The socle series of a Leavitt path algebra

TL;DR

This paper studies the structure of Leavitt path algebras through their socle series, classifies graphs based on this structure, and shows that their Loewy length can be any ordinal up to omega.

Contribution

It classifies graphs for which Leavitt path algebras have a specific socle series and demonstrates the possible Loewy lengths for these algebras.

Findings

Classified graphs with Leavitt path algebras equal to a given socle layer.

Proved that Loewy length can be any ordinal up to omega.

Showed that for row-finite graphs, Loewy length is at most omega.

Abstract

We investigate the ascending Loewy socle series of Leavitt path algebras $L_K(E)$ for an arbitrary graph $E$ and field $K$. We classify those graphs $E$ for which $L_K(E)=S_{\lambda}$ for some element $S_{\lambda}$ of the Loewy socle series. We then show that for any ordinal $\lambda$ there exists a graph $E$ so that the Loewy length of $L_K(E)$ is $\lambda$. Moreover, $\lambda \leq \omega $ (the first infinite ordinal) if $E$ is a row-finite graph.