# Is Whitehead lemma true for super Lie algebras?

Classical Whitehead lemma states that if $\mathfrak g$ is a finite-dimensional complex Lie algebra and $M$ is a finite-dimensional $\mathfrak g$-module, then first cohomology group $H^1(\mathfrak g, M)$ (defined for example as cohomology of the Chevalley-Eilenberg complex) is trivial. I need to know if this is also true for super Lie algebras.

• Did you try reading the proof of the lemma and checking if anything breaks down? – Pedro Tamaroff Aug 18 '18 at 9:37
• Dear Pedro, I looked at it briefly. The problem is that it uses other properties of Lie algebras for which I also don't know if they generalize to graded case. It will be a great exercise to go through all of that, but this will take time. I hope someone can answer before that. Actually I will also appreciate any references on cohomology of super Lie algebras. – Blazej Aug 18 '18 at 9:44

Reference: Representations of algebraic groups, quantum groups and Lie algebras, page $121$.