The Talented Monoid of Higher-Rank Graphs with Applications to Kumjian-Pask Algebras
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| Format: | Preprint |
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2024
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| _version_ | 1866909386006331392 |
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| author | Hazrat, Roozbeh Mukherjee, Promit Pask, David Sardar, Sujit Kumar |
| author_facet | Hazrat, Roozbeh Mukherjee, Promit Pask, David Sardar, Sujit Kumar |
| contents | Given a row-finite higher-rank $k$-graph $Λ$, we define a commutative monoid $T_Λ$ which is a higher-rank analogue of the talented monoid of a directed graph. The talented monoid $T_Λ$ 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 $K_0^{gr}(KP_\mathsf{k}(Λ))$ of the Kumjian-Pask algebra $KP_\mathsf{k}(Λ)$ 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 $T_Λ$ (i.e., if $T_Λ$ has no nonzero periodic element), then we show that the $k$-graph $Λ$ is aperiodic. The converse is also proved to be true provided $Λ$ has no sources and $T_Λ$ 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 $Λ$ without sources, cofinality is equivalent to the simplicity of $T_Λ$ as a $\mathbb{Z}^k$-monoid. In view of this we provide a talented monoid criterion for the Kumjian-Pask algebra $KP_R(Λ)$ of $Λ$ over a unital commutative ring $R$ to be graded basic ideal simple. We also describe the minimal left ideals of $KP_\mathsf{k}(Λ)$ in terms of the aperiodic atoms of $T_Λ$ and thus obtain a monoid theoretic characterization for $Soc(KP_\mathsf{k}(Λ)$) to be an essential ideal. These results help us to characterize semisimple Kumjian-Pask algebras through the lens of $T_Λ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_07582 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Talented Monoid of Higher-Rank Graphs with Applications to Kumjian-Pask Algebras Hazrat, Roozbeh Mukherjee, Promit Pask, David Sardar, Sujit Kumar Rings and Algebras Operator Algebras Given a row-finite higher-rank $k$-graph $Λ$, we define a commutative monoid $T_Λ$ which is a higher-rank analogue of the talented monoid of a directed graph. The talented monoid $T_Λ$ 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 $K_0^{gr}(KP_\mathsf{k}(Λ))$ of the Kumjian-Pask algebra $KP_\mathsf{k}(Λ)$ 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 $T_Λ$ (i.e., if $T_Λ$ has no nonzero periodic element), then we show that the $k$-graph $Λ$ is aperiodic. The converse is also proved to be true provided $Λ$ has no sources and $T_Λ$ 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 $Λ$ without sources, cofinality is equivalent to the simplicity of $T_Λ$ as a $\mathbb{Z}^k$-monoid. In view of this we provide a talented monoid criterion for the Kumjian-Pask algebra $KP_R(Λ)$ of $Λ$ over a unital commutative ring $R$ to be graded basic ideal simple. We also describe the minimal left ideals of $KP_\mathsf{k}(Λ)$ in terms of the aperiodic atoms of $T_Λ$ and thus obtain a monoid theoretic characterization for $Soc(KP_\mathsf{k}(Λ)$) to be an essential ideal. These results help us to characterize semisimple Kumjian-Pask algebras through the lens of $T_Λ$. |
| title | The Talented Monoid of Higher-Rank Graphs with Applications to Kumjian-Pask Algebras |
| topic | Rings and Algebras Operator Algebras |
| url | https://arxiv.org/abs/2411.07582 |