Modules for Leavitt path algebras of bi-separated graphs via representations graphs

TL;DR

This paper introduces a new method to construct modules for Leavitt path algebras of bi-separated graphs using representation graphs, including simple modules that generalize Chen modules.

Contribution

It develops the concept of representation graphs for bi-separated graphs to systematically build modules for their Leavitt path algebras, extending known module constructions.

Findings

Constructed modules for Leavitt path algebras of bi-separated graphs.

Identified a class of simple modules within these algebras.

Recovered Chen simple modules in the case of Cuntz-Krieger bi-separation.

Abstract

Leavitt path algebras of bi-separated graphs have been recently introduced by R. Mohan and B. Suhas. These algebras provide a common framework for studying various generalisations of Leavitt path algebras. In this paper we obtain modules for the Leavitt path algebra $L(\dot E)$ of a finitely bi-separated graph $\dot{E}=(E,C,D)$ by introducing the notion of a representation graph for $\dot{E}$. Among these modules we find a class of simple modules. If the bi-separation on $E$ is the Cuntz-Krieger bi-separation (and hence $L(\dot{E})$ is isomorphic to the usual Leavitt path algebra $L(E)$), one recovers the celebrated Chen simple modules.