Non abelian cohomology of extensions of Lie algebras as Deligne groupoid

Abstract

In this note we show that the theory of non abelian extensions of a Lie algebra $\mathfrak{g}$ by a Lie algebra $\mathfrak{h}$ can be understood in terms of a differential graded Lie algebra $L$. More precisely we show that the non-abelian cohomology $H_{nab}^2(\mathfrak{g},\mathfrak{h})$ is $\mathcal{MC}(L)$, the ${\pi }_0$ of the Deligne groupoid of $L$.

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