The graded classification conjectures hold for various finite representations of Leavitt path algebras

Wolfgang Bock; Roozbeh Hazrat; Alfilgen Sebandal

doi:10.1016/j.jalgebra.2025.02.035

The graded classification conjectures hold for various finite representations of Leavitt path algebras

Wolfgang Bock, Roozbeh Hazrat, Alfilgen Sebandal

School of Computer, Data & Mathematical Sciences

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Abstract

The Graded Classification Conjecture states that for finite directed graphs E and F, the associated Leavitt path algebras L_k(E) and L_k(F) are graded Morita equivalent, i.e., Gr-L_k(E)≈_grGr-L_k(F), if and only if, their graded Grothendieck groups are isomorphic K₀^gr(L_k(E))≅K₀^gr(L_k(F)) as order-preserving Z[x,x⁻¹]-modules. Furthermore, if under this isomorphism, the class [L_k(E)] is sent to [L_k(F)] then the algebras are graded isomorphic, i.e., L_k(E)≅_grL_k(F). In this note we show that, for finite graphs E and F with no sinks and sources, an order-preserving Z[x,x⁻¹]-module isomorphism K₀^gr(L_k(E))≅K₀^gr(L_k(F)) gives that the categories of locally finite dimensional graded modules of L_k(E) and L_k(F) are equivalent, i.e., gr^Z−L_k(E)≈_grgr^Z−L_k(F). We further obtain that the category of finite dimensional (graded) modules is equivalent, i.e., mod-L_k(E)≈mod-L_k(F) and gr-L_k(E)≈_grgr-L_k(F).

Original language	English
Pages (from-to)	303-333
Number of pages	31
Journal	Journal of Algebra
Volume	672
DOIs	https://doi.org/10.1016/j.jalgebra.2025.02.035
Publication status	Published - Jun 2025

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Keywords

Graded Morita equivalence
Hazrat conjecture
Leavitt path algebra

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10.1016/j.jalgebra.2025.02.035Licence: CC BY

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@article{7c6ecd325c054fd2b46936d33c2de966,

title = "The graded classification conjectures hold for various finite representations of Leavitt path algebras",

abstract = "The Graded Classification Conjecture states that for finite directed graphs E and F, the associated Leavitt path algebras Lk(E) and Lk(F) are graded Morita equivalent, i.e., Gr-Lk(E)≈grGr-Lk(F), if and only if, their graded Grothendieck groups are isomorphic K0gr(Lk(E))≅K0gr(Lk(F)) as order-preserving Z[x,x−1]-modules. Furthermore, if under this isomorphism, the class [Lk(E)] is sent to [Lk(F)] then the algebras are graded isomorphic, i.e., Lk(E)≅grLk(F). In this note we show that, for finite graphs E and F with no sinks and sources, an order-preserving Z[x,x−1]-module isomorphism K0gr(Lk(E))≅K0gr(Lk(F)) gives that the categories of locally finite dimensional graded modules of Lk(E) and Lk(F) are equivalent, i.e., grZ−Lk(E)≈grgrZ−Lk(F). We further obtain that the category of finite dimensional (graded) modules is equivalent, i.e., mod-Lk(E)≈mod-Lk(F) and gr-Lk(E)≈grgr-Lk(F).",

keywords = "Graded Morita equivalence, Hazrat conjecture, Leavitt path algebra",

author = "Wolfgang Bock and Roozbeh Hazrat and Alfilgen Sebandal",

year = "2025",

month = jun,

doi = "10.1016/j.jalgebra.2025.02.035",

language = "English",

volume = "672",

pages = "303--333",

journal = "Journal of Algebra",

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T1 - The graded classification conjectures hold for various finite representations of Leavitt path algebras

AU - Bock, Wolfgang

AU - Hazrat, Roozbeh

AU - Sebandal, Alfilgen

PY - 2025/6

Y1 - 2025/6

N2 - The Graded Classification Conjecture states that for finite directed graphs E and F, the associated Leavitt path algebras Lk(E) and Lk(F) are graded Morita equivalent, i.e., Gr-Lk(E)≈grGr-Lk(F), if and only if, their graded Grothendieck groups are isomorphic K0gr(Lk(E))≅K0gr(Lk(F)) as order-preserving Z[x,x−1]-modules. Furthermore, if under this isomorphism, the class [Lk(E)] is sent to [Lk(F)] then the algebras are graded isomorphic, i.e., Lk(E)≅grLk(F). In this note we show that, for finite graphs E and F with no sinks and sources, an order-preserving Z[x,x−1]-module isomorphism K0gr(Lk(E))≅K0gr(Lk(F)) gives that the categories of locally finite dimensional graded modules of Lk(E) and Lk(F) are equivalent, i.e., grZ−Lk(E)≈grgrZ−Lk(F). We further obtain that the category of finite dimensional (graded) modules is equivalent, i.e., mod-Lk(E)≈mod-Lk(F) and gr-Lk(E)≈grgr-Lk(F).

AB - The Graded Classification Conjecture states that for finite directed graphs E and F, the associated Leavitt path algebras Lk(E) and Lk(F) are graded Morita equivalent, i.e., Gr-Lk(E)≈grGr-Lk(F), if and only if, their graded Grothendieck groups are isomorphic K0gr(Lk(E))≅K0gr(Lk(F)) as order-preserving Z[x,x−1]-modules. Furthermore, if under this isomorphism, the class [Lk(E)] is sent to [Lk(F)] then the algebras are graded isomorphic, i.e., Lk(E)≅grLk(F). In this note we show that, for finite graphs E and F with no sinks and sources, an order-preserving Z[x,x−1]-module isomorphism K0gr(Lk(E))≅K0gr(Lk(F)) gives that the categories of locally finite dimensional graded modules of Lk(E) and Lk(F) are equivalent, i.e., grZ−Lk(E)≈grgrZ−Lk(F). We further obtain that the category of finite dimensional (graded) modules is equivalent, i.e., mod-Lk(E)≈mod-Lk(F) and gr-Lk(E)≈grgr-Lk(F).

KW - Graded Morita equivalence

KW - Hazrat conjecture

KW - Leavitt path algebra

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DO - 10.1016/j.jalgebra.2025.02.035

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AN - SCOPUS:86000667647

SN - 0021-8693

VL - 672

SP - 303

EP - 333

JO - Journal of Algebra

JF - Journal of Algebra

ER -

The graded classification conjectures hold for various finite representations of Leavitt path algebras

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