A review of Lie 2-algebras

TL;DR

This paper reviews the theory of Lie 2-algebras, including their definitions, examples from geometry, cohomology, and integration to Lie 2-groups, providing a comprehensive overview of this categorified algebraic structure.

Contribution

It summarizes multiple definitions, examples, cohomology, and integration methods of Lie 2-algebras, offering a unified overview of the subject.

Findings

Presented four types of Lie 2-algebras from geometric structures.

Analyzed the cohomology theory of Lie 2-algebras.

Discussed the integration of strict Lie 2-algebras to Lie 2-groups.

Abstract

We first recall two equivalent definitions of Lie $2$-algebras, categorification of Lie algebras and $2$-term $L_\infty$-algebras. Then we present four different kinds of Lie $2$-algebras from $2$-plectic manifolds, Courant algebroids, homotopy Poisson manifolds and affine multivector fields on a Lie groupoid respectively. Moreover, we recall the cohomology theory of Lie $2$-algebras and analyze its lower degree cases. The integration of strict Lie $2$-algebras to strict Lie $2$-groups is also discussed.