Differential operators and crossed homomorphisms of Lie 2-algebras

In this paper, we first introduce differential operators of arbitrary weight on Lie 2-algebras and classify skeletal differential Lie 2-algebras using the third cohomology of differential Lie algebras. Then crossed homomorphisms between two Lie 2-algebras are given, which are the formal inverse of relative Rota–Baxter operators and induce new Lie 2-algebra actions. Finally, the cohomology theory of crossed homomorphisms is studied and the first cohomology is applied to characterize linear deformations.