The injective envelope of simple modules over Leavitt path algebras II: simples arising from exclusive cycles

@misc{abrams2026injective,
  title = {{The injective envelope of simple modules over Leavitt path algebras II: simples arising from exclusive cycles}},
  author = {G. Abrams and F. Mantese and A. Tonolo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00952},
  eprint = {2609.00952},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00952},
}

TY  - UNPB
TI  - The injective envelope of simple modules over Leavitt path algebras II: simples arising from exclusive cycles
AU  - G. Abrams
AU  - F. Mantese
AU  - A. Tonolo
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.00952
UR  - https://arxiv.org/abs/2609.00952
AB  - Let K be any field, E any directed graph, and L_K(E) the associated Leavitt path algebra. As described first by Chen, and subsequently generalized by Ara and Rangaswamy, for each cycle c in E one can build the simple left L_K(E)-module V_c^E, and then more generally V_p(x),c^E (where p(x) is an irreducible polynomial in K[x,x^−1]). A cycle c is called {\it exclusive} in case none of the vertices of c is the base of any cycle other than c. In our main result we provide an explicit description of the injective envelope of V_c^E, and then more generally of V_p(x),c^E, for each exclusive cycle c. Our method involves defining an L_K(E)-module structure on an appropriately-built K-vector space of infinite series. Our main result significantly generalizes previous work of the authors, in that the result holds for all graphs (finite or not), and all exclusive cycles.
ER  -