Structure of connected abelian Lie groups

Let $G$ be a connected Lie group that is abelian , and let $\mathfrak{g}$ be its Lie algebra .

Theorem (classification). There exist integers $n,m\ge 0$ such that

where $\mathbb{T}^m$ is an $m$-torus (isomorphic to $(S^1)^m$). In particular, $G$ is determined up to isomorphism by the pair $(n,m)$ with $\dim G = n+m$.

Idea of proof (covering + lattice).

• Since $G$ is connected and abelian, its universal cover $\widetilde{G}$ is a connected simply connected abelian Lie group. Such a group is isomorphic to the additive group of its Lie algebra, so $\widetilde{G}\cong (\mathfrak{g},+) \cong \mathbb{R}^{n+m}$; compare universal covering groups .
• The covering map $\widetilde{G}\to G$ has kernel a discrete subgroup of $\widetilde{G}$, hence a lattice in $\mathbb{R}^{n+m}$ up to linear change of coordinates.
• Splitting $\mathbb{R}^{n+m}$ into directions along the lattice and transverse directions produces $\mathbb{R}^n\times (\mathbb{R}^m/\mathbb{Z}^m)$.

Context. This result explains why the basic connected abelian examples are $\mathbb{R}^n$, circles, and tori; see the circle group and the torus . It also matches the infinitesimal picture: $\mathfrak{g}$ is an abelian Lie algebra , so the exponential map is a local group isomorphism whose global kernel records the lattice.