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

TL;DR

This paper demonstrates that under certain conditions, unital shift equivalence leads to graded isomorphism of Leavitt path algebras, advancing the proof of the Graded Classification Conjecture using new lifting techniques.

Contribution

It introduces a general lifting result for graded rings and applies it to establish a link between shift equivalence and graded isomorphism in Leavitt path algebras, supporting the classification conjecture.

Findings

Unital shift equivalence with an alignment condition induces graded isomorphism.

A new lifting theorem for graded rings is established.

Simplified proofs of the fullness of graded K-theory and non-existence of certain graded homomorphisms are provided.

Abstract

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 Va{\v s} through other means that the graded $K$-theory functor is full, and the other proven by Arnone and Corti\~nas that there is no unital graded homomorphism between a Leavitt algebra and the path algebra of a Cuntz splice.