How many differentiable structures on a lie group?

Question

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

1. compact lie group, or

2. 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."

Answer 1

It is a standard fact of the theory of Lie groups that if $G_1, G_2$ are Lie groups and $f: G_1\to G_2$ is a continuous homomorphism, then $f$ is smooth. See for instance, here. If $f$ is also an isomorphism, then $f^{-1}$ will be smooth as well. From this, you conclude that if $G$ is a Lie group, then it has unique smooth structure compatible with its algebraic and topological structure. More precisely, if $s_1, s_2$ are smooth structures on $G$ (compatible with its algebraic and topological structure) then the identity map $(G,s_1)\to (G,s_2)$ is a diffeomorphism.

For some classes of Lie groups one can do even better: One does not even have to fix in advance the topological structure. This holds for all absolutely simple Lie groups. However, in general, an isomorphism of Lie groups need not be continuous, for instance, the Lie group $({\mathbb R},+)$ admits discontinuous automorphisms.