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

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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.

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Acknowledgements

This work was supported by grants from the Simons Foundation (#527708 to Mark Tomforde and #567380 to Efren Ruiz).

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Authors and Affiliations

  1. Department of Mathematics, University of Colorado, 1420 Austin Bluffs Parkway, Colorado Springs, CO, 80918, USA

    Gene Abrams & Mark Tomforde

  2. Department of Mathematics, University of Hawaii, 200 W. Kawili St., Hawaii, Hilo, 96720-4091, USA

    Efren Ruiz

Authors
  1. Gene Abrams
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  2. Efren Ruiz
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  3. Mark Tomforde
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Correspondence to Efren Ruiz.

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Competing interests

The authors have no relevant financial or non-financial interests to disclose. This work was supported by grants from the Simons Foundation (#527708 to Mark Tomforde and #567380 to Efren Ruiz)

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Presented by: Milen Yakimov

Dedicated to the memory of Professor Iain Raeburn. The collaboration of the coauthors on this article is a direct result of Iain’s profound influence on the field in general, and on the three of us individually. He will be deeply missed.

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Abrams, G., Ruiz, E. & Tomforde, M. Recasting the Hazrat Conjecture: Relating Shift Equivalence to Graded Morita Equivalence. Algebr Represent Theor (2024). https://doi.org/10.1007/s10468-024-10266-w

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  • DOI: https://doi.org/10.1007/s10468-024-10266-w

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