Kumjian–Pask algebras of finitely aligned higher-rank graphs

Abstract

We extend 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–Krieger 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–Krieger uniqueness theorem and also to characterize simplicity and basic simplicity.