Graphs with disjoint cycles, classification via the talented monoid

Roozbeh Hazrat; Alfilgen N. Sebandal; Jocelyn P. Vilela

doi:10.1016/j.jalgebra.2021.11.022

Graphs with disjoint cycles, classification via the talented monoid

Roozbeh Hazrat, Alfilgen N. Sebandal, Jocelyn P. Vilela

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

We characterise directed graphs consisting of disjoint cycles via their talented monoids. We show that a graph E consists of disjoint cycles precisely when its talented monoid T_E has a certain Jordan-Hölder composition series. These are graphs E whose associated Leavitt path algebras L_k(E) have finite Gelfand-Kirillov dimension (GKdim). We show that this dimension can be determined as the length of certain ideal series of the talented monoid. Since T_E is the positive cone of the graded Grothendieck group K₀^gr(L_k(E)), we conclude that for graphs E and F, if K₀^gr(L_k(E))≅K₀^gr(L_k(F)) then GKdimL_k(E)=GKdimL_k(F), thus providing more evidence for the Graded Classification Conjecture for Leavitt path algebras.

Original language	English
Pages (from-to)	319-340
Number of pages	22
Journal	Journal of Algebra
Volume	593
DOIs	https://doi.org/10.1016/j.jalgebra.2021.11.022
Publication status	Published - 1 Mar 2022

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© 2021 Elsevier Inc.

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title = "Graphs with disjoint cycles, classification via the talented monoid",

abstract = "We characterise directed graphs consisting of disjoint cycles via their talented monoids. We show that a graph E consists of disjoint cycles precisely when its talented monoid TE has a certain Jordan-H{\"o}lder composition series. These are graphs E whose associated Leavitt path algebras Lk(E) have finite Gelfand-Kirillov dimension (GKdim). We show that this dimension can be determined as the length of certain ideal series of the talented monoid. Since TE is the positive cone of the graded Grothendieck group K0gr(Lk(E)), we conclude that for graphs E and F, if K0gr(Lk(E))≅K0gr(Lk(F)) then GKdimLk(E)=GKdimLk(F), thus providing more evidence for the Graded Classification Conjecture for Leavitt path algebras.",

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T1 - Graphs with disjoint cycles, classification via the talented monoid

AU - Hazrat, Roozbeh

AU - Sebandal, Alfilgen N.

AU - Vilela, Jocelyn P.

PY - 2022/3/1

Y1 - 2022/3/1

N2 - We characterise directed graphs consisting of disjoint cycles via their talented monoids. We show that a graph E consists of disjoint cycles precisely when its talented monoid TE has a certain Jordan-Hölder composition series. These are graphs E whose associated Leavitt path algebras Lk(E) have finite Gelfand-Kirillov dimension (GKdim). We show that this dimension can be determined as the length of certain ideal series of the talented monoid. Since TE is the positive cone of the graded Grothendieck group K0gr(Lk(E)), we conclude that for graphs E and F, if K0gr(Lk(E))≅K0gr(Lk(F)) then GKdimLk(E)=GKdimLk(F), thus providing more evidence for the Graded Classification Conjecture for Leavitt path algebras.

AB - We characterise directed graphs consisting of disjoint cycles via their talented monoids. We show that a graph E consists of disjoint cycles precisely when its talented monoid TE has a certain Jordan-Hölder composition series. These are graphs E whose associated Leavitt path algebras Lk(E) have finite Gelfand-Kirillov dimension (GKdim). We show that this dimension can be determined as the length of certain ideal series of the talented monoid. Since TE is the positive cone of the graded Grothendieck group K0gr(Lk(E)), we conclude that for graphs E and F, if K0gr(Lk(E))≅K0gr(Lk(F)) then GKdimLk(E)=GKdimLk(F), thus providing more evidence for the Graded Classification Conjecture for Leavitt path algebras.

UR - https://hdl.handle.net/1959.7/uws:67553

U2 - 10.1016/j.jalgebra.2021.11.022

DO - 10.1016/j.jalgebra.2021.11.022

M3 - Article

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VL - 593

SP - 319

EP - 340

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Graphs with disjoint cycles, classification via the talented monoid

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