We explore the (noncommutative) geometry of representations of locally finite Lie algebras. Let L be one of these Lie algebras, and let I in U(L) be the annihilator of a locally simple L-module. We show that for each such I, there is a quiver Q so that locally simple L-modules with annihilator I are parameterized by "points" in the "noncommutative space" corresponding to the path algebra of Q. We classify the quivers that occur and along the way discover a beautiful connection to characters of the symmetric groups $S_n$.