nLab
supergravity Lie 3-algebra

The supergravity Lie 3-algebra $\mathrm{𝔰𝔲𝔤𝔯𝔞}(\mathrm{10,1})$ is a shifted central extension of super ∞-Lie algebras

$$0\to b^2𝔲(1)\to \mathrm{𝔰𝔲𝔤𝔯𝔞}(\mathrm{10,1})\to \mathrm{𝔰𝔦𝔰𝔬}(\mathrm{10,1})\to 0$$0 \to b^2 \mathfrak{u}(1) \to \mathfrak{sugra}(10,1) \to \mathfrak{siso}(10,1) \to 0

of the super Poincare Lie algebra $\mathrm{𝔰𝔦𝔰𝔬}(\mathrm{10,1})$ in 10+1 dimensions induced by a super Lie algebra 4-cocycle ${\mu }_4$.

This is the same mechanism by which the String Lie 2-algebra is a shifted central extensiono of $\mathrm{𝔰𝔬}(n)$.

In the D'Auria-Fre formulation of supergravity it is realized (more or less explicitly) that maximal 11-dimensional supergravity as a classical field theory is a higher gauge theory with coefficients in the supergravity Lie 3-algebra.