Classification
If G is a semisimple group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations. The irreducibles are indexed by highest weight; the allowable (dominant) highest weights satisfy a suitable positivity condition. In particular, there exists a set of fundamental weights, indexed by the vertices of the Dynkin diagram of G, such that dominant weights are simply non-negative integer linear combinations of the fundamental weights. The characters of the irreducible representations are given by the Weyl character formula.
If G is a commutative Lie group, then its irreducible representations are simply the continuous characters of G: see Pontryagin duality for this case.
A quotient representation is a quotient module of the group ring.
Read more about this topic: Representation Of A Lie Group