On some relationships between Leavitt path algebras and Brauer configuration algebras
4 affiliations
- 1Ciencias Básicas, Área Físico-Matemáticas, Universidad EIA, Envigado, Colombia
- 2Departamento de Matemáticas, Universidad Nacional de Colombia, Edificio Yu Takeuchi 404, Kra 30 No. 45-03, Bogotá, Colombia
- 3Programa Académico de Matemáticas, Universidad Distrital Francisco José de Caldas, Bogotá, Colombia
- 4Departamento de Matemáticas, Universidad Nacional de Colombia, Medellín, Colombia
Abstract
We compare Cuntz–Krieger and Brauer relations inside the same path algebra when every configuration vertex has valency two. Choosing one arrow from each successor pair produces a directed graph, and the Cuntz–Krieger quotient is its Leavitt path algebra. The Brauer quotient is the usual Brauer configuration algebra; it belongs to the Brauer graph class when all polygons are 2-gons. We also determine the two mixed quotients. One can be described for relaxed data and disappears under the standard Brauer axioms, while the other is always zero. In the converse direction, any finite directed graph can be recovered from a standard Brauer configuration and a cut, with isolated vertices accounting for scalar summands. The choice of cut affects more than growth: for diamond-tail and complete binary-tree configurations it changes the Grothendieck group. In the balanced cases we obtain closed spectral formulas and a determinant recurrence for the orders of the finite groups that occur.
Related
- On the existence of divisible non-injective modules over Leavitt path algebras
- Growth functions of algebras and an application to Leavitt path algebras
- Graded classification of Leavitt path algebras in terms of strong shift equivalence
- Baer-Type Ring Characterizations of Steinberg Algebras, with Applications to Leavitt Path Algebras
- The injective envelope of simple modules over Leavitt path algebras I: simple left ideals
BibTeX
@article{andres2026some,
title = {{On some relationships between Leavitt path algebras and Brauer configuration algebras}},
author = {Sarrazola-Alzate Andrés and Moreno Cañadas Agustín and Cifuentes Vargas Veronica and Rodríguez-Nieto José Gregorio and Salazar-Díaz Olga Patricia},
journal = {Open Mathematics},
year = {2026},
month = oct,
doi = {10.1515/math-2025-0285},
url = {https://doi.org/10.1515/math-2025-0285},
note = {CC BY},
}
RIS
TY - JOUR
TI - On some relationships between Leavitt path algebras and Brauer configuration algebras
AU - Sarrazola-Alzate Andrés
AU - Moreno Cañadas Agustín
AU - Cifuentes Vargas Veronica
AU - Rodríguez-Nieto José Gregorio
AU - Salazar-Díaz Olga Patricia
PY - 2026
DA - 2026/10/26
JO - Open Mathematics
DO - 10.1515/math-2025-0285
UR - https://doi.org/10.1515/math-2025-0285
AB - We compare Cuntz–Krieger and Brauer relations inside the same path algebra when every configuration vertex has valency two. Choosing one arrow from each successor pair produces a directed graph, and the Cuntz–Krieger quotient is its Leavitt path algebra. The Brauer quotient is the usual Brauer configuration algebra; it belongs to the Brauer graph class when all polygons are 2-gons. We also determine the two mixed quotients. One can be described for relaxed data and disappears under the standard Brauer axioms, while the other is always zero. In the converse direction, any finite directed graph can be recovered from a standard Brauer configuration and a cut, with isolated vertices accounting for scalar summands. The choice of cut affects more than growth: for diamond-tail and complete binary-tree configurations it changes the Grothendieck group. In the balanced cases we obtain closed spectral formulas and a determinant recurrence for the orders of the finite groups that occur.
SN - 2391-5455
ER -