Properties of the 𝒱-Monoid of Weighted Leavitt Path Algebras
Abstract
For a row-finite weighted graph (E,w), Preusser showed that the monoid 𝒱(Lk(E,w)) of finitely generated projective modules over the weighted Leavitt path algebra Lk(E,w) is isomorphic to a combinatorially defined weighted graph monoid ℳ(E,w). We study two structural properties of ℳ(E,w): confluence and cancellativity. We introduce a reduction system on the free commutative monoid presenting ℳ(E,w), obtain sufficient conditions for non-confluence by constructing explicit non-confluent triples, and provide a complete confluence characterization for certain classes of weighted graphs. Turning to cancellativity, we work within Preusser's class of weighted graphs satisfying Condition (LPA), for which Lk(E,w) is isomorphic to an unweighted Leavitt path algebra Lk(F) via a two-step construction. We introduce an auxiliary graph associated to the intermediate step of this construction and use it to give a graph-theoretic characterization of when ℳ(E,w) is cancellative. Finally, under Condition (LPA), we show that Preusser's construction upgrades to a graded isomorphism Lk(E,w) ≅gr Lk(F) with respect to the standard ℤλ(E,w)-grading of weighted Leavitt path algebras, yielding 𝒱gr(Lk(E,w)) ≅ 𝒱gr(Lk(F)) as ℤλ(E,w)-monoids.
Related
- On the existence of divisible non-injective modules over Leavitt path algebras
- Growth functions of algebras and an application to Leavitt path algebras
- A note on the ideal lattice of a weighted Leavitt path algebra
- Graded classification of Leavitt path algebras in terms of strong shift equivalence
- On some relationships between Leavitt path algebras and Brauer configuration algebras
BibTeX
@misc{goswami2026properties,
title = {{Properties of the $\mathcal V$-Monoid of Weighted Leavitt Path Algebras}},
author = {Rishabh Goswami and Alfilgen Sebandal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.31087},
eprint = {2608.31087},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.31087},
}
RIS
TY - UNPB
TI - Properties of the 𝒱-Monoid of Weighted Leavitt Path Algebras
AU - Rishabh Goswami
AU - Alfilgen Sebandal
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2608.31087
UR - https://arxiv.org/abs/2608.31087
AB - For a row-finite weighted graph (E,w), Preusser showed that the monoid 𝒱(L_k(E,w)) of finitely generated projective modules over the weighted Leavitt path algebra L_k(E,w) is isomorphic to a combinatorially defined weighted graph monoid ℳ(E,w). We study two structural properties of ℳ(E,w): confluence and cancellativity. We introduce a reduction system on the free commutative monoid presenting ℳ(E,w), obtain sufficient conditions for non-confluence by constructing explicit non-confluent triples, and provide a complete confluence characterization for certain classes of weighted graphs. Turning to cancellativity, we work within Preusser's class of weighted graphs satisfying Condition (LPA), for which L_k(E,w) is isomorphic to an unweighted Leavitt path algebra L_k(F) via a two-step construction. We introduce an auxiliary graph associated to the intermediate step of this construction and use it to give a graph-theoretic characterization of when ℳ(E,w) is cancellative. Finally, under Condition (LPA), we show that Preusser's construction upgrades to a graded isomorphism L_k(E,w) ≅_gr L_k(F) with respect to the standard ℤ^λ(E,w)-grading of weighted Leavitt path algebras, yielding 𝒱^gr(L_k(E,w)) ≅ 𝒱^gr(L_k(F)) as ℤ^λ(E,w)-monoids.
ER -