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Kumjian-Pask algebras of locally convex higher-rank graphs

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 $\tildeΛ$ without sources such that the Kumjian-Pask algebras of $\tildeΛ$ 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.