arXiv subjects

Promit Mukherjee

Publications and source records attributed to Promit Mukherjee.

4 recordsLinked to original sources

arXiv2026

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

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.

math.RA↗

arXiv2026

Higher-Dimensional Symbolic Dynamics: A Textile Framework For 3-graphs

Textile systems are best known to model two-dimensional shifts of finite type. In this article, we associate a discrete algebra with a textile system and provide a groupoid model for it. When the textile system is left-resolving, this algebra coincides with the Kumjian--Pask algebra of the associated $2$-graph. The main objective of this paper is to extend textile systems to dimension $3$ so that the resulting structures can, on the one hand, capture all three-dimensional shifts of finite type and, on the other hand, provide a textile-like framework for $3$-graphs extending the well-known connection between $2$-graphs and left-resolving textile systems. We introduce a model of a three-dimensional textile system and investigate the interplay between such textile systems and $3$-graphs. In particular, our investigation shows that the conditions required to form a $3$-graph from a $3$-colored graph (including the delicate associativity condition on tricolored paths), can be encoded in terms of simple pullback diagrams arising from the textile data. We also define homology groups for three-dimensional textiles and prove that these groups coincide with the homology groups of the associated $3$-graphs, thus establishing that our construction is homologically consistent with $3$-graphs.

math.DS↗

arXiv2025

Higher-rank graphs and the graded $K$-theory of Kumjian-Pask algebras

This paper lays 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 $H_0^{gr}(\mathcal{G}_\Lambda)$ of the infinite path groupoid $\mathcal{G}_\Lambda$ and the graded Grothendieck group $K_0^{gr}(KP_\mathsf{k}(\Lambda))$ of the Kumjian-Pask algebra $KP_\mathsf{k}(\Lambda)$, 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. 2022) 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 $\Omega$ 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 $K_0^{gr}(KP_\mathsf{k}(\Lambda))$ and $K_0^{gr}(KP_\mathsf{k}(\Omega))$ to a unital graded ring homomorphism between $KP_\mathsf{k}(\Lambda)$ and $KP_\mathsf{k}(\Omega)$. For this we adopt, in the setting of $k$-graphs, the bridging bimodule technique recently introduced by Abrams, Ruiz and Tomforde (Algebr. Represent. Theory 2024).

math.KT↗

arXiv2024

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

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 $T_\Lambda$.

math.RA↗