Example: the torus $T^n$

The -torus is a connected abelian Lie group with Lie algebra and exponential map .

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 ).

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}).

This subgroup is closed (a subtorus) iff the coordinates of $a$ are rationally related; otherwise its image is dense in 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 ).