Skip to content
DraconiticResearch
Search Sources API
179,152 studies31 sourcesUpdated 53 min ago
PreprintarXiv licencearXiv

The injective envelope of simple modules over Leavitt path algebras I: simple left ideals

  • G. Abrams
  • F. Mantese
  • A. Tonolo
Open at sourcePDFDOICite

Abstract

Let K be any field and E any directed graph. We characterize up to isomorphism the simple (i.e., minimal) left ideals of the Leavitt path algebra LK(E). Then, for each simple %(i.e., minimal) left ideal I of %the Leavitt path algebra LK(E), we explicitly construct the injective envelope of I. This result generalizes to all graphs E and all simple left ideals in LK(E) the construction presented previously by the three authors for the specific case of the Jacobson algebra R=K⟨ X,Y | XY=1⟩ and the simple left R-ideal R(1−YX). Our method involves defining an LK(E)-module structure on a K-vector space of infinite series. We conclude the article by showing how our construction directly gives a description of the injective envelope of simple LK(E)-modules arising from two types of infinite emitters in E.

Related

  1. The injective envelope of simple modules over Leavitt path algebras II: simples arising from exclusive cycles
    G. Abrams, F. Mantese, A. Tonolo
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. On the existence of divisible non-injective modules over Leavitt path algebras
    Mara Barban, Alberto Tonolo
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. simple cartan-free modules of finite rank over the super-virasoro algebras
    Yang Chen, Yongqi Luo, Yueqiang Zhao
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  4. Extensions of Restricted Simple Modules for Restricted Lie Algebras of Cartan Type
    Chao Ma
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. STRONGLY ϕ-FLAT MODULES, STRONGLY NONNIL-INJECTIVE MODULES AND THEIR HOMOLOGICAL DIMENSIONS
    Xiaolei Zhang, Shiqi Xing, Wei Qi
    Rocky Mountain Journal of MathematicsPeer-reviewedLicence not statedFull text6 citedAlgebra & number theoryOpenAlex

Cite

BibTeXRIS
BibTeX
@misc{abrams2026injective,
  title = {{The injective envelope of simple modules over Leavitt path algebras I: simple left ideals}},
  author = {G. Abrams and F. Mantese and A. Tonolo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00947},
  eprint = {2609.00947},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00947},
}
RIS
TY  - UNPB
TI  - The injective envelope of simple modules over Leavitt path algebras I: simple left ideals
AU  - G. Abrams
AU  - F. Mantese
AU  - A. Tonolo
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.00947
UR  - https://arxiv.org/abs/2609.00947
AB  - Let K be any field and E any directed graph. We characterize up to isomorphism the simple (i.e., minimal) left ideals of the Leavitt path algebra L_K(E). Then, for each simple %(i.e., minimal) left ideal I of %the Leavitt path algebra L_K(E), we explicitly construct the injective envelope of I. This result generalizes to all graphs E and all simple left ideals in L_K(E) the construction presented previously by the three authors for the specific case of the Jacobson algebra R=K⟨ X,Y | XY=1⟩ and the simple left R-ideal R(1−YX). Our method involves defining an L_K(E)-module structure on a K-vector space of infinite series. We conclude the article by showing how our construction directly gives a description of the injective envelope of simple L_K(E)-modules arising from two types of infinite emitters in E.
ER  -