Abelian Lie algebra

This article defines a property for a Lie algebra

ANALOGY: This is an analogue in Lie algebra of a property encountered in group. Specifically, it is a Lie algebra property analogous to the group property: Abelian group
View other analogues of Abelian group | View other analogues in Lie algebras of group properties (OR, View as a tabulated list)

Definition

A Lie algebra is said to be Abelian if the Lie bracket of any two elements in it is zero.

Relation with the Lie group

Lie algebra of an Abelian Lie group is Abelian

If we start with an Abelian Lie group, then its Lie algebra is also Abelian. This follows from the fact that for a Lie group, the Lie bracket of two tangent vectors $v$ and $w$ is the limit of a certain difference expression of conjugates.

More explicitly, there is a natural action of the Lie group on its Lie algebra, called the adjoint representation, and the commutator $[v,w]$ can be defined as the differential at $t=0$ of $\exp{tv}.w - w$. When the group is Abelian, the adjoint representation is trivial so $\exp{tv}.w = w$ and hence $\exp{tv}.w - w = 0$ identically. Thus $[v,w]=0$ (as the differential of a constant function).

If the Lie algebra is Abelian then the group is nearly so

The Lie algebra being Abelian does not force the group to be Abelian. However, we can certainly conclude that the connected component containing the identity is Abelian. In other words, the group is an Abelian-by-discrete group.