Recasting the Hazrat Conjecture: Relating Shift Equivalence to Graded Morita Equivalence

Gene Abrams; Efren Ruiz; Mark Tomforde

paper

Recasting the Hazrat Conjecture: Relating Shift Equivalence to Graded Morita Equivalence

arXiv:2311.02896

Abstract

Let $E$ and $F$ be finite graphs with no sinks, and $k$ any field. We show that shift equivalence of the adjacency matrices $A_{E}$ and $A_{F}$ , together with an additional compatibility condition, implies that the Leavitt path algebras $L_{k} (E)$ and $L_{k} (F)$ are graded Morita equivalent. Along the way, we build a new type of $L_{k} (E)$ -- $L_{k} (F)$ -bimodule (a bridging bimodule), which we use to establish the graded equivalence.

36 pages