Editing Lie algebra (section)

===Goals of representation theory===
One important aspect of the study of Lie algebras (especially semisimple Lie algebras, as defined below) is the study of their representations. Although Ado's theorem is an important result, the primary goal of representation theory is not to find a faithful representation of a given Lie algebra <math>\mathfrak{g}</math>. Indeed, in the semisimple case, the adjoint representation is already faithful. Rather, the goal is to understand all possible representations of <math>\mathfrak{g}</math>. For a semisimple Lie algebra over a field of characteristic zero, [[Weyl's theorem on complete reducibility|Weyl's theorem]]<ref name="reducibility">{{harvnb|Hall|2015|loc=Theorem 10.9.}}</ref> says that every finite-dimensional representation is a direct sum of irreducible representations (those with no nontrivial invariant subspaces). The finite-dimensional irreducible representations are well understood from several points of view; see the [[representation theory of semisimple Lie algebras]] and the [[Weyl character formula]].