Derivations and central extensions of some infinite dimensional simple Lie algebras in characteristic zero

Question

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.

Answer 1

The fact that all derivations of the one-sided Witt algebras $W_n$ are inner follows from Theorem 2 of Ikeda - Kawamoto: On the derivations of generalized Witt algebras over a field of characteristic zero, once one observes that the stated result holds with the same proof if one replaces $G=\mathbb{Z}^n$ by $H=\mathbb{N}^n$, yielding the desired conclusion.

Moreover, the fact that every central extension of $W_n$ splits is contained in Lemma 3.2 of Su - Zhao: Second cohomology group of generalized type Lie algebras and certain representations.

Finally, the vanishing of $H^1(K_{2r+1},K_{2r+1})$ and $H^2(K_{2r+1},\mathbb{F})$ follows, respectively, from Theorem 3.1 and Theorem 4.6 in Song - Su: Derivations and 2-Cocycles of Contact Lie Algebras Related to Locally-Finite Derivations, provided I am correctly interpreting the somewhat heavy notation used in that paper.