The Talented Monoid of Higher-Rank Graphs with Applications to Kumjian-Pask Algebras

Abstract

Given a row-finite higher-rank $k$-graph $\Lambda$, we define a commutative monoid $T_{\Lambda }$ which is a higher-rank analogue of the talented monoid of a directed graph. The talented monoid $T_{\Lambda }$ is canonically a ${\mathbb{Z}}^k$-monoid with respect to the action of state shift. This monoid coincides with the positive cone of the graded Grothendieck group $K_0^{gr}(K{P}_{\mathsf{k}}(\Lambda ))$ of the Kumjian-Pask algebra $K{P}_{\mathsf{k}}(\Lambda )$ with coefficients in a field $\mathsf{k}$. The aim of the paper is to investigate this ${\mathbb{Z}}^k$-monoid as a capable invariant for classification of Kumjian-Pask algebras. If ${\mathbb{Z}}^k$ acts freely on $T_{\Lambda }$ (i.e., if $T_{\Lambda }$ has no nonzero periodic element), then we show that the $k$-graph $\Lambda$ is aperiodic. The converse is also proved to be true provided $\Lambda$ has no sources and $T_{\Lambda }$ is atomic. Moreover in this case, we provide a talented monoid characterization for strongly aperiodic $k$-graphs. We prove that for a row-finite $k$-graph $\Lambda$ without sources, cofinality is equivalent to the simplicity of $T_{\Lambda }$ as a ${\mathbb{Z}}^k$-monoid. In view of this we provide a talented monoid criterion for the Kumjian-Pask algebra $K{P}_R(\Lambda )$ of $\Lambda$ over a unital commutative ring $R$ to be graded basic ideal simple. We also describe the minimal left ideals of $K{P}_{\mathsf{k}}(\Lambda )$ in terms of the aperiodic atoms of $T_{\Lambda }$ and thus obtain a monoid theoretic characterization for $Soc(K{P}_{\mathsf{k}}(\Lambda )$) to be an essential ideal. These results help us to characterize semisimple Kumjian-Pask algebras through the lens of $T_{\Lambda }$.