# Leavitt path algebras: Graded direct-finiteness and graded $\Sigma$-injective simple modules

## TL;DR

This paper characterizes Leavitt path algebras that are graded $ ext{Sigma}$-$V$ rings, linking algebraic properties to graph structures and exploring differences between graded and non-graded cases.

## Contribution

It provides a complete characterization of graded $ ext{Sigma}$-$V$ Leavitt path algebras and relates algebraic properties to graph-theoretic conditions, including new insights into graded direct-finiteness.

## Key findings

- Leavitt path algebra is a graded $ ext{Sigma}$-$V$ ring iff it is a subdirect product of matrix rings over $K$ or $K[x,x^{-1}]$
- For finite graphs, graded $ ext{Sigma}$-$V$ rings are equivalent to being graded directly-finite, bounded nilpotence, and graded semi-simple
- Examples show these equivalences do not hold for infinite graphs

## Abstract

In this paper, we give a complete characterization of Leavitt path algebras which are graded $\Sigma $-$V$ rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we show that a Leavitt path algebra $L$ over an arbitrary graph $E$ is a graded $\Sigma $-$V$ ring if and only if it is a subdirect product of matrix rings of arbitrary size but with finitely many non-zero entries over $K$ or $K[x,x^{-1}]$ with appropriate matrix gradings. We also obtain a graphical characterization of such a graded $\Sigma $-$V$ ring $L$% . When the graph $E$ is finite, we show that $L$ is a graded $\Sigma $-$V$ ring $\Longleftrightarrow L$ is graded directly-finite $\Longleftrightarrow L $ has bounded index of nilpotence $\Longleftrightarrow $ $L$ is graded semi-simple. Examples show that the equivalence of these properties in the preceding statement no longer holds when the graph $E$ is infinite. Following this, we also characterize Leavitt path algebras $L$ which are non-graded $\Sigma $-$V$ rings. Graded rings which are graded directly-finite are explored and it is shown that if a Leavitt path algebra $L$ is a graded $\Sigma$-$V$ ring, then $L$ is always graded directly-finite. Examples show the subtle differences between graded and non-graded directly-finite rings. Leavitt path algebras which are graded directly-finite are shown to be directed unions of graded semisimple rings. Using this, we give an alternative proof of a theorem of Va\v{s} \cite{V} on directly-finite Leavitt path algebras.