Purely infinite simple Kumjian-Pask algebras

TL;DR

This paper characterizes when simple Kumjian-Pask algebras, a higher-rank generalization of Leavitt path algebras, are purely infinite, establishing conditions based on graph structure and a dichotomy for simplicity.

Contribution

It provides new criteria for pure infiniteness in simple Kumjian-Pask algebras and establishes a dichotomy for their structure.

Findings

Purely infinite simple Kumjian-Pask algebras occur when each vertex is reached from a generalized cycle with an entrance.

A dichotomy shows simple Kumjian-Pask algebras are either purely infinite or locally matricial under certain conditions.

The results cover all unital simple Kumjian-Pask algebras.

Abstract

Given any finitely aligned higher-rank graph $\Lambda$ and any unital commutative ring $R$, the Kumjian-Pask algebra $\mathrm{KP}_R(\Lambda)$ is known as the higher-rank generalization of Leavitt path algebras. After characterizing simple Kumjian-Pask algebras by L.O. Clark and Y.E.P. Pangalela (and others), we focus in this article on the purely infinite simple ones. Briefly, we show that if $\mathrm{KP}_R(\Lambda)$ is simple and every vertex of $\Lambda$ is reached from a generalized cycle with an entrance, then $\mathrm{KP}_R(\Lambda)$ is purely infinite. We next prove a standard dichotomy for simple Kumjian-Pask algebras: in the case that each vertex of $\Lambda$ is reached only from finitely many vertices and $\mathrm{KP}_R(\Lambda)$ is simple, then $\mathrm{KP}_R(\Lambda)$ is either purely infinite or locally matritial. This result covers all unital simple Kumjian-Pask algebras.