Higher-rank graphs and the graded $K$-theory of Kumjian-Pask Algebras

Abstract: 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 $\Lambda$ without sources, we show that there exists a $\mathbb{Z}[\mathbb{Z}^k]$-module isomorphism between the graded zeroth (integral) homology $H_0^{gr}(\mathcal{G}_\Lambda)$ of the infinite path groupoid $\mathcal{G}\Lambda$ and the graded Grothendieck group $K_0^{gr}(KP\mathsf{k}(\Lambda))$ of the Kumjian-Pask algebra $KP_\mathsf{k}(\Lambda)$, 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 $\Lambda$ and $\Omega$ without sources and with finite object sets, we obtain a sufficient criterion for lifting a pointed order-preserving $\mathbb{Z}[\mathbb{Z}^k]$-module homomorphism between $K_0^{gr}(KP_\mathsf{k}(\Lambda))$ and $K_0^{gr}(KP_\mathsf{k}(\Omega))$ to a unital graded ring homomorphism between $KP_\mathsf{k}(\Lambda)$ and $KP_\mathsf{k}(\Omega)$. For this we adapt, in the setting of $k$-graphs, the bridging bimodule technique recently introduced by Abrams, Ruiz and Tomforde (Algebr. Represent. Theory 2024).