Prime and primitive Kumjian-Pask algebras

TL;DR

This paper characterizes prime and primitive Kumjian-Pask algebras of row-finite $k$-graphs over rings, providing graph-theoretic and algebraic criteria, especially for strongly aperiodic graphs over fields.

Contribution

It offers a complete characterization of prime and primitive Kumjian-Pask algebras using graph-theoretic and algebraic methods, including quotient $k$-graphs.

Findings

Prime and primitive Kumjian-Pask algebras characterized by graph properties.

Prime and primitive ideals described via quotient $k$-graphs.

Complete determination of prime and primitive ideals when $ ext{Λ}$ is strongly aperiodic and $R$ is a field.

Abstract

In this paper, prime as well as primitive Kumjian-Pask algebras $\mathrm{KP}_R(\Lambda)$ of a row-finite $k$-graph $\Lambda$ over a unital commutative ring $R$ are completely characterized in graph-theoretic and algebraic terms. By applying quotient $k$-graphs, these results describe prime and primitive graded basic ideals of Kumjian-Pask algebras. In particular, when $\Lambda$ is strongly aperiodic and $R$ is a field, all prime and primitive ideals of a Kumjian-Pask algebra $\mathrm{KP}_R(\Lambda)$ are determined.