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.

Definition

The notion of strict Lie 2-algebra is the special case of a general notion Lie 2-algebra for which the Jacobi identity does hold (and not just up to nontrivial isomorphism).

More precisely: a strict Lie 2-algebra is an ∞-Lie algebra with generators just in degree 1 and 2 and at most the unary and binary brackets being nontrivial.

Equivalently, this is a dg-Lie algebra with generators in the lowest two degrees.

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 $Der(\mathfrak{g})$ of an ordinary Lie algebra $\mathfrak{g}$. This is the one coming from the differential crossed module $(\mathfrak{g} \stackrel{ad}{\to} der(\mathfrak{g}))$.

Related concepts

• Lie algebra, Lie group

• Lie 2-algebra, Lie 2-group

• L-infinity algebra, smooth ∞-group

• Lie algebroid, Lie groupoid

• L-∞ algebroid, smooth ∞-groupoid