Example: the torus $T^n$

The $n$ -torus is the Lie group

T^n := (S^1)^n \cong (\mathbb R/2\pi\mathbb Z)^n.

It is a connected abelian Lie group.

Lie algebra and exponential (explicit)

The Lie algebra is

\mathrm{Lie}(T^n)\cong \mathbb R^n

as an abelian Lie algebra. Writing a vector $\theta=(\theta_1,\dots,\theta_n)\in\mathbb R^n$ , the exponential map is

\exp(\theta) = (e^{i\theta_1},\dots,e^{i\theta_n})\in (S^1)^n.

The kernel is the lattice

\ker(\exp)= (2\pi\mathbb Z)^n,

so $\mathbb R^n \to T^n$ is the universal covering homomorphism (compare covering Lie groups and universal cover).

One-parameter subgroups (calculation)

Given $a=(a_1,\dots,a_n)\in\mathbb R^n$ , the associated one-parameter subgroup is

\gamma_a(t)=\exp(ta)=(e^{ita_1},\dots,e^{ita_n}).

Its image is closed exactly when there is $T>0$ such that $Ta\in(2\pi\mathbb Z)^n$ . In general, its closure is a subtorus.

Context. The torus is the model for maximal tori in compact Lie groups (see maximal torus theorem) and for the structure of connected abelian Lie groups (see connected abelian Lie group structure).