# Visualizing Lie groups.

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?

• You are realizing the groups by their left/right actions on themselves. I am not quite sure what you expect to see for other groups since, barring trivial examples, they are at least three-dimensional and curved. Of course, it's useful to study these actions for any group, I just wouldn't talk about visualization. – Marek Sep 29 '13 at 18:16
• I for one would really like a visualization, even if its just a list of patterns. Too bad this question is so old and the one answer is hardly a visualization. – BAR Oct 2 '15 at 22:05

Here are a few more examples you can easily "visualize":

1. $SO(3)$ 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;

2. $SO(4)$ is isometric to $S^3\times S^3/\mathbb Z_2$, the quotient of the product of two $3$-spheres by an involution;

3. $SU(2)$ 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.

• What is standard metrics on Lie group? – tom Oct 1 '13 at 19:13
• @tom: In the above list the "standard" metric is a bi-invariant metric. These metrics always exist when the group is compact, and tend to be "preferred" metrics because of their large isometry group (the Lie group endowed with this metric is a symmetric space). – Renato G. Bettiol Oct 1 '13 at 20:32
• What would be a visualisation of the $sl(2)$ group? – Millardo Peacecraft Feb 9 '14 at 23:44
• Could you point to some numerical exposition of the metric preserving nature of first pair of spaces? That will help calculate distance between rotation matrices. – user_1_1_1 Jul 12 '16 at 7:37