# Group gradations on Leavitt path algebras

## TL;DR

This paper investigates the gradation properties of Leavitt path algebras over arbitrary groups, establishing conditions for various gradation strengths and providing new proofs and criteria related to their algebraic structure.

## Contribution

It introduces the concept of nearly epsilon-strongly G-graded Leavitt path algebras and characterizes when these algebras are epsilon-strongly or strongly G-graded, including new proofs and criteria.

## Key findings

- L_R(E) is always nearly epsilon-strongly G-graded.
- If E is finite, then L_R(E) is epsilon-strongly G-graded.
- L_R(E) is strongly Z-graded iff E has no sink when E is row-finite with no source.

## Abstract

Given a directed graph $E$ and an associative unital ring $R$ one may define the Leavitt path algebra with coefficients in $R$, denoted by $L_R(E)$. For an arbitrary group $G$, $L_R(E)$ can be viewed as a $G$-graded ring. In this article, we show that $L_R(E)$ is always nearly epsilon-strongly $G$-graded. We also show that if $E$ is finite, then $L_R(E)$ is epsilon-strongly $G$-graded. We present a new proof of Hazrat's characterization of strongly $\mathbb{Z}$-graded Leavitt path algebras, when $E$ is finite. Moreover, if $E$ is row-finite and has no source, then we show that $L_R(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ has no sink. We also use a result concerning Frobenius epsilon-strongly $G$-graded rings, where $G$ is finite, to obtain criteria which ensure that $L_R(E)$ is Frobenius over its identity component.