Questions about $su(2)$.

Question

Edit: In physics, it seems that people usually study $su(2)$ but not only $sl(2)$? Why people study $su(2)$ but not only $sl(2)$?

Answer 1

The form of the question is somewhat iffy, but may be intended to ask a question that has a sense that I understand: since most representations are on complex vector spaces, a representation of a Lie algebra is indistinguishable from a representation of its complexification. The complexifications of su(2) and sl(2,R) are both sl(2,C)... often unhelpfully written merely as sl(2), not acknowledging the real-versus-complex game.

Perhaps another iteration of the question would make clearer what other/remaining issues might be...

Edit: The Lie groups $SU(2)$ and $SL_2(\mathbb R)$ are easier to distinguish: the former is compact and the latter is not, which has as consequence that the irreducible unitary repns of the former are all finite-dimensional (indexed by highest weight), while the irreducible unitaries of $SL_2(\mathbb R)$ are all infinite-dimensional (apart from the trivial repn). Appreciating this point about the Lie groups is probably necessary in order to appreciate distinctions about the Lie algebras which otherwise might seem capricious.

Answer 2

I can try to answer. Elements of $SL(n)$ preserve only volume, but elements of $SU(n)$ also preserve distance.