Examples of familiar, easy-to-visualize manifolds that admit Lie group structures

Question

I have a trouble learning Lie groups --- I have no canonical example to imagine while thinking of a Lie group. When I imagine a manifold it is usually some kind of a $2$D blanket or a circle/curve or a sphere, a torus etc.

However I have a problem visualizing a Lie group. The best one I thought is $SO(2)$ which as far as I understand just a circle. But a circle apparently lacks distinguished points so I guess there is no way to canonically prescribe a neutral element to turn a circle into a group $SO(2)$.

Examples I saw so far start from a group, describe it as a group of matrices to show that the group is endowed with the structure of a manifold. I would appreciate the other way --- given a manifold show that it is naturally a group. And such a manifold should be easily imaginable.

Answer 1

A good example is the $3$-sphere $\mathbb{S}^3$. In my mind, I think of this first and foremost as a geometric entity, and certainly people were considering spheres before they started to think about groups. As you may know, $\mathbb{S}^3$ can be made into a group by identifying it with the set of unit quaternions

$$\{ q \in \mathbb{H}: \lVert q \rVert = 1\}$$

Then, the group structure on $\mathbb{S}^3$ is inherited from the group structure of the quaternions, making $\mathbb{S}^3$ a Lie group.

Answer 2

Think of $SO_2$ as the group of $2\times 2$ rotation matrices:

$$ \left[\begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array} \right]$$

or the group of complex numbers of unit length $e^{i\theta}$.

You can convince yourself directly from definitions that either of these objects is a group under the appropriate multiplication, and that they are isomorphic to each other.

This group (which is presented in two ways) is a 1-manifold because it admits smooth parametrization in one variable ($\theta$ here).

Why does this group represent the circle $S^1$? Well, the matrices of the above form are symmetries of circles about the origin in $\mathbb{R}^2$, and the trace of $e^{i\theta}$ is the unit circle in the complex plane. I think it is best conceptually to think of Lie groups first as groups, and then to develop geometric intuition to "flavor" your algebraic construction.