1. The Talented Monoid of Higher-Rank Graphs with Applications to Kumjian-Pask Algebras
- Author
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Hazrat, Roozbeh, Mukherjee, Promit, Pask, David, and Sardar, Sujit Kumar
- Subjects
Mathematics - Rings and Algebras ,Mathematics - Operator 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 $K_0^{gr}(KP_\mathsf{k}(\Lambda))$ of the Kumjian-Pask algebra $KP_\mathsf{k}(\Lambda)$ 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 $KP_R(\Lambda)$ of $\Lambda$ over a unital commutative ring $R$ to be graded basic ideal simple. We also describe the minimal left ideals of $KP_\mathsf{k}(\Lambda)$ in terms of the aperiodic atoms of $T_\Lambda$ and thus obtain a monoid theoretic characterization for $Soc(KP_\mathsf{k}(\Lambda)$) to be an essential ideal. These results help us to characterize semisimple Kumjian-Pask algebras through the lens of $T_\Lambda$.
- Published
- 2024