nLab
strict Lie 2-algebra

Idea

A strict Lie 2-algebra is the infinitesimal approximation to a smooth strict 2-group in generalization of how an ordinary Lie algebra is the infinitesimal approximation to a Lie group.

It is the special case of a general Lie 2-algebra for which the Jacobi identity does hold (and not just up to nontrivial isomorphism). Said more generally: a strict Lie 2-algebra is an $L_{\mathrm{\infty }}$-algebra with generators just in degree 1 and 2 and at most the unary and binary brackets being nontrivial.

In direct analogy to how strict 2-groups are equivalently encoded in a smooth crossed module of groups, a strict Lie 2-algebra is equivalently encoded in a differential crossed module of ordinary Lie algebras.

Examples

Derivation Lie 2-algebra

The Lie version of a smooth automorphism 2-group is the derivation Lie 2-algebra $\mathrm{Der}(𝔤)$ of an ordinary Lie algebra $𝔤$. This is the one coming from the differential crossed module $(𝔤\stackrel{\mathrm{ad}}{\to }\mathrm{der}(𝔤))$.