Modules for Leavitt path algebras via extended algebraic branching systems

TL;DR

This paper introduces extended algebraic branching systems for graphs, classifies them, and demonstrates how they induce modules for Leavitt path algebras, including a class of nonsimple modules with field endomorphism rings.

Contribution

It generalizes the concept of algebraic branching systems and establishes a classification that leads to new modules for Leavitt path algebras.

Findings

Classified extended $E$-algebraic branching systems.

Constructed modules for Leavitt path algebras from these systems.

Identified nonsimple modules with field endomorphism rings.

Abstract

For a graph $E$, we introduce the notion of an extended $E$-algebraic branching system, generalising the notion of an $E$-algebraic branching system introduced by Gon\c{c}alves and Royer. We classify the extended $E$-algebraic branching systems and show that they induce modules for the corresponding Leavitt path algebra $L(E)$. Among these modules we find a class of nonsimple modules whose endomorphism rings are fields.