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 $\mathfrak{s}\mathfrak{l}(n^{\mathrm{\infty }})$, $\mathfrak o(n^\infty)$,and$\mathfrak{sp}(n^\infty)$\mathrm{.}Let$\mathfrak g_\infty$ be one of these Lie algebras, and let $I\subseteq U({\mathfrak{g}}_{\mathrm{\infty }})$ be the nonzero annihilator of a locally simple ${\mathfrak{g}}_{\mathrm{\infty }}$-module. We show that for each such $I$, there is a quiver $Q$ so that locally simple $\mathfrak g_\infty$-moduleswithannihilator$I$ are parameterised 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.