Group gradations on Leavitt path algebras

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.