Any lie group is a differentiable manifold.
How many differentiable structures on a lie group can there be?
For example a $SU(2)$ is diffeomorphic to $S^3$, it only has a unique differentiable structure.
How about other lie groups?
For
compact lie group, or
non-compact lie group like $\mathbf{R}^d$?
do we limit the types of differentiable structures by specifying the properties of Lie groups (compactness, connected or simply connected etc.)?
Edit: Here I meant: "smooth structures compatible with the given algebraic structure of Lie algebra and Lie group."