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

Gabriel’s Theorem states that the quivers which have finitely many isomorphism classes of indecomposable representations are exactly those with underlying graph one of the ADE Dynkin diagrams and that the indecomposables are in bijection with the positive roots of this graph. When the underlying graph is \(\varvec{A_n}\), these indecomposable representations are thin (either 0 or 1 dimensional at every vertex) and in bijection with the connected subquivers. Using linear algebraic methods we show that every (possibly infinite-dimensional) representation of a quiver with underlying graph \(\varvec{A_{\infty , \infty }}\) is infinite Krull-Schmidt, i.e. a direct sum of indecomposables, as long as the arrows in the quiver eventually point outward. We furthermore prove that these indecomposable are again thin and in bijection with both the connected subquivers and the limits of the positive roots of \(\varvec{A_{\infty , \infty }}\) with respect to a certain uniform topology on the root space. Finally we give an example of an \(\varvec{A_{\infty , \infty }}\) quiver which is not infinite Krull-Schmidt and hence necessarily is not eventually-outward.