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.

  • $\begingroup$ Did you try reading the proof of the lemma and checking if anything breaks down? $\endgroup$ – Pedro Tamaroff Aug 18 '18 at 9:37
  • $\begingroup$ 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. $\endgroup$ – Blazej Aug 18 '18 at 9:44

The Whitehead Lemmas no longer hold for simple Lie superalgebras.

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

  • $\begingroup$ Technically this answers my question, so I will accept your answer eventually. May I first ask if you are aware of some special class of super Lie algebras for which the first whitehead lemma is true? $\endgroup$ – Blazej Aug 18 '18 at 13:32
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    $\begingroup$ Concerning references for cohomology of Lie superalgebras, see the paper by Scheunert, and the references given there. $\endgroup$ – Dietrich Burde Aug 24 '18 at 11:46

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