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

Gene Abrams; Efren Ruiz; Mark Tomforde

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

Rings and Algebras 2023-11-07 v1

Authors: Gene Abrams , Efren Ruiz , Mark Tomforde

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.

Keywords

hazrat conjecture shift equivalence graded morita equivalence leavitt path algebras adjacency matrices bridging bimodule

Cite

@article{arxiv.2311.02896,
  title  = {Recasting the Hazrat Conjecture: Relating Shift Equivalence to Graded Morita Equivalence},
  author = {Gene Abrams and Efren Ruiz and Mark Tomforde},
  journal= {arXiv preprint arXiv:2311.02896},
  year   = {2023}
}

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36 pages

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

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