Cohomological characterizations of non-abelian extensions of strict Lie 2-algebras

Abstract

In this paper, we study non-abelian extensions of strict Lie 2-algebras via the cohomology theory. A non-abelian extension of a strict Lie 2-algebra $\g$ by $\frkh$ gives rise to a strict homomorphism from $\g$ to $\SOut(\frkh)$. Conversely, we prove that the obstruction of existence of non-abelian extensions of strict Lie 2-algebras associated to a strict Lie 2-algebra homomorphism from $\g$ to $\SOut(\frkh)$ is given by an element in the third cohomology group. We further prove that the isomorphism classes of non-abelian extensions of strict Lie 2-algebras are classified by the second cohomology group.