Kumjian-Pask algebras of locally convex higher-rank graphs

TL;DR

This paper extends the theory of Kumjian-Pask algebras to a broader class of higher-rank graphs with sources, establishing key theorems and analyzing their structure through Morita equivalence and desourcification.

Contribution

It introduces a new definition for Kumjian-Pask algebras of locally convex higher-rank graphs with sources and proves fundamental theorems for this class.

Findings

Established graded-uniqueness and Cuntz-Krieger theorems for these algebras.

Showed Morita equivalence between algebras of graphs with and without sources.

Analyzed ideal structure via desourcification and Morita equivalence.

Abstract

The Kumjian-Pask algebra of a higher-rank graph generalises the Leavitt path algebra of a directed graph. We extend the definition of Kumjian-Pask algebra to row-finite higher-rank graphs $\Lambda$ with sources which satisfy a local-convexity condition. After proving versions of the graded-uniqueness theorem and the Cuntz-Krieger uniqueness theorem, we study the Kumjian-Pask algebra of rank-2 Bratteli diagrams by studying certain finite subgraphs which are locally convex. We show that the desourcification procedure of Farthing and Webster yields a row-finite higher-rank graph $\tilde{\Lambda}$ without sources such that the Kumjian-Pask algebras of $\tilde{\Lambda}$ and $\Lambda$ are Morita equivalent. We then use the Morita equivalence to study the ideal structure of the Kumjian-Pask algebra of $\Lambda$ by pulling the appropriate results across the equivalence.