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Graded classification of Leavitt path algebras in terms of strong shift equivalence

  • Boris Bilich
  • Adam Dor-On
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Abstract

Given two finite essential adjacency matrices A and B, Hazrat's graded classification conjectures posit that an order preserving ℤ[x,x−1]-module isomorphism of K0 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 K0 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.

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Cite

BibTeXRIS
BibTeX
@misc{bilich2026graded,
  title = {{Graded classification of Leavitt path algebras in terms of strong shift equivalence}},
  author = {Boris Bilich and Adam Dor-On},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11834},
  eprint = {2609.11834},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11834},
}
RIS
TY  - UNPB
TI  - Graded classification of Leavitt path algebras in terms of strong shift equivalence
AU  - Boris Bilich
AU  - Adam Dor-On
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11834
UR  - https://arxiv.org/abs/2609.11834
AB  - Given two finite essential adjacency matrices A and B, Hazrat's graded classification conjectures posit that an order preserving 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.
ER  -