I like to visualize lie groups as flows on some manifold.
For example:
$SO(2)$ can be visualized as rotations of $S^1$ and it's lie algebra as constant vector fields on $S^1$.
Or $SO(1,1)$ can be visualized as flows on hyperbola $\{ (x,y) : x^2-y^2 = 1 \}$.
In general I visualize Lie group as subgroup of diffeomorphisms of some manifold and elements of lie algebra as vector field on this manifold
From these visualizations one can see that $SO(2)$ is connected and its exponential mapping is onto but one-to-one. And that $O(2)$ has two components. Or that $O(1,1)$ has four components.
So I would like to know if I can visualize other groups like Heisenberg group, symplectic group?
Is there a way how can I see that Lie group is simply connected?
Best Answer
Here are a few more examples you can easily "visualize":
$SO(3)$ is isometric to the projective space $\mathbb R P^3$, when both are equipped with the standard metrics. Its Lie algebra $(\mathfrak{so}(3),[,])$ is isomorphic to $(\mathbb R^3,\times)$ endowed with the cross product;
$SO(4)$ is isometric to $S^3\times S^3/\mathbb Z_2$, the quotient of the product of two $3$-spheres by an involution;
$SU(2)$ is isometric to the $3$-sphere $S^3$, when both have the standard metrics. Also the symplectic group $Sp(1)$ is isomorphic to $SU(2)$. As such, $SU(2)$ and $Sp(1)$ are the (universal) double cover of $SO(3)=\mathbb R P^3$.
In particular, all of the above are compact and connected Lie groups.