$$ of ${\mathrm{Vect}}(M)$ acting on ${{\mathcal{D}}}({{\mathcal{S}}})$.

Note that for $M=S^1$ this computation has been done in [@LO1; @BO] see also [@FF] for the case of the Lie algebra of formal vector fields on ${\mathbb{R}}$.

Modules of differential operators on tensor densities
-----------------------------------------------------

Let ${{\mathcal{F}}}_\l(M)$ (or ${{\mathcal{F}}}_\l$ in short) be the space of tensor densities of degree $\l$ on $M$ (i.e. the space of sections of the line bundle $\Delta_\l(M)=\mod{\Lambda^nT^*M}^{\otimes\l}$ over $M$). Clearly, ${{\mathcal{F}}}_0\cong{}C^\infty(M)$ as a ${\mathrm{Vect}}(M)$-module, any two ${\mathrm{Vect}}(M)$-modules of tensor densities are non-isomorphic (see also [@Fuc]).

Denote ${{\mathcal{D}}}_\l$ the space ${{\mathcal{D}}}({{\mathcal{F}}}_\l)$ of linear differential operators on ${{\mathcal{F}}}_\l$. This space is an associative (and, therefore, a Lie) algebra with the filtration by the order of differentiation: $$\label{filtr1}
{{\mathcal{D}}}^0_\l\subset{{\mathcal{D}}}^1_\l\subset\cdots\subset{{\mathcal{D}}}^k_\l\subset\cdots$$ The algebra ${{\mathcal{S}}}$ is naturally identified with the associated graded algebra ${\mathrm{gr}}({{\mathcal{D}}}_\l)$ that is, $$\label{principal}
{{\mathcal{D}}}^k_\l/{{\mathcal{D}}}^{k-1}_\l\cong{{\mathcal{S}}}_k\,.$$ The corresponding projection $\s_k:{{\mathcal{D}}}^k_\l\to{{\mathcal{S}}}_k$ is called the (principal) *symbol*.

The associative algebra ${{\mathcal{D}}}_\l$ can be naturally interpreted as a non-trivial deformation of ${{\mathcal{S}}}$ and constitutes one of the main objects considered in *deformation quantization*.

We will be interested, however, only in the ${\mathrm{Vect}}(M)$-module structure on ${{\mathcal{D}}}_\l$ rather than in the whole associative (or Lie algebra) structure. The (tautological) Lie algebra embedding ${\mathrm{Vect}}(M)\hookrightarrow{{\mathcal{D}}}_\l$ $$\label{StandEmbed}
X\mapsto{}L^\l_X,$$ where $L^\l_X$ is the Lie derivative on ${{\mathcal{F}}}_\l$, defines a ${\mathrm{Vect}}(M)$-module structure on ${{\mathcal{D}}}_\l$.

[If $M$ is oriented by a volume form $\Omega$, then $$\label{LieDerDens}
L^\l_X=L_X+\l\,{\mathrm{div}}_\Omega{X}.$$ Moreover, ${{\mathcal{D}}}_\l$ and ${{\mathcal{D}}}_\m$ are isomorphic associative algebras. However, as ${\mathrm{Vect}}(M)$-modules they are isomorphic if and only if $\l+\m=1$ [@DO; @LMT]. ]{}

The Main Theorem
----------------

The space ${{\mathcal{D}}}({{\mathcal{S}}})$ is decomposed, as a ${\mathrm{Vect}}(M)$-module, into the direct sum: $$\label{Razlozhenie}
{{\mathcal{D}}}({{\mathcal{S}}})
=\bigoplus_{k,\ell}
{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell),$$ where ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)\subset{\mathrm{Hom}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$. It would then suffice to compute the cohomology (\[DefCohom\]) with coefficients in each of these modules. Our main result is the following

\[CohomThm\] If $\dim M\geq2$, then $$\label{CohomResult}
{\mathrm{H}}^1({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell))
=
\left\{
\begin{array}{lcl}
{\mathbb{R}}& , & \hbox{if} \quad k-\ell=2\\[6pt]
{\mathbb{R}}& , & \hbox{if} \quad k-\ell=1, \ell\neq0\\[6pt]
{\mathbb{R}}\oplus{\mathrm{H}}^1_{\rm DR}(M) & , & \hbox{if} \quad k-\ell=0\\[6pt]
0 & , & \hbox{otherwise}
\end{array}
\right.$$ where ${\mathrm{H}}^1_{\rm DR}(M)$ is the first space of the de Rham cohomology of $M$.

The proof will be given in Section \[TheProof\].

From now on we assume that $\dim{}M\geq2$.

Differentiability {#peetre}
-----------------

As a first step towards the proof of Theorem \[CohomThm\], we will prove now that any 1-cocycle on ${\mathrm{Vect}}(M)$ with values in the space of differential operators ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ is locally differentiable. Due to the well-known Peetre Theorem [@Pee], this means that for any $\g\in{\rm Z}^1({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell))$, the bilinear map $(X,P)\mapsto\g(X)(P)$, where $X\in{\mathrm{Vect}}(M)$ and $P\in{{\mathcal{S}}}_k$, is local: $$\label{Sup}
{\mathrm{Supp}}\,\g(X)(P)
\subset{\mathrm{Supp}}\,{X}\cap\,{\mathrm{Supp}}\,{P}$$

\[PeetrePro\] Any 1-cocycle $\g$ on ${\mathrm{Vect}}(M)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ is local.

Let $U\subset{}M$ be open and $X\in{\mathrm{Vect}}(M)$ vanish on $U$. We have to show that $
\g(X)_{|_U}=0.
$ Let $x_0$ be any point in $U$. As well-known, there exists a neighborhood $V\subset{}U$ of $x_0$ and vector fields $X_i,X_i'$, $i=1,\ldots,r$ on $V$ such that $$X=
\sum_{1\leq{}i\leq{}r}
[X_i,X_i']$$ and $${X_i}
_{|_V}={X_i'}
_{|_V}=0,$$ where $r$ depends only on the dimension of $M$. One has, using the fact that $\g$ is a 1-cocycle $$\g(X)_{|_V}=
\sum_{1\leq{}i\leq{}r}
\left(
L_{X_i}\g(X_i')_{|_V}-L_{X_i'}\g(X_i)_{|_V}
\right)=0.$$

Non-trivial cohomology classes {#FormSection}
==============================

Let us now describe a natural basis of the above cohomology spaces (\[CohomResult\]).

Case $k=\ell$
-------------

Since ${\mathrm{Id}}\in{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_k)$ is ${\mathrm{Vect}}(M)$-invariant, $c\mapsto{}c\,{\mathrm{Id}}$ maps any cocycle $c$ to a cocycle and thus induces a homomorphism ${\mathrm{H}}({\mathrm{Vect}}(M);C^\infty(M))\to{\mathrm{H}}({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_k))$. Theorem \[CohomThm\] states that it is an isomorphism in degree one.

Recall that ${\mathrm{H}}({\mathrm{Vect}}(M);C^\infty(M))$ is well-known (see [@Fuc]). In particular, given a covariant derivation $\nabla$, the 1-cocycles are the maps $$\label{OneCocycle}
c_{a,\omega}:
X
\mapsto
a\,{\mathrm{div}}_\nabla(X)+i_X\omega,$$ where $a\in{\mathbb{R}}$ and $\omega$ is a closed 1-form, ${\mathrm{div}}_\nabla$ being the divergence associated to $\nabla$. The cocycle (\[OneCocycle\]) is a coboundary if and only if $a=0$ and $\omega$ is exact.

Case $k=\ell+1$, $\ell\neq0$ {#Konstrukciya1}
----------------------------

Consider the exact sequence of ${\mathrm{Vect}}(M)$-modules $$\label{exact1}
\begin{CD}
0 @> >>{{\mathcal{D}}}^{k-1}_\l @> >>{{\mathcal{D}}}_\l^k @> >>{{\mathcal{S}}}_k @> >>0.
\end{CD}$$ Dividing out by ${{\mathcal{D}}}_\l^{k-2}$ leads to the exact sequence $$\label{exact1First}
\begin{CD}
0 @> >>{{\mathcal{S}}}_{k-1} @> >>{{\mathcal{D}}}_\l^k/{{\mathcal{D}}}_\l^{k-2} @> >>{{\mathcal{S}}}_k @> >>0
\end{CD}$$ Assume $k\neq1$ and $\l\neq1/2$. The sequence (\[exact1First\]) does not split [@LO]. Its cohomology class is a non-zero element in ${\mathrm{H}}^1({\mathrm{Vect}}(M);{\mathrm{Hom}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1}))$ (see Appendix). This class admits a representative with values in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1})$, since the ${\mathrm{Vect}}(M)$-actions in (\[exact1First\]) are differential. It thus defines a non-trivial class in ${\mathrm{H}}^1({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1}))$ which, by Theorem \[CohomThm\], is a basis of this space.

Case $k=\ell+2$ {#RelationSec}
---------------

If $\l=1/2$, it is shown in [@LO] that the sequence (\[exact1First\]) is split and that the sequence $$\label{exact2}
\begin{CD}
0@> >>{{\mathcal{S}}}_{k-2}@> >>
{{\mathcal{D}}}_{1/2}^{k}/{{\mathcal{D}}}_{1/2}^{k-3}@> >>
{{\mathcal{D}}}_{1/2}^{k}/{{\mathcal{D}}}_{1/2}^{k-2}@> >>0
\end{CD}$$ is not. Moreover, the splitting of (\[exact1First\]) is given by differential projectors. Since (\[exact1First\]) is split, the class $[{{\mathcal{D}}}^{k-1}_{1/2},{{\mathcal{D}}}_{1/2}^k]$ of (\[exact1\]) belongs to ${\mathrm{H}}^1({\mathrm{Vect}}(M);{\mathrm{Hom}}({{\mathcal{S}}}_{k},{{\mathcal{D}}}_{1/2}^{k-2}))$. Since (\[exact2\]) is not split, its projection ${\s_{k-2}}_\sharp\,[{{\mathcal{D}}}^{k-1}_{1/2},{{\mathcal{D}}}_{1/2}^k]$ is non-zero (see Lemma \[SequenceLemma\] from Appendix).

As in the previous case, this projection is easily seen to admit a representative with values in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-2})$. Hence, it provides a basis of ${\mathrm{H}}^1({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-2}))$.

\[RemRes\] [In the above Subsections \[Konstrukciya1\] and \[RelationSec\] we have associated non-trivial cohomology classes to the exact sequences (\[exact1First\]) and (\[exact2\]). It is important to note that these classes are “natural” in the following sense. For any open subset $U\subset{}M$ their restrictions to $U$ are precisely the classes associated to the same sequences upon $U$. ]{}

Projectively equivariant cohomology {#ProjectCohom}
===================================

Throughout this section we put $M\cong{\mathbb{R}}^n$ and $n\geq2$.

The Lie algebra of infinitesimal projective transformations {#Introducing}
-----------------------------------------------------------

The main idea of our proof of Theorem \[CohomThm\] is to use the filtration with respect to the Lie subalgebra $$\label{Definitionsl2}
{\mathrm{sl}}(n+1,{\mathbb{R}})\subset{\mathrm{Vect}}({\mathbb{R}}^n).$$ It is suggested by the fact that the exact sequence (\[exact1\]) that generate our cohomology is split as a sequence of ${\mathrm{sl}}(n+1,{\mathbb{R}})$-modules [@LO]. In some sense, this Lie subalgebra plays the same rôle in our approach as the linear subalgebra ${\mathrm{gl}}(n,{\mathbb{R}})$ in the traditional one (cf. [@Fuc]).

Recall that the standard action of the Lie algebra ${\mathrm{sl}}(n+1,{\mathbb{R}})$ on ${\mathbb{R}}^n$ is generated by the vector fields $$\label{sl2}
X_i=
\frac{\partial}{\partial{}x^i}\,, 
\qquad 
X_{ij}=
x^i\frac{\partial}{\partial{}x^j}\,, 
\qquad
\bar{X}_i=
x^i{\cal E}\,,$$ where $${\cal E} = x^i\frac{\partial}{\partial{}x^i}\,.
\label{Euler}$$ Observe in particular that $X_i$ and $X_{ij}$ generate an action of the Lie algebra ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$.

Computing the relative cohomology space
---------------------------------------

In this section we will compute the first space of the so-called relative cohomology of ${\mathrm{Vect}}({\mathbb{R}}^n)$, i.e. the cohomology of the complex of ${\mathrm{Vect}}({\mathbb{R}}^n)$-cochains vanishing on the subalgebra ${\mathrm{sl}}(n+1,{\mathbb{R}})$. We will prove the following

\[RelativeTheorem\] If $n\geq2$, then $$\label{RelatCohom}
{\mathrm{H}}^1({\mathrm{Vect}}({\mathbb{R}}^n),{\mathrm{sl}}(n+1,{\mathbb{R}});{{\mathcal{D}}}(S_k,{{\mathcal{S}}}_\ell))=
\left\{
\begin{array}{lcl}
{\mathbb{R}}& , & \hbox{if} \quad k-\ell=2\\[6pt]
{\mathbb{R}}& , & \hbox{if} \quad k-\ell=1, \ell\neq0\\[6pt]
0 & , & \hbox{otherwise}
\end{array}
\right.$$

Equivariance property {#EquivSection}
---------------------

We begin the proof with a simple observation.

Let ${\mathfrak{h}}\subset{\mathfrak{g}}$ be a Lie subalgebra and $V$ a ${\mathfrak{g}}$-module. If $c:{\mathfrak{g}}\to{}V$ is a 1-cocycle such that $c_{|_{\mathfrak{h}}}\equiv0$, then it is *equivariant* with respect to ${\mathfrak{h}}$ i.e. $$\label{equivCoc}
L_X(c(Y))=
c([X,Y]),
\qquad X\in{\mathfrak{h}},$$ where $L$ stays for the ${\mathfrak{g}}$-action on the module $V$.

Consequently, our strategy to compute the space of relative cohomology (\[RelatCohom\]) consists, first, in classifying the ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant linear maps $c:{\mathrm{Vect}}({\mathbb{R}})\to{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ vanishing on ${\mathrm{sl}}(n+1,{\mathbb{R}})$ and, second, to isolate among them the 1-cocycles.

Commutant of the affine Lie algebra
-----------------------------------

Consider the space of polynomials ${\mathbb{C}}[x,\xi]={\mathbb{C}}[x^1,\ldots,x^n,\xi_1,\ldots,\xi_n]$ as a submodule of ${{\mathcal{S}}}$ under the action of ${\mathrm{sl}}(n+1,{\mathbb{R}})$. We need to compute the *commutant* of the subalgebra ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$, i.e. the algebra of differential operators on ${\mathbb{C}}[x,\xi]$ commuting with the ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$-action.

The differential operators on ${\mathbb{C}}[x,\xi]$ given by $$\label{Div}
{\mathrm{E}}=\xi_i\,
\frac{\partial}{\partial\xi_i},\qquad
{\mathrm{D}}=\frac{\partial}{\partial x^i}\,
\frac{\partial}{\partial\xi_i}$$ commute with the ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$-action. Let us recall the classical result of the Weyl invariant theory (see [@Wey]).

\[WeylPro\] The algebra of differential operators on ${\mathbb{C}}[x,\xi]$ commuting with the action of the affine Lie algebra, is generated by ${\mathrm{E}}$ and ${\mathrm{D}}$.

We will call the operators (\[Div\]) the Euler operator and the divergence operator respectively. The eigenspaces of ${\mathrm{E}}$ are obviously consist of homogeneous polynomials in $\xi$.

\[NiceCor\] The operator ${\mathrm{D}}^{k-\ell}$ is the unique (up to a constant) ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$-equivariant differential operator from ${{\mathcal{S}}}_k$ to ${{\mathcal{S}}}_\ell$.

Any differential operator on ${{\mathcal{S}}}_k$ is indeed determined by its values on the subspace ${\mathbb{C}}[x,\xi]$.

The Euler operator ${\mathrm{E}}$ is clearly equivariant with respect to the whole ${\mathrm{Vect}}({\mathbb{R}}^n)$. We will need the commutation relations of the operator ${\mathrm{D}}$ with the quadratic generators of ${\mathrm{sl}}(n+1,{\mathbb{R}})$.

For $\bar{X}_i$ as in (\[sl2\]), one has $$[L_{\bar{X}_i},{\mathrm{D}}]
=
\Big(2{\mathrm{E}}+(n+1)\Big)\circ\frac{\partial}{\partial{\xi_i}}\,,
\label{relation}$$

: straightforward.

Bilinear ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$-invariant operators
---------------------------------------------------------------------------------

We also need to classify the bilinear ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$-invariant differential operators. For that purpose, let us use a natural identification $$\label{IdentBin}
{\mathbb{C}}[x,\xi]\otimes{\mathbb{C}}[y,\eta]
\cong
{\mathbb{C}}[x,\xi,y,\eta].$$ There are, obviously, four invariant differential operators ${\mathrm{D}}_{(x,\xi)},{\mathrm{D}}_{(y,\eta)}$ (the divergence operators with respect to the first and the second arguments) and ${\mathrm{D}}_{(x,\eta)},{\mathrm{D}}_{(y,\xi)}$ (the operators of contraction in terms of tensors). Applying again [@Wey] one gets the following

\[AffPro\] Every bilinear differential operator $$\label{Bilin1}
{{\mathcal{S}}}_j\otimes{{\mathcal{S}}}_k\to{{\mathcal{S}}}_\ell$$ invariant with respect to the action of the affine Lie algebra, is a homogeneous polynomial in ${\mathrm{D}}_{(x,\xi)},{\mathrm{D}}_{(x,\eta)},{\mathrm{D}}_{(y,\xi)}$ and ${\mathrm{D}}_{(y,\eta)}$ of degree $j+k-\ell$.

We are now ready to start the proof of Theorem \[RelativeTheorem\].

Bilinear ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant operators
----------------------------------------------------------------

In view of Section \[EquivSection\], we will now classify the ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant linear differential maps $$\label{LinearEquiv}
c:{\mathrm{Vect}}({\mathbb{R}}^n)\to{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$$ vanishing on the subalgebra ${\mathrm{sl}}(n+1,{\mathbb{R}})\subset{\mathrm{Vect}}({\mathbb{R}}^n)$. We can, equivalently, consider the equivariant bilinear maps $$\label{Bilin}
C:{{\mathcal{S}}}_1\otimes{{\mathcal{S}}}_k\to{{\mathcal{S}}}_{k-p},$$ where $p=k-\ell$.

By Proposition \[AffPro\], any such operator is of the form $$\label{Ansatz}
\begin{array}{rcl}
C&=&
\displaystyle\sum_{s=0}^{p+1}
\left(
\frac{\a_s}{s!(p-s+1)!}\,
{\mathrm{D}}_{(x,\eta)}^{s}{\mathrm{D}}_{(y,\eta)}^{p-s+1}\right.\\[12pt]
&&\qquad
\left.\displaystyle+\frac{\b_s}{(s-1)!(p-s+1)!}\,
{\mathrm{D}}_{(x,\xi)}{\mathrm{D}}_{(x,\eta)}^{s-1}{\mathrm{D}}_{(y,\eta)}^{p-s+1}\right)_{
\Big|\!\begin{array}{l}
y=x \\
\eta= \xi
\end{array}
}\\[12pt] &&
{\displaystyle+\sum_{s=0}^p
\frac{\g_s}{s!(p-s)!}\,
{\mathrm{D}}_{(y,\xi)}{\mathrm{D}}_{(x,\eta)}^s{\mathrm{D}}_{(y,\eta)}^{p-s}}_{\Big|\!
\begin{array}{l}
y=x \\
\eta= \xi
\end{array}}
\end{array}$$ where $\a_s,\b_s,\g_s\in{\mathbb{R}}$.

Moreover, $$\label{NulForSmall}
\a_s=\b_s=\g_s=0
\qquad
\hbox{for}
\qquad
s<2,$$ since $C$ vanishes on the affine subalgebra and $$\label{Zero}
(k-p)\a_2+(n+1)\b_2+(p-1)\g_2=0,$$ since $C$ vanishes on the quadratic generators $\bar{X}_i$ of ${\mathrm{sl}}(n+1,{\mathbb{R}})$.

For $k=p$, we have not to take into account the coefficients $\a_s$ in the expression (\[Ansatz\]) because the corresponding terms vanish when applied to ${{\mathcal{S}}}_1\otimes{{\mathcal{S}}}_k$.

It is quite easy, using (\[relation\]) and analogous relations with the operators ${\mathrm{D}}_{(x,\eta)},{\mathrm{D}}_{(y,\xi)}$ and ${\mathrm{D}}_{(y,\eta)}$, to obtain the necessary and sufficient condition for the coefficients in (\[Ansatz\]) for $C$ to be equivariant. One gets the following recurrence relations: $$\begin{aligned}
(s-1)\,\a_{s+1}-(2k+n-p+s-1)\,\a_s-\g_s
&=&0
\label{sys1}
\\
(s-1)\,\b_{s+1}-(2k+n-p+s-1)\,\b_s-\g_s
&=&0
\label{sys2}
\\
(s-2)\,\g_s-(2k+n-p+s-1)\,\g_{s-1}
&=&0
\label{sys3}
\\
(k-p)\,\a_{s+1}+(n+1)\,\b_{s+1}+(p-s)\,\g_{s+1}+(k-p+s)\,\g_s
&=&0
\label{sys4}\end{aligned}$$ where $2\leq{}s\leq{}p$. (For $k=p$, equation (\[sys1\]) has not to be taken into account.)

Now, to solve the system (\[Zero\]-\[sys4\]), we need the following technical

\[Simplify\] If $\a_s,\b_s,\g_c$ verify the equations (\[sys1\]-\[sys3\]) and (\[Zero\]), then $\a_s,\b_s,\g_c$ verify the equation (\[sys4\]).

(A similar result holds true when $k=p$.)

Check that for $s=1$, the equation (\[sys4\]) coincides with (\[Zero\]), the result follows then by induction.

It is now very easy to get the complete solution of the system (\[Zero\]-\[sys3\]). One has the following four cases.

\(a) For $p=0$ and for $(p=1,k=1)$ there is no solution.

\(b) For $(p=1,k\geq2)$ the system has a one-dimensional space of solutions spanned by $$\label{FirstCocycle}
C_1={\frac{1}{2}}\,{\mathrm{D}}^2_{(x,\eta)}+
\frac{k-1}{n+1}\,{\mathrm{D}}_{(x,\xi)}{\mathrm{D}}_{(x,\eta)}\,,$$ which is, in fact, a just a solution of the equation (\[Zero\]).

[One readily checks that the operator $c(X)\in{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1})$ given by (\[FirstCocycle\]) coincides (up to a constant) with the operator of contraction with the tensor field $$\label{FirstCocycleTensor}
c_1(X)=
\left(
\frac{\partial}{\partial{}x^i}
\frac{\partial}{\partial{}x^j}(X^\ell)
+
\frac{2}{n+1}\,
\d_j^\ell
\frac{\partial}{\partial{}x^i}
\frac{\partial}{\partial{}x^s}(X^s)
\right)
dx^i{}dx^j
\otimes\xi_\ell$$ This expression is obviously a 1-cocycle. The expression (\[FirstCocycleTensor\]) is known in the literature as the Lie derivative of a flat *projective connection* (cf., e.g., [@kh]). ]{}

\(c) For $p\geq2,k>p$, the system (\[sys1\]-\[sys3\]) under the condition (\[Zero\]), has a two-dimensional space of solutions parametrized by $(\a_2,\b_2)$.

\(d) For $p=k\geq2$, the equation (\[sys1\]) should be discarded. The space of solutions is again one-dimensional.

Projectively invariant cocycles
-------------------------------

We will now determine which of the ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant maps (\[LinearEquiv\]) classified in the preceding section are 1-cocycles. Let us examine separately the cases (b)-(d).

\(b) In the simplest case, $p=1$, one easily checks that the unique ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant map (\[FirstCocycle\]), indeed, defines a 1-cocycle on ${\mathrm{Vect}}({\mathbb{R}}^n)$ with values in ${{\mathcal{D}}}^0({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1})$.

\(c) The cocycle relation adds the equation $\b_3=2\b_2$ to the general system (\[sys1\]-\[sys3\]).

In the case $p=2$, one checks by a straightforward computation, that the solutions are the constant multiples of the solution given by $$\label{CocSecondDef}
\begin{array}{rcl}
\a_2 &=& 2,\\[4pt]
\a_3 &=& 2k+n+1,\\[4pt]
\b_2 &=& 1, \\[4pt]
\b_3 &=& 2, \\[4pt]
\d_2 &=& \!\! -(2k+n-3).
\end{array}$$ In the case $p>2$, the only solution of the system (\[sys1\]-\[sys3\]) together with the equation $\b_3=2\b_2$ is zero.

\(d) If $k=p$, then the non-trivial solutions of the system are cocycles if and only if $k=p=2$. This cocycle is precisely of the form (\[CocSecondDef\]) disregarding $\a_2$ and $\a_3$.

\[NontrivPro\] The 1-cocycles on ${\mathrm{Vect}}({\mathbb{R}}^n)$ defined by the formul[æ]{} (\[FirstCocycle\]) and (\[CocSecondDef\]) are non-trivial.

This follows immediately from Sections \[Konstrukciya1\], \[RelationSec\] and the fact that the sequence (\[exact1\]) is split when restricted to ${\mathrm{sl}}(n+1,{\mathbb{R}})$, see [@LO]. Let us also give an elementary proof.

Recall that a 1-cocycle on ${\mathrm{Vect}}({\mathbb{R}}^n)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ is a coboundary if it is of the form $X\mapsto{}[L_X,B]$ for some $B\in{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$. Moreover, the 1-cocycle vanishes on ${\mathrm{sl}}(n+1,{\mathbb{R}})$ if and only if $B$ is ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant.

(cf. [@Lec1]). \[LemInv\] If $k\neq\ell$, there is no ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant operators $B\in{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ different from zero.

In virtue of Corollary \[NiceCor\], the property of ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariance implies, in particular, that $B$ has to be proportional to ${\mathrm{D}}^{k-\ell}$. Now, the commutation relation (\[relation\]) shows that this operator can never be ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant.

Proposition \[NontrivPro\] follows.

Proof of Theorem \[RelativeTheorem\]
------------------------------------

We have shown that there exist unique (up to a constant) 1-cocycles $c_1$ and $c_2$ on ${\mathrm{Vect}}({\mathbb{R}}^n)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1})$ and ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-2})$ respectively, vanishing on ${\mathrm{sl}}(n+1,{\mathbb{R}})$. These cocycles define non-trivial classes of relative cohomology.

Theorem \[RelativeTheorem\] is proven.

Proof of Theorem \[CohomThm\] {#TheProof}
=============================

Using the filtration with respect to the subalgebra ${\mathrm{sl}}(n+1,{\mathbb{R}})$, we will first prove Theorem \[CohomThm\] in the case when $M$ is a vector space and then extend it to an arbitrary manifold. To that end, we need some more information about the cohomology of ${\mathrm{sl}}(n+1,{\mathbb{R}})$.

Cohomology of ${\mathrm{sl}}(n+1,{\mathbb{R}})$
-----------------------------------------------

The cohomology of the Lie algebra ${\mathrm{sl}}(n+1,{\mathbb{R}})$ with coefficients in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ has been computed in [@Lec1].

\[SlThm\] The space of cohomology ${\mathrm{H}}({\mathrm{sl}}(n+1,{\mathbb{R}});{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell))$ is trivial for $k\neq\ell$, for $k=\ell$ it is isomorphic to the Grassman algebra of invariant functionals on ${\mathrm{gl}}(n,{\mathbb{R}})$: $$\label{CohSl}
{\mathrm{H}}({\mathrm{sl}}(n+1,{\mathbb{R}});{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_k))
=
\left(
\bigwedge\textstyle{\mathrm{gl}}(n,{\mathbb{R}})^*
\right)^{{\mathrm{gl}}(n,{\mathbb{R}})}$$

In particular, $$\label{CohSlParticul}
{\mathrm{H}}^1({\mathrm{sl}}(n+1,{\mathbb{R}});{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell))=
\left\{
\begin{array}{rl}
{\mathbb{R}},& k=\ell \\
0,& \hbox{otherwise}
\end{array}
\right.$$ and the class of the 1-cocycle $X\mapsto{\mathrm{div}}(X){\mathrm{Id}}$ spans that space in the case $k=\ell$. (In fact, it corresponds to the invariant function ${\mathrm{tr}}:{\mathrm{gl}}(n,{\mathbb{R}})\to{\mathbb{R}}$.) Note that this cocycle is just the restriction to ${\mathrm{sl}}(n+1,{\mathbb{R}})$ of the cocycle $c_{1,0}$, see (\[OneCocycle\]).

The case of ${\mathbb{R}}^n$ {#VectorSection}
----------------------------

The restriction of a 1-cocycle $c:{\mathrm{Vect}}({\mathbb{R}}^n)\to{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ to ${\mathrm{sl}}(n+1,{\mathbb{R}})$ is a 1-cocycle on ${\mathrm{sl}}(n+1,{\mathbb{R}})$. If $k\neq\ell$, then this restriction is trivial and, therefore, $c$ is cohomological to a 1-cocycle on ${\mathrm{Vect}}({\mathbb{R}}^n)$ vanishing on ${\mathrm{sl}}(n+1,{\mathbb{R}})$; if $k=\ell$, then the restriction of $c$ to ${\mathrm{sl}}(n+1,{\mathbb{R}})$ is cohomological to $c_{1,0}$ and so $c-c_{1,0}$ is, again, cohomological to a 1-cocycle on ${\mathrm{Vect}}({\mathbb{R}}^n)$ vanishing on ${\mathrm{sl}}(n+1,{\mathbb{R}})$. The result then follows from Theorem \[RelativeTheorem\].

Theorem \[CohomThm\] is proven for the special case $M={\mathbb{R}}^n$.

The general case
----------------

Let us now prove Theorem \[CohomThm\] for an arbitrary manifold $M$. Consider a 1-cocycle $c$ on ${\mathrm{Vect}}(M)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$.

\(a) If $k-\ell\neq0,1,2$, then in any domain of chart $U\cong{\mathbb{R}}^n$, the restriction $c_{|_U}$ is a coboundary, that is $c(X)_{|_U}=L_X(S_U),$ where $S_U\in{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ is some operator on $U$. But, on $U\cap{}V$, one has $c(X)_{|_{U\cap{}V}}=L_X(S_U)=L_X(S_V)$ and so the operator $S_U-S_V$ is invariant. Lemma \[LemInv\] implies $S_U-S_V=0$. Therefore, the $S_U$’s are the restrictions of some globally defined $S\in{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ and $c$ is its coboundary.

\(b) If $k-\ell=1$ or $2$, it follows from Theorem \[CohomThm\] for $M={\mathbb{R}}^n$ that the class of $c_{|_U}$ is determined up to a constant. In view of Remark \[RemRes\], one has thus $$\label{LocalEquation}
c_{|_U}
=
\a_U\,\g_{|_U}+L_X(S_U),$$ for some $\a_U\in {\mathbb{R}}$ and $S_U$ as above, where $\g$ is a representative of one of the classes associated to the sequences (\[exact1First\]) and (\[exact2\]) respectively. On $U\cap{}V$ one obviously has $\a_U=\a_V$ and $S_U=S_V$ since $$\left(
\a_U-\a_V
\right)
\g_{|_{U\cap{}V}}
=
\partial
\left(
S_U-S_V
\right),$$ $\g_{|_{U\cap{}V}}$ is non-trivial and, as above, $S_U-S_V$ is invariant.

\(c) If $k-\ell=0$, one has $$\label{LocalEquationBis}
c_{|_U}
=
\a_U\,{c_{1,0}}_{|_U}+L_X(S_U).$$ Once again, $\a_U=\a_V\,(:=a)$ and $S_U-S_V$ is invariant, but any invariant operator in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_k)$ is proportional to the identity so that $S_U-S_V=\b_{UV}\,{\mathrm{Id}}$, where $\b_{UV}$ is a constant. It is clear that the $\b_{UV}$’s define a $\check{\rm C}$ech 1-cocycle. If now $\omega$ is a closed 1-form representing the corresponding de Rham class, one easily sees that $c$ is cohomologous to $c_{a,\omega}$.

Theorem \[CohomThm\] is proven.

Cocycles associated to a connection {#ConnectCocycles}
===================================

Using a torsion free covariant derivation $\nabla$, it is possible to construct globally defined cocycles spanning ${\mathrm{H}}^1({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell))$ for $k-\ell=1,2$.

Lie derivative of a connection {#Konstrukciya1}
------------------------------

For each vector field $X$, the Lie derivative $$L_X(\nabla) : (Y,Z) \mapsto [X,\nabla_Y Z] -
\nabla_{[X,Y]} Z-\nabla_Y[X,Z]$$ of $\nabla$ is well-known to be a symmetric $(1,2)$-tensor field. It yields a non-trivial 1-cocycle $$X
\mapsto
L_X(\nabla)$$ on ${\mathrm{Vect}}(M)$ with values in $\Gamma(\bigotimes_2^1TM)$. Therefore, for $k\geq2$, the contraction $$\label{TheFirstCoc1}
\g^\nabla_1(X)(P)
=
\langle
P,L_X(\nabla)
\rangle,
\qquad
P\in{{\mathcal{S}}}_k,$$ defines a 1-cocycle on ${\mathrm{Vect}}(M)$ with values in ${{\mathcal{D}}}^0({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1})$.

Second-order cohomology class and the Vey cocycle {#Konstrukciya2}
-------------------------------------------------

The last case, $\ell=k-2$, is directly related to deformation quantization.

For any symplectic manifold $V$, there exists a non-zero class in ${\mathrm{H}}^2(C^\infty(V);C^\infty(V))$. It is given by so-called *Vey cocycle* usually denoted $S^3_\Gamma$ (see [@WL] and [@Rog] for explicit construction using a connection $\Gamma$ on $V$).

In the particular, if $V=T^*M$ one can choose the connection so that $S^3_\Gamma$ is homogeneous of weight $-3$, namely, restricted to ${{\mathcal{S}}}\subset{}C^\infty(T^*M)$, $$\label{S3}
S^3_\Gamma:
{{\mathcal{S}}}_k\otimes{{\mathcal{S}}}_\ell\to{{\mathcal{S}}}_{k+\ell-3},$$ see [@WL1] (e.g. choosing $\Gamma$ as a lift of $\nabla$ to $T^*M$). It follows easily from (\[S3\]) that the map ${\mathrm{Vect}}(M)\to{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-2})$ defined by $$\label{VeyCoc1}
\g_2^\nabla(X)(P)=
S^3_\Gamma(X,P),
\qquad
P\in{{\mathcal{S}}}_k.$$ is a 1-cocycle.

Appendix: approximations of the class of a short exact sequence of modules {#Appendix1}
==========================================================================

Class of a short exact sequence of ${\mathfrak{g}}$-modules {#ShortExact}
-----------------------------------------------------------

We will need some general information about short exact sequences of filtered modules.

Let ${\mathfrak{g}}$ be a Lie algebra. Consider an exact sequence of ${\mathfrak{g}}$-modules $$\label{s1}
\begin{CD}
0 @> >>A @>{i}>> B @>{j}>> C @> >>0
\end{CD}$$ It is characterized by an element of ${\mathrm{H}}^1({\mathfrak{g}};{\mathrm{Hom}}(C,A))$ (cf. [@Fuc], Sec. 1.4.5). It will be convenient to denote it $[A,B]$. Recall that if $\tau : C \to B$ is a section of $j$, then $[A,B]$ is the class of the 1-cocycle $\g^\tau: {\mathfrak{g}}\to{\mathrm{Hom}}(C,A)$ given by $$\label{AbstractCoc}
\g^\tau(X)(T) 
= 
i^{-1}(X.\tau(T) - \tau(X.T)),$$ where $X\in{\mathfrak{g}}$ and $T\in{}C$ (this expression is well defined since $X.\tau(T) -
\tau(X.T)\in\ker{j}$).

Given a submodule $V$ of $A$, one has the following commutative diagram: $$\label{SubmodDiagramm}
\begin{CD}
@. 0 \\
@. @VV V \\
@. V \\
@.@VV{i_V}V\\
0 @> >>A @> >> B @> >> C @> >>0 \\
@.@VV{\pi_A}V @VV{\pi_B}V @VV{{\rm Id}}V \\
0 @> >>A/V @> >> B/V @> >> C @> >>0 \\
@. @VV V \\
@. 0
\end{CD}$$ where $i_V$ is the injection of $V$ into $A$ and $\pi_A$, $\pi_B$ are the projections.

One has the relation $[A/V,B/V] = {\pi_A}_\sharp\,[A,B]$. Moreover, the left vertical of (\[SubmodDiagramm\]) leads to the exact triangle $$\label{TriangleDiagramm}
\begin{array}{cl}
\mathrm{H}({\mathfrak{g}};\mathrm{Hom}(C,V)) \\
& \searrow {i_V}_\sharp \\
\Delta\;\big\uparrow & \mathrm{H}({\mathfrak{g}};\mathrm{Hom}(C,A)) \\
& \swarrow {\pi_A}_\sharp \\
\mathrm{H}({\mathfrak{g}};\mathrm{Hom}(C,A/V))
\end{array}$$ where $\Delta$ is the connecting homomorphism. One easily obtains the following

\[SequencePro\] (i) The class $[A/V,B/V]$ vanishes if and only if $[A,B]\in\mathrm{im}\,{i_V}_\sharp$. (ii) If $[A/V,B/V]=0$ then the class of the exact sequence $$\label{ExactB}
\begin{CD}
0 @> >>V @>{i \circ i_V} >> B @> >> B/V @> >>0
\end{CD}$$ is $[V,B] = [V,A] + [A,B]$ and vanishes if and only if $[V,A]=[A,B]=0$.

Case of a filtered module {#Filtr}
-------------------------

Consider now a flag of filtered ${\mathfrak{g}}$-modules $A_0\subset{}A_1\subset\cdots\subset{}A_r\subset\cdots$ and put $S_r =A_r/A_{r-1}$. Let us study the classes $[A_r,A_{r+1}]$ of the sequences $$\label{BolshayaSeq}
\begin{CD}
0 @> >>A_r @> >> A_{r+1} @> >> S_{r+1} @> >>0.
\end{CD}$$ The quotient by $V= A_{r-1}$, leads to its “first approximation”: $$\label{s3}
\begin{CD}
0 @> >>S_{r} @> >> A_{r+1}/A_{r-1} @> >> S_{r+1} @> >>0
\end{CD}$$ If the sequence (\[s3\]) is split, then $[A_r,A_{r+1}]\in{\mathrm{H}}^1({\mathfrak{g}};{\mathrm{Hom}}(S_{r+1},A_{r-1}))$ and so $$\label{SplitClass}
[A_{r-1},A_{r+1}] = [A_{r-1},A_r]+[A_r,A_{r+1}]$$ by Proposition \[SequencePro\].

The next approximation is a result of the quotient by $A_{r-2}$. Let $\pi_r:A_r\to{}A_r/A_{r-1}$ be the projection to the quotient-module.

\[SequenceLemma\] If the sequence (\[s3\]) is split for all $r>0$, but the sequences $$\label{SecondApprox}
\begin{CD}
0 @> >>S_{r-1} @> >> A_{r+1}/A_{r-2} @> >>A_{r+1}/A_{r-1} @> >>0,
\end{CD}$$ for $r>1$ are not split, then the class ${\pi_{r-1}}_\sharp[A_r,A_{r+1}]$ does not vanish.

Since the sequence (\[s3\]) is split, one has ${\pi_{r-1}}_\sharp[A_{r-1},A_r]=0$. If in addition ${\pi_{r-1}}_\sharp[A_r,A_{r+1}]
= 0$, then by Proposition \[SequencePro\] $$[A_{r-1}/A_{r-2},A_{r+1}/A_{r-2}]=
{\pi_{r-1}}_\sharp
[A_{r-1},A_{r+1}]=
{\pi_{r-1}}_\sharp
\left(
[A_{r-1},A_r]
+[A_r,A_{r+1}]
\right)
= 0$$ and the sequence (\[SecondApprox\]) is split.

[*Acknowledgments*]{}. It is a pleasure to acknowledge numerous fruitful discussions with C. Duval. We are also thankful to M. De Wilde, V. Fock and C. Roger for helpful suggestions.

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[^1]: Institute de Mathématiques, Université de Liège, Sart Tilman, Grande Traverse, 12 (B 37), B-4000 Liège, BELGIUM, mailto:plecomte@ulg.ac.be

[^2]: C.N.R.S., Centre de Physique Théorique, Luminy – Case 907, F–13288 Marseille, Cedex 9, FRANCE, mailto:ovsienko@cpt.univ-mrs.fr
