Year: 2026
Type: preprint
Abstract: Given two finite essential adjacency matrices $A$ and $B$, Hazrat's graded classification conjectures posit that an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism of $K_0$ groups implies graded Morita equivalence of the Leavitt path algebras of $A$ and $B$, while the pointed version predicts a graded isomorphism of the Leavitt path algebras when the $K_0$ group isomorphism additionally preserves the class of the regular module. For any field $k$, we show that the Leavitt path algebras over $k$ of $A$ and $B$ are graded Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. By appealing to counterexamples of Kim and Roush from symbolic dynamics, this shows that Hazrat's graded classification conjectures are false.
Source: arXiv (Cornell University)
Authors: Bilich, Boris, Dor-On, Adam
Institutions: University of Göttingen, University of Haifa
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: NSF-BSF: Semigroup Operator Algebras - Representations, Boundaries and Dynamics, Gruppoidmodelle für Diagramme von Gruppoidkorrespondenzen und ihre C*Algebren, Collaborative Research: NSF-BSF - Dynamics Through the Viewpoint of Self-Similar Algebras: Topological Markov Chains and Self-Similar Groups.
OpenAlex ID: W7212248305