Higher-rank graphs and the graded K-theory of Kumjian–Pask algebras

This paper lays out the foundations of graded K-theory for Leavitt algebras associated with higher-rank graphs, also known as Kumjian-Pask algebras, establishing it as a potential tool for their classification. For a row-finite k-graph $Λ$ without sources, we show that there exists a Z[Zᵏ]-module isomorphism between the graded zeroth (integral) homology H₀ᵍʳ(G_Λ) of the infinite path groupoid G_Λ and the graded Grothendieck group K₀ᵍʳ(KPₖ(Λ)) of the Kumjian-Pask algebra KPₖ(Λ), which respects the positive cones (i.e., the talented monoids). We demonstrate that the k-graph moves of in-splitting and sink deletion defined by Eckhardt et al. (Canad. J. Math. 2022) preserve the graded K-theory of associated Kumjian-Pask algebras and produce algebras which are graded Morita equivalent, thus providing evidence that graded K-theory may be an effective invariant for classifying certain Kumjian-Pask algebras. We also determine a natural sufficient condition regarding the fullness of the graded Grothendieck group functor. More precisely, for two row-finite k-graphs $Λ$ and $Ω$ without sources and with finite object sets, we obtain a sufficient criterion for lifting a pointed order-preserving Z[Zᵏ]-module homomorphism between K₀ᵍʳ(KPₖ(Λ)) and K₀ᵍʳ(KPₖ(Ω)) to a unital graded ring homomorphism between KPₖ(Λ) and KPₖ(Ω). For this we adopt, in the setting of k-graphs, the bridging bimodule technique recently introduced by Abrams, Ruiz and Tomforde (Algebr. Represent. Theory 2024).