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arxiv: 1907.02817 · v1 · pith:JK5NU76Hnew · submitted 2019-07-05 · 🧮 math.RA

Weighted Leavitt path algebras that are isomorphic to unweighted Leavitt path algebras

Raimund Preusser This is my paper

Pith reviewed 2026-05-25 01:42 UTC · model grok-4.3

classification 🧮 math.RA
keywords Leavitt path algebrasweighted graphsrow-finite graphsisomorphismsNoetherian ringsArtinian ringsvon Neumann regular ringsGelfand-Kirillov dimension
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The pith

Row-finite weighted graphs yield weighted Leavitt path algebras isomorphic to unweighted ones exactly when their weights meet a specific condition, and several common algebraic properties force the isomorphism.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper characterises the row-finite weighted graphs (E,w) for which the weighted Leavitt path algebra L_K(E,w) over a field K is isomorphic to some unweighted Leavitt path algebra. It further proves that whenever L_K(E,w) is locally finite, Noetherian, Artinian, von Neumann regular, or has finite Gelfand-Kirillov dimension, the isomorphism to an unweighted algebra necessarily holds. A reader would care because these results collapse the weighted case to the unweighted case under hypotheses that arise frequently in ring theory. The work therefore supplies a reduction that simplifies the study of these algebras when the listed properties are present.

Core claim

We characterise the row-finite weighted graphs (E,w) such that the weighted Leavitt path algebra L_K(E,w) is isomorphic to an unweighted Leavitt path algebra. Moreover, we prove that if L_K(E,w) is locally finite, or Noetherian, or Artinian, or von Neumann regular, or has finite Gelfand-Kirillov dimension, then L_K(E,w) is isomorphic to an unweighted Leavitt path algebra.

What carries the argument

The weighted Leavitt path algebra L_K(E,w) associated to a row-finite weighted graph (E,w), which extends the ordinary Leavitt path algebra by assigning positive integer weights to the edges.

If this is right

  • If L_K(E,w) is locally finite then it is isomorphic to an unweighted Leavitt path algebra.
  • If L_K(E,w) is Noetherian then it is isomorphic to an unweighted Leavitt path algebra.
  • If L_K(E,w) is Artinian then it is isomorphic to an unweighted Leavitt path algebra.
  • If L_K(E,w) is von Neumann regular then it is isomorphic to an unweighted Leavitt path algebra.
  • If L_K(E,w) has finite Gelfand-Kirillov dimension then it is isomorphic to an unweighted Leavitt path algebra.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit characterisation may permit direct construction of the target unweighted graph from the weighted data when the weights are bounded or periodic.
  • Questions about module categories or K-theory for weighted algebras meeting the finiteness conditions can be transferred to the unweighted literature without change.
  • The restriction to row-finite graphs leaves open whether the same characterisation or implications survive when infinite out-degrees are allowed.

Load-bearing premise

All results are stated only for row-finite weighted graphs.

What would settle it

A concrete row-finite weighted graph (E,w) such that L_K(E,w) is Noetherian yet not isomorphic to any unweighted Leavitt path algebra would falsify the implication theorem.

read the original abstract

Let $K$ be a field. We characterise the row-finite weighted graphs $(E,w)$ such that the weighted Leavitt path algebra $L_K(E,w)$ is isomorphic to an unweighted Leavitt path algebra. Moreover, we prove that if $L_K(E,w)$ is locally finite, or Noetherian, or Artinian, or von Neumann regular, or has finite Gelfand-Kirillov dimension, then $L_K(E,w)$ is isomorphic to an unweighted Leavitt path algebra.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper characterizes the row-finite weighted graphs (E,w) such that the weighted Leavitt path algebra L_K(E,w) is isomorphic to an unweighted Leavitt path algebra L_K(E'). It additionally proves that if L_K(E,w) is locally finite, Noetherian, Artinian, von Neumann regular, or has finite Gelfand-Kirillov dimension, then L_K(E,w) is isomorphic to an unweighted Leavitt path algebra.

Significance. If the results hold, the characterization and implication theorems provide a useful reduction from weighted to unweighted Leavitt path algebras under standard ring-theoretic hypotheses. This could streamline classification and structural results in the area by allowing the extensive literature on unweighted algebras to apply directly in many cases. The explicit scoping to row-finite graphs is a strength.

minor comments (3)
  1. §1 (Introduction): clarify whether the characterization is if-and-only-if or one direction only, and state the precise form of the target unweighted graph E'.
  2. Definition of weighted Leavitt path algebra (likely §2): verify that the relations for weighted edges are stated uniformly with the unweighted case to make the isomorphism claim transparent.
  3. Theorems on ring properties (likely §4–5): add a short remark on whether the proofs rely on row-finiteness in an essential way or could extend.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary accurately reflects the paper's main contributions on characterizing row-finite weighted graphs (E,w) for which L_K(E,w) is isomorphic to an unweighted Leavitt path algebra, along with the implication results under the listed ring-theoretic conditions.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states a direct characterization of row-finite weighted graphs (E,w) for which L_K(E,w) ≅ some unweighted L_K(E') and proves implication theorems under that hypothesis. No equations, fitted parameters, or self-referential constructions appear; the claims rest on standard definitions of weighted/unweighted Leavitt path algebras and graph-theoretic conditions without reducing any result to its own inputs by construction or via load-bearing self-citation. The scope is explicitly limited to row-finite graphs, avoiding any extension claims that could introduce gaps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, ad-hoc axioms, or new entities are mentioned. The work appears to rest on the standard definitions of weighted and unweighted Leavitt path algebras.

pith-pipeline@v0.9.0 · 5610 in / 1129 out tokens · 23822 ms · 2026-05-25T01:42:47.816895+00:00 · methodology

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

  1. [1]

    Abrams, G

    G. Abrams, G. Aranda Pino, The Leavitt path algebra of a graph , J. Algebra 293 (2005), no. 2, 319–334. 1

    work page 2005

  2. [2]

    Abrams, P

    G. Abrams, P. Ara, M. Siles Molina, Leavitt path algebras , Lecture Notes in Mathematics 2191, Springer, 2017. 18

    work page 2017

  3. [3]

    Ara, M.A

    P. Ara, M.A. Moreno, E. Pardo, Nonstable K-theory for graph algebras , Algebr. Represent. Theory 10 (2007), no. 2, 157–178. 1

    work page 2007

  4. [4]

    Bergman, Coproducts and some universal ring constructions , Trans

    G.M. Bergman, Coproducts and some universal ring constructions , Trans. Amer. Math. Soc. 200 (1974), 33–88. 1

    work page 1974

  5. [5]

    Cuntz, Simple C ∗ -algebras generated by isometries , Comm

    J. Cuntz, Simple C ∗ -algebras generated by isometries , Comm. Math. Phys. 57 (1977), no. 2, 173–185. 1

    work page 1977

  6. [6]

    Hazrat, The graded structure of Leavitt path algebras , Israel J

    R. Hazrat, The graded structure of Leavitt path algebras , Israel J. Math. 195 (2013), no. 2, 833–895. 1, 4, 17

    work page 2013

  7. [7]

    Hazrat, R

    R. Hazrat, R. Preusser, Applications of normal forms for weighted Leavitt path alge bras: simple rings and domains , Algebr. Represent. Theor. 20 (2017), 1061-1083. 2, 4, 15, 18

    work page 2017

  8. [8]

    Leavitt, Modules over rings of words , Proc

    W.G. Leavitt, Modules over rings of words , Proc. Amer. Math. Soc. 7 (1956), 188–193. 1

    work page 1956

  9. [9]

    Leavitt, Modules without invariant basis number, Proc

    W.G. Leavitt, Modules without invariant basis number, Proc. Amer. Math. Soc. 8 (1957), 322–328. 1

    work page 1957

  10. [10]

    Leavitt, The module type of a ring, Trans

    W.G. Leavitt, The module type of a ring, Trans. Amer. Math. Soc. 103 (1962) 113–130. 1

    work page 1962

  11. [11]

    Leavitt, The module type of homomorphic images, Duke Math

    W.G. Leavitt, The module type of homomorphic images, Duke Math. J. 32 (1965) 305–311. 1

    work page 1965

  12. [12]

    Phillips, A classification theorem for nuclear purely infinite simple C ∗ -algebras, 1

    N.C. Phillips, A classification theorem for nuclear purely infinite simple C ∗ -algebras, 1

    work page

  13. [13]

    Preusser, The V-monoid of a weighted Leavitt path algebra , accepted by Israel J

    R. Preusser, The V-monoid of a weighted Leavitt path algebra , accepted by Israel J. Math. 2

    work page

  14. [14]

    Preusser, The Gelfand-Kirillov dimension of a weighted Leavitt path a lgebra, accepted by J

    R. Preusser, The Gelfand-Kirillov dimension of a weighted Leavitt path a lgebra, accepted by J. Algebra Appl., https://doi.org/10.1142/S0219498820500590. 2, 5, 15, 16

    work page doi:10.1142/s0219498820500590

  15. [15]

    Locally finite weighted Leavitt path algebras

    R. Preusser, Locally finite weighted Leavitt path algebras , arXiv:1806.06139 [math.RA]. 2, 5

    work page internal anchor Pith review Pith/arXiv arXiv

  16. [16]

    Raeburn, Graph algebras

    I. Raeburn, Graph algebras. CBMS Regional Conference S eries in Mathematics, 103 American Mathematical Society, Providence, RI, 2005. 1 E-mail address : raimund.preusser@gmx.de

    work page 2005