I'm currently studying differential geometry on smooth manifolds using differential forms and I'm trying to apply it to what I have learned earlier about Lie groups, but something doesn't seem to quite work out.
More specifically, I'm looking at the smooth manifold of a Lie group and have a homomorphism to $\mathbb{R}^n$ from where I would like to pull back and push forward the differential geometric properties to the group. When I push-forward the coordinate vector fields $\partial_\mu$ I do get directional partial derivatives with respect to the generators, i.e. $\frac{\partial}{\partial G_\mu}$, so these seem to span the tangent space.
However from Lie group theory I know and understand that the generators $G_\mu$ themselves span the tangent space, and it's not quite clear to me how to match that with the partial derivatives. Is this a different definition of tangent vectors that is used here?
Next, I'm not sure how the coordinate differentials $dx_\mu$ are pulled back to the group to give sensible results. In the end what I would like to achieve is to transfer the lie algebra of the group to an algebra on the representation with vector fields and differentials. How is that done?
I'm probably missing something very fundamental here, but my attempts to understand only made my confusion greater. My textbooks are very unclear on these aspects.