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

## TL;DR

This paper develops a cohomological framework to classify and analyze non-abelian extensions of strict Lie 2-algebras, establishing connections between extensions, homomorphisms, and cohomology groups.

## Contribution

It introduces a cohomology-based approach to characterize and classify non-abelian extensions of strict Lie 2-algebras, linking extensions to homomorphisms and cohomology classes.

## Key findings

- Extensions correspond to strict homomorphisms to Sout(rkh)
- Obstructions to extensions are in the third cohomology group
- Isomorphism classes of extensions are classified by the second cohomology group

## 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.