Path Algebras of Quivers and Representations of Locally Finite Lie Algebras

We explore the (noncommutative) geometry of locally simple representations of the diagonal locally finite Lie algebras sl(n(infinity)), o(n(infinity)), and sp(n(infinity)). Let g(infinity) be one of these Lie algebras, and let I subset of U(g(infinity)) be the non-zero annihilator of a locally simple g(infinity)-module. We show that for each such I, there is a quiver Q so that locally simple g(infinity)-modules with annihilator I are parameterized by "points" in the "noncommutative space" corresponding to the path algebra of Q. Methods of noncommutative algebraic geometry are key to this correspondence. We classify the quivers that arise and relate them to characters of symmetric groups.