Path algebras of quivers and representations of locally finite Lie algebras

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.

arXiv comment30 pages; v2 typos and minor errors corrected. To appear in IMRN

References in corpus (1)

• Branching laws for tensor modules over classical locally finite Lie algebras