Can we fully understand the structure of infinite-dimensional representations of certain algebraic objects? This study investigates the decomposition of infinite-dimensional representations of a quiver with underlying graph $$A_{\infty , \infty }$$. Gabriel’s Theorem classifies quivers with finitely many indecomposable representations. This research demonstrates that every representation of a quiver with underlying graph $$A_{\infty , \infty }$$ is infinite Krull-Schmidt (a direct sum of indecomposables) if the arrows eventually point outward. These indecomposables are thin and correspond to connected subquivers and limits of positive roots. The study provides an example of an $$A_{\infty , \infty }$$ quiver that is not infinite Krull-Schmidt and is therefore not eventually outward. By using linear algebraic methods, the paper characterizes the structure of these representations and contributes to the understanding of infinite-dimensional representation theory.
Published in Algebras and Representation Theory, this study aligns with the journal’s focus on algebraic structures and their representations. By examining the decomposition of infinite-dimensional quiver representations, it contributes to the journal's scope of advancing knowledge in representation theory and related areas of algebra.
| Category | Category Repetition |
|---|---|
| Science: Mathematics | 3 |
| Technology: Technology (General): Industrial engineering. Management engineering: Applied mathematics. Quantitative methods | 2 |