Simple modules for Kumjian-Pask algebras

TL;DR

This paper develops a framework for modules over Kumjian-Pask algebras using representation k-graphs, characterizes simple modules, and explores their categorical structure via higher-rank graph coverings.

Contribution

It introduces the concept of representation k-graphs for higher-rank graphs and characterizes which yield simple modules for Kumjian-Pask algebras.

Findings

Representation k-graphs produce modules for Kumjian-Pask algebras.

Simple modules are characterized via representation k-graphs.

The category of representation k-graphs is analyzed using higher-rank graph coverings.

Abstract

The paper introduces the notion of a representation $k$-graph $(\Delta,\alpha)$ for a given $k$-graph $\Lambda$. It is shown that any representation $k$-graph for $\Lambda$ yields a module for the Kumjian-Pask algebra $KP(\Lambda)$, and the representation $k$-graphs yielding simple modules are characterised. Moreover, the category $RG(\Lambda)$ of representation $k$-graphs for $\Lambda$ is investigated using the covering theory of higher-rank graphs.