Let be $A$ the Lie algebra of polynomial vector fields. A p-cochain $C$ of $A$ is a p-linear alternate map from $A^p$ to $A$. For $p=0$, $C$ is an element of $A$. The coboundary operator $\eth$ is defined as follow: \begin{multline} \eth C (X_0,...,X_p) = \sum _{0\leq i \leq p } (-1)^i [X_i,C(X_0,...,\hat{X_i},...,X_p)] +\\ \sum _{0 \leq i < j \leq p} (-1)^{i+j} C ([X_i,X_j],...,\hat{X_i},...,\hat{X_j},...,X_p), \end{multline} $C$ a p-cochain and $X_0,...,X_n \in A$. As we can see $\eth C$ is a p+1-cochain if $C$ is a p-cochain.
Let be $Z^p (A)$ the space of all p-cocyle $C$ ($\eth C=0$) and $B^p(A)$ the set of all p-coboundaries $C$ ($C=\eth C'$). Since $\eth ^2 =0$, $B^p (A)$ is a subspace of $Z^p(A)$ and $H^p(A)=Z^p(A)/B^p(A)$ is called $p^{th}$ cohomology space of $A$.
My goal is to compute $H^2(A)$, the 2-cocycles are: \begin{align*} Z^2(A)=\{ C:A\times A \rightarrow A : [X_0,C(X_1,X_2)] -[X_1,C(X_0,X_2)] + [X_2,C(X_0,X_1)] \\ -C([X_0,X_1],X_2 ) + C([X_0,X_2],X_1 ) - C([X_1,X_2],X_0 ) =0\}, \end{align*} and the 2-coboundary are: $$B^2(A)= \{ B : A\times A \rightarrow A : B (X,Y)= [X,CY] + [CX,Y]-C[X,Y] \} $$ In wich condition two 2-cocycle $C_1$ and $C_2$ have the same class in $H^2(A)$? On other hand $C_1-C_2 \in B^2(A)$.
