Let A and B be finite essential adjacency matrices, and let L_k(A) and L_k(B) denote the corresponding Leavitt path algebras over a field k. Hazrat's graded classification conjectures assert: (1) if there is an order-preserving Z[x,x^-1]-module isomorphism K_0^gr(L_k(A))cong K_0^gr(L_k(B)), then L_k(A) and L_k(B) are graded Morita equivalent; and (2) if such an isomorphism also preserves the classes of the regular modules, [L_k(A)]mapsto [L_k(B)], then L_k(A) and L_k(B) are graded-isomorphic.

Hazrat’s graded classification conjectures

Let $A$ and $B$ be finite essential adjacency matrices, and let $L_k(A)$ and $L_k(B)$ denote the corresponding Leavitt path algebras over a field $k$ . Hazrat's graded classification conjectures assert: (1) if there is an order-preserving $\mathbb{Z}[x,x^{-1}]$ -module isomorphism $K_0^{\mathrm{gr}}(L_k(A))\cong K_0^{\mathrm{gr}}(L_k(B))$ , then $L_k(A)$ and $L_k(B)$ are graded Morita equivalent; and (2) if such an isomorphism also preserves the classes of the regular modules, $[L_k(A)]\mapsto [L_k(B)]$ , then $L_k(A)$ and $L_k(B)$ are graded-isomorphic.

References

Primary source

arXiv

Additional references

Graded classification of Leavitt path algebras in terms of strong shift equivalence — arXiv — Boris Bilich, Adam Dor-On

algebra symbolic-dynamics

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 paper claims to refute the proposed classification tests and replace them with an exact criterion, but the claim is unrefereed.

The conjectures concern classification of Leavitt path algebras using graded invariants, including $K_0$ data. A new paper by Boris Bilich and Adam Dor-On claims that the proposed criteria fail and gives a replacement criterion based on strong shift equivalence.

Known results

A 2025 paper proves that unital shift equivalence is characterized by isomorphic unital dimension data, identified with pointed graded $K_0$ -data, and derives graded algebra isomorphisms in a restricted setting.
A 2026 paper proves the conjectural equivalences for meteor graphs of length three with pairwise coprime cycle lengths.
Kim–Roush constructed matrices that are shift equivalent but not strongly shift equivalent, providing the obstruction used in the new claim.

September 2026 claimed refutation

On September 10, 2026, Bilich and Dor-On’s article Graded classification of Leavitt path algebras in terms of strong shift equivalence reported that the proposed $K_0$ criteria are insufficient and supplied an exact alternative criterion. This is a claimed complete resolution, not yet independently verified.

Current status (as of September 2026): The general conjectures are claimed to be refuted and replaced, but this claim remains unverified; restricted cases and the underlying Kim–Roush obstruction are established.

Sources

No solutions have been posted yet.