I move the question here.
Whitehead's lemma (Lie algebras) is: Let $\mathfrak{g}$ be a semisimple Lie algebra over a field of characteristic zero, $V$ a finite-dimensional module over it and $f$: $\mathfrak{g} \to V$ a linear map such that $f([x, y]) = xf(y) - yf(x)$. The lemma states that there exists a vector $v$ in $V$ such that $f(x) = xv$ for all $x$.
If we let $\mathfrak{g}$ be a reductive Lie algebra (for example, let $\mathfrak{g} = \mathfrak{gl}_n$), the conclusion of Whitehead's lemma is still true or not (or we need to add some other conditions)? Are there some references about this? Any help will be greatly appreciated!
Edit: I would like to know in particular the case of $\mathfrak{gl}_n$. Do we also have the following: all Lie bialgebra structure on $gl_n$ are coboundary (see also the post)?
The following result is proved in Bourbaki's book on Lie algebras:
Theorem (A converse to the First Whitehead Lemma). Any finite-dimensional Lie algebra over the field of characteristic zero such that its first cohomology with coefficients in any finite-dimensional module vanishes, is semisimple.
Since $\mathfrak{gl}(n)$ is not semisimple (it has a nontrivial center), Whitehead's first Lemma must fail for $\mathfrak{gl}(n)$ and some finite-dimenisonal module $V$ (and it does).