Abelian Lie group - Imported Knowlpedia Content

Let $G$ be a Lie group.

Definition. $G$ is abelian if its multiplication is commutative:

gh=hg \quad \text{for all } g,h\in G.

Equivalent characterizations

Equivalently, the commutator subgroup $[G,G]$ is trivial, or (for connected $G$ ) its Lie algebra $\mathfrak{g}=\mathrm{Lie}(G)$ is an abelian Lie algebra.

Remarks

Motivation. For abelian $G$ , the Baker–Campbell–Hausdorff formula collapses to ordinary addition in the Lie algebra: in exponential coordinates, local multiplication is just $X+Y$ . This is one reason connected abelian Lie groups admit an explicit classification (see structure of connected abelian Lie groups).