Decompositions of Infinite-Dimensional $A_{ \infty,\infty}$ Quiver Representations

TL;DR

This paper proves that all possibly infinite-dimensional representations of the $A_{\infty,\infty}$ quiver are Krull-Schmidt under certain arrow conditions, extending classical finite-dimensional results to an infinite setting.

Contribution

It establishes the Krull-Schmidt property for infinite-dimensional $A_{\infty,\infty}$ quiver representations with outward-pointing arrows, using linear algebraic methods.

Findings

All such representations are Krull-Schmidt.

The Krull-Schmidt property holds for infinite-dimensional cases under specified arrow conditions.

The results extend finite-dimensional representation theory to infinite quivers.

Abstract

Using linear algebraic methods we show that every (possibly infinite-dimensional) representation of a quiver with underlying graph $A_{\infty, \infty}$ is Krull-Schmidt, as long as the arrows in the quiver eventually point outward.