Simply Connected Lie Groups
Definition
A matrix Lie group $G$ being simply connected means that $G$ is connected and every closed curve on $G$ can be continuously contracted to a point in $G$.
Explanation
Mathematically, the statement that a closed curve can be continuously contracted to a point in $G$ is equivalent to satisfying the following condition. For any path on $G$ that is $A(0) = A(1)$, there exists a continuous function $A(t, s)$ $(0 \le s, t \le 1)$ such that
- Closed curve: $A(0, s) = A(1, s)$ $(s \in [0, 1])$
- Contraction start point: $A(t, 0) = A(t)$ $(t \in [0, 1])$
- Contraction end point: $A(t, 1) = A(0, 1)$ $(t \in [0, 1])$
In more technical language, simply connected means every closed curve is null-homotopic. The $A(t, s)$ above is called a homotopy.
Examples
The following matrix Lie groups are simply connected.
- Special unitary group $\operatorname{SU}(n)$