nLab
Lie 2-algebra

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Lie 22-algebras

Idea

A Lie 2-algebra is to a Lie 2-group as a Lie algebra is to a Lie group. Thus, it is a vertical categorification of a Lie algebra.

Definition

Semistrict case

A (“semistrict”) Lie 2-algebra 𝔤\mathfrak{g} is an L-∞-algebra with generators concentrated in the lowest two degrees.

This means that it is

The Jacobiator identity equivalently expresses the commutativity of the following diagram in the given 2-vector space (analogous to the pentagon identity)

(graphics grabbed from Baez-Crans 04, p. 19)

Strict case

If the trinary bracket [,,][-,-,-] in a Lie 2-algebra is trivial, one speaks of a strict Lie 2-algebra. Strict Lie 2-algebras are equivalently differential crossed modules (see there for details).

Examples

References