Example: U(1) (the circle group)

The circle group is the compact Lie group

U(1)=\{z\in\mathbb C:|z|=1\}\cong S^1.

It is connected and abelian.

Lie algebra

Its Lie algebra (as a real Lie algebra) is

\mathfrak u(1)= i\mathbb R,

with trivial bracket, so it is an abelian Lie algebra (compare unitary Lie algebra for general $U(n)$ ).

Exponential map and universal cover (explicit calculation)

Identifying $\mathfrak u(1)\cong\mathbb R$ via $t\mapsto it$ , the exponential map is

\exp(it)=e^{it}.

Its kernel is

\ker(\exp)=\{2\pi k\,i : k\in\mathbb Z\}\cong 2\pi\mathbb Z,

so the map

p:\mathbb R\to U(1),\qquad p(t)=e^{it},

One-parameter subgroups

Every one-parameter subgroup of $U(1)$ has the form $t\mapsto e^{iat}$ for some $a\in\mathbb R$ .

$U(1)$ is the basic building block of tori and the prototypical example where the exponential map is surjective but not injective.