nLab Lie 3-algebra

Definition

A Lie 3-algebra or $3$-Lie algebra is a 3-truncated L-∞-algebra. Under Lie integration is the infinitesimal approximation to a Lie 3-group.

Models by 2-crossed modules

A semistrict model for Lie 3-algebras, analogous to that of 3-groupoids that are Gray groupoids is given by differential 2-crossed modules.

Examples

• 11d supergravity Lie 3-algebra

• 4d supergravity Lie 3-algebra

References

Lie 3-algebras coming from differential 2-crossed modules and parallel transport for Lie 3-algebra valued forms is discussed in

• João Faria Martins, Roger Picken, The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module (arXiv:0907.2566)

On weak Lie 3-algebras:

• Malte Dehling, On weak Lie 3-algebras [arXiv:1710.11104]