I've had a brief look through similar threads on this topic to see if my question has already been answered, but I didn't find quite what I was looking for, perhaps it is because I'm finding it hard to put words on my question and I hope that you will be able to help me ask it clearly.
I'm trying to link what I know from mathematics to what we are writing in physics for Noether's theorem. If I understand correctly we are looking at the symmetries of the action i.e. under which symmetry groups is it invariant. Noether's theorem allows us to calculate a conserved current in the case of a continuous symmetry (Lie groups), by means of so-called infinitesimal symmetries which I believe to be elements of the Lie algebra (so the tangent space to the neutral element) of the Lie group.
I'm guessing that if the action is invariant under the symmetry then it's "variation" should be 0 when we vary the system (space-time coordinate, or field) using this symmetry; it is exactly this step that I would like to understand better, how can I formalize this step correctly mathematically? How should I understand this variation, and how does it's calculation give rise to the elements of the Lie algebra?