nLab
abelian Lie algebra

Definition

A Lie algebra $𝔤$ is abelian if its bracket is identically 0, in that for all $x,y\in 𝔤$ we have

$$[x,y]=0.$$[x,y] = 0 \,.

Examples

Every vector space has a (necessarily unique) abelian Lie algebra structure. As such, we may identify an abelian Lie algebra with its underlying vector space.

A $0$-dimensional or $1$-dimensional Lie algebra must be abelian. The $0$-dimensional Lie algebra is the trivial Lie algebra. The $1$-dimensional Lie algebra is a simple object in LieAlg, but it is traditionally not considered a simple Lie algebra.

Lie integration

Under Lie integration abelian Lie algebras integrate to abelian Lie group?s.