Irreducible representations of Leavitt algebras

TL;DR

This paper constructs and characterizes irreducible representations of Leavitt path algebras using representation graphs, providing a comprehensive classification of simple and indecomposable modules for these algebras.

Contribution

It introduces a novel graph-based framework for understanding irreducible modules of Leavitt algebras, unifying previous constructions and extending to new classes of modules.

Findings

Classification of simple modules via representation graphs

Construction of irreducible modules for Leavitt algebras $L_K(n,m)$

Identification of non-simple indecomposable modules

Abstract

For a weighted graph $E$, we construct representation graphs $F$, and consequently, $L_K(E)$-modules $V_F$, where $L_K(E)$ is the Leavitt path algebra associated to $E$, with coefficients in a field $K$. We characterise representation graphs $F$ such that $V_F$ are simple $L_K(E)$-modules. We show that the category of representation graphs of $E$, $RG(E)$, is a disjoint union of subcategories, each of which contains a unique universal object $T$ which gives an indecomposable $L_K(E)$-module $V_T$ and a unique irreducible representation graph $S$, which gives a simple $L_K(E)$-module $V_S$. Specialising to graphs with one vertex and $m$ loops of weight $n$, we construct irreducible representations for the celebrated Leavitt algebras $L_K(n,m)$. On the other hand, specialising to graphs with weight one, we recover the simple modules of Leavitt path algebras constructed by Chen via infinite paths or sinks and give a large class of non-simple indecomposable modules.