Let $F$ be a field of characteristic zero and consider the polynomial algebra $A=F[x_1,x_2,\ldots,x_n]$ in $n$ indeterminates over $F$. I recall that the derivations of $A$ form a simple Lie algebra $W_n$ of infinite dimension over $F$, called a $\textit{Witt algebra}$. Moreover, consider $$K_{2r+1}=\{D\in W_{2r+1}\mid D(\omega)\in A\omega\},$$ where $\omega=\mathrm{d} x_{2r+1}+\sum\limits_{j=1}^rx_j\mathrm{d} x_{j+r}-x_{j+r} \mathrm{d} x_j$. Then $K_{2r+1}$ is also an infinite dimensional simple Lie algebra over $F$, called a $\textit{contact algebra}$. (For more detalis, see e.g. Section 0.1.3 of the celebrated Mathieu's paper https://link.springer.com/article/10.1007/BF02100615)

It seems to be a known fact that all derivations of $W_n$ and $K_{2r+1}$ are inner and, moreover, the central extensions of these Lie algebras are trivial. Thus, in cohomological terms, all the groups $H^1(W_n,W_n)$, $H^1(K_{2r+1},K_{2r+1})$, $H^2(W_n,F)$, $H^2(K_{2r+1},F)$ are zero.

However, it is not clear to me where the mentioned results can be found. Is there any good reference? I will be grateful for any suggestion.

Let $F$ be a field of characteristic zero and consider the polynomial algebra $A=F[x_1,x_2,\ldots,x_n]$ in $n$ indeterminates over $F$. I recall that the derivations of $A$ form a simple Lie algebra $W_n$ of infinite dimension over $F$, called a Witt algebra. Moreover, consider $$K_{2r+1}=\{D\in W_{2r+1}\mid D(\omega)\in A\omega\},$$ where $\omega=\mathrm{d} x_{2r+1}+\sum\limits_{j=1}^rx_j\mathrm{d} x_{j+r}-x_{j+r} \mathrm{d} x_j$. Then $K_{2r+1}$ is also an infinite dimensional simple Lie algebra over $F$, called a contact algebra. (For more details, see e.g. Section 0.1.3 of the celebrated Mathieu's paper Classification of simple graded Lie algebras of finite growth.)

It seems to be a known fact that all derivations of $W_n$ and $K_{2r+1}$ are inner and, moreover, the central extensions of these Lie algebras are trivial. Thus, in cohomological terms, all the groups $H^1(W_n,W_n)$, $H^1(K_{2r+1},K_{2r+1})$, $H^2(W_n,F)$, $H^2(K_{2r+1},F)$ are zero.

However, it is not clear to me where the mentioned results can be found. Is there any good reference? I will be grateful for any suggestion.

Let $K$ be a field of characteristic zero and consider the polynomial algebra $A=K[x_1,x_2,\ldots,x_n]$ in $n$ indeterminates over $K$. I recall that the derivations of $A$ form a simple Lie algebra $W_n$ of infinite dimension over $K$, called a $\textit{Witt algebra}$. Moreover, consider $$K_{2r+1}=\{D\in W_{2r+1}\mid D(\omega)\in A\omega\},$$ where $\omega=\mathrm{d} x_{2r+1}+\sum\limits_{j=1}^rx_j\mathrm{d} x_{j+r}-x_{j+r} \mathrm{d} x_j$. Then $K_{2r+1}$ is also an infinite dimensional simple Lie algebra over $K$, called a $\textit{contact algebra}$. (For more detalis, see e.g. Section 0.1.3 of the celebrated Mathieu's paper https://link.springer.com/article/10.1007/BF02100615)

It seems to be a known fact that all derivations of $W_n$ and $K_n$ are inner and, moreover, the central extensions of these Lie algebras are trivial. Thus, in cohomological terms, all the groups $H^1(W_n,W_n)$, $H^1(K_{2r+1},K_{2r+1})$, $H^2(W_n,K)$, $H^2(K_{2r+1},K)$ are zero.

However, it is not clear to me where the mentioned results can be found. Is there any good reference? I will be grateful for any suggestion.

Let $F$ be a field of characteristic zero and consider the polynomial algebra $A=F[x_1,x_2,\ldots,x_n]$ in $n$ indeterminates over $F$. I recall that the derivations of $A$ form a simple Lie algebra $W_n$ of infinite dimension over $F$, called a $\textit{Witt algebra}$. Moreover, consider $$K_{2r+1}=\{D\in W_{2r+1}\mid D(\omega)\in A\omega\},$$ where $\omega=\mathrm{d} x_{2r+1}+\sum\limits_{j=1}^rx_j\mathrm{d} x_{j+r}-x_{j+r} \mathrm{d} x_j$. Then $K_{2r+1}$ is also an infinite dimensional simple Lie algebra over $F$, called a $\textit{contact algebra}$. (For more detalis, see e.g. Section 0.1.3 of the celebrated Mathieu's paper https://link.springer.com/article/10.1007/BF02100615)

It seems to be a known fact that all derivations of $W_n$ and $K_{2r+1}$ are inner and, moreover, the central extensions of these Lie algebras are trivial. Thus, in cohomological terms, all the groups $H^1(W_n,W_n)$, $H^1(K_{2r+1},K_{2r+1})$, $H^2(W_n,F)$, $H^2(K_{2r+1},F)$ are zero.

However, it is not clear to me where the mentioned results can be found. Is there any good reference? I will be grateful for any suggestion.