I'm introducing myself to Lie Groups theory, and I read Ado's Theorem says that every finite real Lie algebra is isomorphic to a Lie subalgebra of $\mathfrak{gl}(n,\mathbb{R})$.
I found the following problem as a consequence of Ado's Theorem:
"Give infinite examples of non isomorphic Lie groups with Lie algebra $\mathfrak{so}(3)$."
Because of the following statement:
"If a Lie group has Lie algebra $\mathfrak{g}$, then it is isomorphic to $G/\Gamma$, where $G$ is the simply connected Lie group with Lie algebra $\mathfrak{g}$, and $\Gamma$ is a discrete subgroup of the center of $G$, $Z(G)$."
I think I must work with a group $G$ non simpy connected, but I don't achieve the examples. Could you help me?
Best Answer
One cannot find an infinite number of connected Lie groups with Lie algebra $\mathfrak{so}(3)$ because such a Lie group is either $\mathrm{SO}(3)$, either $\mathbb{S}^3$, the universal cover of $\mathrm{SO}(3)$. This is because $\mathbb{S}^3$ is a simply connected Lie group with Lie algebra $\mathfrak{so}(3)$ and has center $\{\pm 1\}$, thus the only discrete subgroups of its center are $\{1\}$ and $\{1,-1\}$, and the results follows from what you stated.
On the other hand, there are infinitely many non-connected Lie groups with Lie algebra $\mathfrak{so}(3)$. For example, $\mathbb{S}^3 \times \mathbb{Z}/n\mathbb{Z}$ for all $n \geqslant 2$.