Decomposable Leavitt path algebras for arbitrary graphs

TL;DR

This paper characterizes when Leavitt path algebras, associated with arbitrary graphs over any field, are indecomposable and describes their decomposition into simpler components, providing a clear structural understanding.

Contribution

It provides a complete characterization of indecomposable Leavitt path algebras for arbitrary graphs and describes their unique decompositions under finiteness conditions.

Findings

Characterization of indecomposable Leavitt path algebras

Decomposition of algebras into sums of simpler Leavitt path algebras

Existence of unique indecomposable decompositions under certain conditions

Abstract

For any field $K$ and for a completely arbitrary graph $E$, we characterize the Leavitt path algebras $L_K(E)$ that are indecomposable (as a direct sum of two-sided ideals) in terms of the underlying graph. When the algebra decomposes, it actually does so as a direct sum of Leavitt path algebras for some suitable graphs. Under certain finiteness conditions, a unique indecomposable decomposition exists.