Brauer configuration algebras defined by snake graphs and Kronecker modules

Agustín Moreno Cañadas; Pedro Fernando Fernández Espinosa; Natalia Agudelo Muñetón; Agustín Moreno Cañadas; Pedro Fernando Fernández Espinosa; Natalia Agudelo Muñetón

doi:10.3934/era.2022157

Electronic Research Archive

2022, Volume 30, Issue 8: 3087-3110. doi: 10.3934/era.2022157

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Research article

Brauer configuration algebras defined by snake graphs and Kronecker modules

Agustín Moreno Cañadas ¹,
Pedro Fernando Fernández Espinosa ^{2
,
,},
Natalia Agudelo Muñetón ¹

1.
Departamento de Matemáticas, Universidad Nacional de Colombia, Edificio Yu Takeuchi 404, Kra 30 No 45–03, Bogotá 11001000, Colombia
2.
Escuela de Ingeniería Electrónica, Universidad Pedagógica y Tecnológica de Colombia, Calle 4 Sur No. 15–134, Sogamoso, Boyacá 152210, Colombia

Received: 30 March 2022 Revised: 11 May 2022 Accepted: 30 May 2022 Published: 09 June 2022

Recently, Çanakçi and Schroll proved that associated with a string module $ M(w) $ there is an appropriated snake graph $ \mathscr{G} $. They established a bijection between the corresponding perfect matching lattice $ \mathscr{L}(\mathscr{G}) $ of $ \mathscr{G} $ and the canonical submodule lattice $ \mathscr{L}(M(w)) $ of $ M(w) $. We introduce Brauer configurations whose polygons are defined by snake graphs in line with these results. The developed techniques allow defining snake graphs, which after suitable procedures, build Kronecker modules. We compute the dimension of the Brauer configuration algebras and their centers arising from the different processes. As an application, we estimate the trace norm of the canonical non-regular Kronecker modules and some families of trees associated with some snake graphs classes.
- Auslander-Reiten quiver,
- Brauer configuration algebra,
- Kronecker algebra,
- Snake graph,
- trace norm
Citation: Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa, Natalia Agudelo Muñetón. Brauer configuration algebras defined by snake graphs and Kronecker modules[J]. Electronic Research Archive, 2022, 30(8): 3087-3110. doi: 10.3934/era.2022157

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Abstract

Recently, Çanakçi and Schroll proved that associated with a string module $ M(w) $ there is an appropriated snake graph $ \mathscr{G} $. They established a bijection between the corresponding perfect matching lattice $ \mathscr{L}(\mathscr{G}) $ of $ \mathscr{G} $ and the canonical submodule lattice $ \mathscr{L}(M(w)) $ of $ M(w) $. We introduce Brauer configurations whose polygons are defined by snake graphs in line with these results. The developed techniques allow defining snake graphs, which after suitable procedures, build Kronecker modules. We compute the dimension of the Brauer configuration algebras and their centers arising from the different processes. As an application, we estimate the trace norm of the canonical non-regular Kronecker modules and some families of trees associated with some snake graphs classes.

References

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