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I'm looking for a proof of the $1^{st}$ Whitehead lemma.

$1^{st}$ Whitehead lemma:

If $L$ is a Lie Algebra which is semisimple, then for any finite dimensional representation $(V,\rho)$ of $L$, $H^1(L,V,\rho) = \{0\}$

I cannot find any proof on the internet. If someone knows where I can find one...

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user405156
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2 Answers2

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There is a proof here. I did not read it, though.

José Carlos Santos
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Here is a proof using Weyl's Theorem. And the proof of Weyl's Theorem can be found on the internet, even on MSE. So we know that every finite-dimensional $L$-module of a semisimple Lie algebra over $K$ is semisimple, i.e., has a module complement. Hence all module extensions are trivial, and we have $$ 0 = {\rm Ext}(B,A) = H^1(L; {\rm Hom}(B,A)) $$ for each pair of finite-dimensional $L$-moduls $A$ and $B$. In particualr, we obtain for $B=K$, the trivial module, $$ H^1(L,A)=0 $$ for all finite-dimensional $L$-modules $A$.

Dietrich Burde
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