Kumjian-Pask algebras of finitely-aligned higher-rank graphs

TL;DR

This paper extends Kumjian-Pask algebras to finitely aligned higher-rank graphs, establishing their universal properties, graded and Cuntz-Krieger uniqueness theorems, and characterizations of simplicity using a groupoid approach.

Contribution

It introduces a new class of Kumjian-Pask algebras for finitely aligned higher-rank graphs and proves key structural theorems using groupoid and Steinberg algebra techniques.

Findings

Kumjian-Pask algebras are universally defined for these graphs.

Established graded and Cuntz-Krieger uniqueness theorems.

Characterized simplicity and basic simplicity of the algebras.

Abstract

We extend the the definition of Kumjian-Pask algebras to include algebras associated to finitely aligned higher-rank graphs. We show that these Kumjian-Pask algebras are universally defined and have a graded uniqueness theorem. We also prove the Cuntz-Kreiger uniqueness theorem; to do this, we use a groupoid approach. As a consequence of the graded uniqueness theorem, we show that every Kumjian-Pask algebra is isomorphic to the Steinberg algebra associated to its boundary path groupoid. We then use Steinberg algebra results to prove the Cuntz-Kreiger uniqueness theorem and also to characterize simplicity and basic simplicity.