Path algebras of quivers and representations of locally finite Lie algebras

TL;DR

This paper investigates the noncommutative geometric structure of locally simple representations of certain infinite-dimensional Lie algebras, linking their annihilators to quivers and path algebras via noncommutative algebraic geometry.

Contribution

It establishes a correspondence between locally simple modules of these Lie algebras and points in noncommutative spaces defined by quivers, classifying the arising quivers and connecting them to symmetric group characters.

Findings

Each annihilator corresponds to a specific quiver.

Classified all quivers arising from these Lie algebras.

Connected quivers to symmetric group characters.

Abstract

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.