Kumjian–Pask algebras of locally convex higher-rank graphs

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 Λ 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 Λ˜ without sources such that the Kumjian–Pask algebras of Λ˜ and Λ are Morita equivalent. We then use the Morita equivalence to study the ideal structure of the Kumjian–Pask algebra of Λ by pulling the appropriate results across the equivalence.