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The Graded Classification Conjecture for Leavitt path algebras

From papers

Let EE and FF be finite graphs, and let KK be a field. For a

-graded ring $A$ with identity, write $K_0^{\operatorname{gr}}(A)$ for the graded Grothendieck group, equipped in the

-graded case with its natural Z[x,x−1]\mathbb{Z}[x,x^{-1}]-module structure. Let TET_E denote the talented monoid of EE. The Graded Classification Conjecture. The following statements are equivalent: (1) the Leavitt path algebras LK(E)L_K(E) and LK(F)L_K(F) are graded Morita equivalent; (2) there is an order-preserving Z[x,x−1]\mathbb{Z}[x,x^{-1}]-module isomorphism

K0gr⁡(L(E))⟶K0gr⁡(L(F));K_0^{\operatorname{gr}}(L(E))\longrightarrow K_0^{\operatorname{gr}}(L(F));

(3) the talented monoids TET_E and TFT_F are Z\mathbb{Z}-isomorphic. This conjecture proposes a complete graded invariant for Leavitt path algebras of finite graphs; its precise status is not established in the supplied text.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The graded classification conjecture for Leavitt path algebras

    Let KK be a 2-proper, ∗*-pythagorean field, and let LK(E)L_K(E) denote the Leavitt path algebra of a directed graph EE over KK, equipped with its natural grading and involution. Write K0gr⁡(LK(E))K^{\operatorname{gr}}_0(L_K(E)) for its graded K0K_0-group.

    Graded classification conjecture. The invariant K0gr⁡K^{\operatorname{gr}}_0 completely classifies Leavitt path algebras over KK as graded ∗*-algebras.

    This conjecture extends the graded classification results known for particular classes of Leavitt path algebras. The paper proves the corresponding statement for a specified class of countable, row-finite, no-exit graphs, while the conjecture is presented for Leavitt path algebras over any 2-proper, ∗*-pythagorean field.

    source: Roozbeh Hazrat and Lia Vas, “K-theory Classification of Graded Ultramatricial Algebras with Involution”, arXiv:1604.07797 (2018).

Sources & referencesView supporting material

Primary source

Tran Quang Do, Roozbeh Hazrat and Tran Giang Nam, “Williams' conjecture holds for graphs of Gelfand-Kirillov dimension three”, arXiv:2504.11342 (2025).

algebraic-k-theoryclassificationgraded-k-theorygraded-ringsleavitt-path-algebras

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