2026/04/29 by Jun Jiang, Kanghe Xu, KangHe Xu
Mathematics · #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology #Advanced Algebra and Geometry
paper · pdf · doi:10.1016/j.geomphys.2026.105936
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 (\mathfrakg, D)-kernel, respectively. Moreover, within this unified framework, we address the following existence problem: given a non-abelian extension of Lie algebras 0\longrightarrow\mathfrakh\overseti\longrightarrow\mathfrakg\oversetp\longrightarrow\mathfrakg\longrightarrow 0, let (K,D)\inDer(\mathfrakh)\timesDer(\mathfrakg) be a pair of derivations of \mathfrakh and \mathfrakg respectively. When does there exist a derivation D of \mathfrakg such that D|_\mathfrakh=K and D∘ p=p∘D. We provide an obstruction class for the existence of such a lift.