Graded Naimark's Problem for Leavitt Path Algebras

TL;DR

This paper characterizes graded Leavitt path algebras over arbitrary graphs, showing they are isomorphic to graded matrix rings over specific fields and identifying conditions on the graph structure for these properties.

Contribution

It provides a complete characterization of graded Leavitt path algebras with certain module isomorphism properties, linking algebraic structure to graph-theoretic conditions.

Findings

Leavitt path algebra is graded isomorphic to graded infinite matrices over  or [x,x^{-1}].

Graph E must be row-finite, downward directed, with a single hereditary saturated vertex.

Leavitt path algebras with countably many graded-simple module classes are characterized.

Abstract

In this paper we study the graded version of Naimark's problem for Leavitt path algebras considering them as $\mathbb{Z}$-graded algebras. Several characterizations are obtained of a Leavitt path algebra $L$ of an arbitrary graph $E$ over a field $\mathbb{K}$ over which any two graded-simple modules are graded isomorphic. Such a Leavitt path algebra $L$ is shown to be graded isomorphic to the algebra of graded infinite matrices having at most finitely many non-zero entries from the ring $R$ where $R=\mathbb{K}$ or $R=\mathbb{K}[x,x^{-1}]$. Equivalently, $L$ is a graded-simple ring which is graded-semisimple, that is, $L$ is a graded direct sum of graded-isomorphic graded-simple left $L$-modules. Graphically, the graph $E$ is shown to be row-finite, downward directed and the vertex set $E^{0}$ is the hereditary saturated closure of a single vertex $v$ which is either a line point or lies on a cycle without exits. We also characterize Leavitt path algebras possessing at most countably many isomorphism classes of graded-simple left modules. Examples are constructed illustrating these results.