Leavitt path algebras with bounded index of nilpotence and simple modules over them

TL;DR

This paper characterizes Leavitt path algebras with bounded nilpotence index, showing that all simple modules are graded -injective precisely when the underlying graph has no cycles and bounded path lengths and path counts.

Contribution

It provides a complete graphical description of Leavitt path algebras with bounded nilpotence index and characterizes when all simple modules are -injective.

Findings

Leavitt path algebras with bounded nilpotence index are characterized graphically.

All simple modules are -injective iff the graph has no cycles and bounded path lengths and counts.

The paper establishes a precise graph-theoretic criterion for -injectivity of simple modules.

Abstract

In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence and show that each graded simple module $S$ over a Leavitt path algebra with bounded index of nilpotence is graded $\Sigma$-injective, that is, $S^{(\alpha)}$ is graded injective for any cardinal $\alpha$. Furthermore, we characterize Leavitt path algebras over which each simple module is $\Sigma$-injective. We have shown that each simple module over a Leavitt path algebra $L_K(E)$ is $\Sigma$-injective if and only if the graph $E$ contains no cycles, and there is a positive integer $d$ such that the length of any path in $E$ is less than or equal to $d$ and the number of distinct paths ending at any vertex $v$ (including $v$) is less than or equal to $d$.