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.