Suppose we have a finite dimensional Lie algebra $g$， Is there a machinery to describe all the irreducible representation of $g$.

Consider toy example: $sl_{2}$ or $sl_{3}$, how do we describe all the irreducible representations of them.

Further, consider quantum case, Is there a machinery way(like algorithm)describing all the irreducible representations of $U_{q}(sl_{2})$

EDIT: What I am looking for is an "mechanical" and canonical machinery describing **all** the irreducible representations(of course, not only finite dimensional representations,not only unitary representations)

EDIT2: What I am looking for is some reference to describe them in explicitly(such as $sl_{3}$）

anyalgebraic structure? $\endgroup$