What is the Lie group infinitesimal generator? Our prof gave us the definition of the Lie group infinitesimal generator and it's kth-coordinate, he also explained why the kth-coordinate is called "the kth-coordinate", but nonetheless he didn't want to explain the Lie group infinitesimal generator because "we will understand it later as the course will continue".
I'm fine with that explanation but nevertheless I'd really like to at least know why we are defining the Lie group infinitesimal generator in the first place, and where it is used. Maybe nothing too deep as we just started working with Lie groups. But I want to grasp the intuition behind the Lie group infinitesimal generator
 A: A Lie group is a group that is also a manifold. Formally the generators are a basis for the tangent space of the identity element in the group.
Informally you can think of the generators as corresponding to small group transformations which are "near" the identity. Every element in the connected component of the group can be "generated" by repeatedly applying these transformations.
As an example consider the positive real numbers with multiplications as the group operation. The identity element of this group is the real number $1$. The tangent space at $1$ is just the real line, and the tangent vector $\sigma=+1$ can be taken as a basis for this 1D space.
Now consider the "infinitesimal transformation" which is "in the neighborhood of the identity element" of our group,
$g = 1+\epsilon \sigma,$
where $\epsilon$ is some "small" real number (can be positive or negative).
For a finite epsilon we can approximately represent most members of our group from repeated multiplication of $g$.
For instance suppose $\epsilon=0.001$ and we want to represent the group element $2$.
Then our approximate $g$ would be $1.001$ and we could represent $2$ by $g^{693}=1.999013$. Now of course we don't need to be so approximate. Using calculus we can get an exact representation.
Let $g_\theta$ be a member of the group labelled by a real number $\theta$. Then we can express $g_\theta$ in terms of our generator $g$ by the following limit.
$$g_\theta = \lim_{N\rightarrow \infty} (1+\theta \cdot \sigma /N)^N = e^{\theta \cdot \sigma}$$
Here I have replaced $\epsilon$ with $\theta/N$ and taken the limit as the exponent $N$ approaches infinity.
Now we have a way of representing our entire group in terms of this function $exp(\theta \cdot \sigma)$ where $\theta$ labels the different elements of the group.
The essential idea of Lie groups is to represent the connected component of your group-manifold in terms of the exponential map applied to a linear combination of your generators. The relations between the generators themselves can reveal a lot about the structure of the group.
