Non-abelian Extensions of Lie algebras with derivations

Abstract

In this paper, we investigate non-abelian extensions of Lie algebras with derivations from several different perspectives. We show that the theory of non-abelian extensions of a Lie algebra with a derivation can be characterized by means of the second non-abelian cohomology, the Deligne groupoid, the homotopy category of strict Lie $2$-algebras with strict derivations, and the notion of a $(\mathfrak{g},D)$-kernel, respectively. Moreover, within this unified framework, we address the following existence problem: given a non-abelian extension of Lie algebras $$0⟶\mathfrak{h}\stackrel{i}{⟶}\hat{\mathfrak{g}}\stackrel{p}{⟶}\mathfrak{g}⟶0,$$ let $(K,D)\in \mathrm{Der}(\mathfrak{h})\times \mathrm{Der}(\mathfrak{g})$ be a pair of derivations of $\mathfrak{h}$ and $\mathfrak{g}$ respectively. When does there exist a derivation $\hat{D}$ of $\hat{\mathfrak{g}}$ such that $\hat{D}{|}_{\mathfrak{h}}=K$ and $D\circ p=p\circ \hat{D}.$ We provide an obstruction class for the existence of such a lift.

27pages, comments are welcome