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<p>We prove that a unital shift equivalence induces a graded isomorphism of Leavitt path algebras when the shift equivalence satisfies an alignment condition. This yields another step towards confirming the Graded Classification Conjecture. Our proof uses the bridging bimodule developed by Abrams, the fourth-named author and Tomforde, as well as a general lifting result for graded rings that we establish here. This general result also allows us to provide simplified proofs of two important recent results: one independently proven by Arnone and Vaš through other means that the graded <i>K</i>-theory functor is full, and the other proven by Arnone and Cortiñas that there is no unital graded homomorphism between a Leavitt algebra and the path algebra of a Cuntz splice.</p>

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Unital aligned shift equivalence and the graded classification conjecture for Leavitt path algebras

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月7日

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  • Kevin Aguyar Brix,
  • Adam Dor-On,
  • Roozbeh Hazrat,
  • Efren Ruiz

摘要

We prove that a unital shift equivalence induces a graded isomorphism of Leavitt path algebras when the shift equivalence satisfies an alignment condition. This yields another step towards confirming the Graded Classification Conjecture. Our proof uses the bridging bimodule developed by Abrams, the fourth-named author and Tomforde, as well as a general lifting result for graded rings that we establish here. This general result also allows us to provide simplified proofs of two important recent results: one independently proven by Arnone and Vaš through other means that the graded K-theory functor is full, and the other proven by Arnone and Cortiñas that there is no unital graded homomorphism between a Leavitt algebra and the path algebra of a Cuntz splice.