Abelian Lie algebra: Difference between revisions

Revision as of 13:46, 9 June 2007

This article defines a property for a Lie algebra

This article defines the analogue in Lie algebra of the following group property: [[Abelian group]][[Category:Analogues in other algebraic structures of Abelian group]]

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 $\exp t v . w - w$ . When the group is Abelian, the adjoint representation is trivial so $\exp t v . w = w$ and hence $\exp t v . w - w = 0$ 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.

Relation with other properties

Weaker properties

@@ Line 13: / Line 13: @@
 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 <math>v</math> and <math>w</math> 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 <math>[v,w]</math> can be defined as <math>d/dt (g.w - w)</math> where <math>g = \exp (tv)</math>.
+More explicitly, there is a natural action of the Lie group on its Lie algebra, called the [[adjoint representation]], and the commutator <math>[v,w]</math> can be defined as the differential at <math>t=0</math> of <math>\exp{tv}.w - w</math>. When the group is Abelian, the adjoint representation is trivial so <math>\exp{tv}.w = w</math> and hence <math>\exp{tv}.w - w = 0</math> identically. Thus <math>[v,w]=0</math> (as the differential of a constant function).
 ===If the Lie algebra is Abelian then the group is nearly so===