I'm currently attempting to understand a little bit about how the exponential map works in general. I'll try to lay out what it is I think I've understood and where I think the problem lies.
If I have a 1 dimensional Lie group (or subgroup) $g$ that is parametrized by some parameter $\alpha$, then the corresponding Lie algebra will be spanned by the vector $\frac{\partial}{\partial\alpha}\big|_e\in T_eg$, which is then called the generator. I can then recover part of the Lie group around the identity via the exponential map, which gives me a smooth curve $X(t) = \exp(t\frac{\partial}{\partial\alpha}\big|_e)$.
But how does one actually compute the exponential map in this case? Is it similar to the case with matrix representations where it's just the exponential series, such that $$\exp(t\frac{\partial}{\partial\alpha}\bigg|_e) = \left[\sum_{n=0}^\infty \frac{1}{n!}t^n\partial_\alpha^{(n)}\right]_e$$ or would it be completely different?
This came up in a physics context where I was trying to figure out why an action being invariant wrt the infinitesimal transformations implies that it is also invariant under finite transformations. It makes sense when the algebra can be represented by matrices, since if $T$ maps to $0$, then so does $T^n$, but how does it work with differential operators?