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We explore the (noncommutative) geometry of locally simple representations ofthe diagonal locally finite Lie algebras $\mathfrak{sl}(n^\infty)$, $\mathfrako(n^\infty)$, and $\mathfrak{sp}(n^\infty)$. Let $\mathfrak g_\infty$ be one ofthese Lie algebras, and let $I \subseteq U(\mathfrak g_\infty)$ be the nonzeroannihilator of a locally simple $\mathfrak g_\infty$-module. We show that foreach such $I$, there is a quiver $Q$ so that locally simple $\mathfrakg_\infty$-modules with annihilator $I$ are parameterised by "points" in the"noncommutative space" corresponding to the path algebra of $Q$. Methods ofnoncommutative algebraic geometry are key to this correspondence. We classifythe quivers that arise and relate them to characters of symmetric groups.

Path algebras of quivers and representations of locally finite Lie algebras