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

TL;DR

This paper explores the graded structure of Leavitt path algebras by viewing them as partial skew group rings, establishing equivalences among various graded algebraic properties.

Contribution

It introduces a novel perspective by using the isomorphism to partial skew group rings to analyze the graded properties of Leavitt path algebras.

Findings

Graded cleanness, unit-regularity, and strong gradeness are equivalent in these algebras.

The algebra is endowed with an F-gradation derived from the free group on edges.

The approach links algebraic properties to the graded structure via 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.