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arXiv · 2608.12798

Some more talents of the talented monoid of a higher-rank graph

Roozbeh Hazrat·Huanhuan Li·Promit Mukherjee
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Abstract

In this paper, we explore the idea that the graded Grothendieck group $K_0^{gr}$, or equivalently its positive cone, the talented monoid, can detect the structural type of higher-rank graph algebras (i.e., higher-rank graph $C^*$-algebras and Kumjian--Pask algebras). We show that the talented monoid captures some of the essential geometric information of a higher-rank graph, including the existence of cycles with and without entrances. In turn, we show that the graded $K$-theory can effectively distinguish the class of locally finite Kumjian--Pask algebras, and also the class of crossed product Kumjian--Pask algebras. We also derive talented monoid criteria for higher-rank graph algebras to be purely infinite simple, and not to be $AF$ or ultramatricial.

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Roozbeh Hazrat, Huanhuan Li, Promit Mukherjee. 2026-08-13. Some more talents of the talented monoid of a higher-rank graph. https://arxiv.org/abs/2608.12798

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