Physicist here! I have been studying $SU(2)$ and I have a few questions in regards to thinking about the group itself.
Consider the Lie group $SU(2)$.
One way to think about $SU(2)$ as a manifold is the set of unit quaternions, that is, $S^3$.
Q1: If we consider elements of $SU(2)$ as transformations, is there a way to visualize this transformation on the manifold?
We know, from representation theory, that we can find $n$-dimensional representations of $SU(2)$.
Q2: How does our view of the Lie Group changes when we deal with a different representation other than the "usual one" that is the 2D representation? That is, if we use the 10 dimensional representation of $SU(2)$, do we still think of the manifold as being $S^3$?
Q3: What can we say about the geometry of SU(2)? Does it change when we study it in light of a different representation of the group?
Thanks!
Q1: Yes/no/maybe: Give me an action of $SU(2)$ by transformations (of what kind?) on (what space?) and depending on the complexity of the situation I may or may not be able to do a visualization.
Q2: A Lie group $G$ (in your case, $G=SU(2)$) should be treated as a primary object here and it does not change (as a Lie group) if we consider a different representation. Accordingly, the underlying smooth manifold will not change either. A physical analogy would be, say, an electron: You can place it in a different environment which may change, say, its energy, but it will remain an electron.
Q3: It depends. One can put different geometries on $SU(2)$. Some of these will be intrinsic (independent of the group action), some may be extrinsic (depending on the action). The standard example of intrinsic geometry is the biinvariant Riemannian metric given by, say, the Killing form on the Lie algebra. An example of an extrinsic geometry is the chordal metric given by the identification of $SU(2)$ and the unit sphere in ${\mathbb C}^2$.