The graded structure of Leavittt path algebras viewed as partial skew group rings

Abstract

Let $E$ be a directed graph, $\mathbb{K}$ be a field, and $\mathbb{F}$ be the free group on the edges of $E$. In this work, we use the isomorphism between Leavitt path algebras and partial skew group rings to endow $L_{\mathbb{K}}(E)$ with an $\mathbb{F}$-gradation and study some algebraic properties of this gradation. More precisely, we show that graded cleanness, graded unit-regularity, and strong gradeness of $L_{\mathbb{K}}(E)$ are all equivalent.

11 pages