Abstract

We lay 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 $H_0^{\mathrm{gr}<!--FUNCTION\; APPLICATION-->}({\mathcal{𝒢}}_{\Lambda })$ of the infinite path groupoid ${\mathcal{𝒢}}_{\Lambda }$ and the graded Grothendieck group $K_0^{\mathrm{gr}<!--FUNCTION\; APPLICATION-->}({\mathrm{KP}<!--FUNCTION\; APPLICATION-->}_F(\Lambda ))$ of the Kumjian–Pask algebra ${\mathrm{KP}<!--FUNCTION\; APPLICATION-->}_F(\Lambda )$, 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. 74:3 (2022), 655–685) 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 $\mathrm{\Omega }$ 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 $K_0^{\mathrm{gr}<!--FUNCTION\; APPLICATION-->}({\mathrm{KP}<!--FUNCTION\; APPLICATION-->}_F(\Lambda ))$ and $K_0^{\mathrm{gr}<!--FUNCTION\; APPLICATION-->}({\mathrm{KP}<!--FUNCTION\; APPLICATION-->}_F(\mathrm{\Omega }))$ to a unital graded ring homomorphism between ${\mathrm{KP}<!--FUNCTION\; APPLICATION-->}_F(\Lambda )$ and ${\mathrm{KP}<!--FUNCTION\; APPLICATION-->}_F(\mathrm{\Omega })$. For this, we adopt, in the setting of $k$-graphs, the bridging bimodule technique recently introduced by Abrams, Ruiz and Tomforde (Algebr. Represent. Theory 27:2 (2024), 1477–1511).

Keywords

Mathematical Subject Classification

Milestones

Authors