Deformations of Lie 2-algebras

TL;DR

This paper explores the deformation theory of Lie 2-algebras using cohomology, introduces Nijenhuis operators for trivial deformations, and examines abelian extensions through semidirect products.

Contribution

It establishes the correspondence between infinitesimal deformations and 2-cocycles, and introduces Nijenhuis operators to characterize trivial deformations of Lie 2-algebras.

Findings

Infinitesimal deformations correspond to 2-cocycles in cohomology.

Nijenhuis operators characterize trivial deformations.

Abelian extensions relate to deformations of semidirect product Lie 2-algebras.

Abstract

In this paper, we consider deformations of Lie 2-algebras via the cohomology theory. We prove that a 1-parameter infinitesimal deformation of a Lie 2-algebra $\g$ corresponds to a 2-cocycle of $\g$ with the coefficients in the adjoint representation. The Nijenhuis operator for Lie 2-algebras is introduced to describe trivial deformations. We also study abelian extensions of Lie 2-algebras from the viewpoint of deformations of semidirect product Lie 2-algebras.