The cycline subalgebra of a Kumjian-Pask algebra

TL;DR

This paper characterizes a maximal commutative subalgebra within Kumjian-Pask algebras of higher-rank graphs and establishes a generalized Cuntz-Krieger uniqueness theorem linking injectivity of representations to this subalgebra.

Contribution

It identifies the cycline subalgebra in Kumjian-Pask algebras and proves a generalized uniqueness theorem for their representations.

Findings

Identification of the maximal commutative subalgebra in Kumjian-Pask algebras.

Proof of a generalized Cuntz-Krieger uniqueness theorem.

Characterization of injective representations via the subalgebra.

Abstract

Let $\Lambda$ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalgebra $\mathcal{M}$ inside the Kumjian-Pask algebra ${\rm KP}_R(\Lambda)$. We also prove a generalized Cuntz-Krieger uniqueness theorem for Kumjian-Pask algebras which says that a representation of ${\rm KP}_R(\Lambda)$ is injective if and only if it is injective on $\mathcal{M}$.