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