Abstract
Giraldo and Merklen classified the irreducible morphisms in the bounded derived categories of finite dimensional algebras in three classes. The Auslander-Reiten triangles in these categories are made of irreducible morphisms and we classify these triangles in terms of Giraldo and Merklen’s classes. As a byproduct this yields an explicit description of the cone of any irreducible morphism. For tilted algebras this applies to a constructive description of the transjective components of the Auslander-Reiten quiver.
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Acknowledgments
The authors wish to thank Patrick Le Meur for his fruitful discussions. The authors are especially grateful to Ibrahim Assem for his constant and invaluable help. The first author was partially supported by the DMAT-UFPR and CNPq-Universal 477880/2012-6. The second author was partially supported by the DMA-UFV. The third author was partially supported by the CODI, Estrategia de Sostenibilidad 2019-2020 (Universidad de Antioquia), and COLCIENCIAS-ECOPETROL (Contrato RC. No. 0266-2013).
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Presented by: Christof Geiss
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Alvares, E.R., Fernandes, S.M. & Giraldo, H. Shapes of Auslander-Reiten Triangles. Algebr Represent Theor 23, 2257–2274 (2020). https://doi.org/10.1007/s10468-019-09937-w
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DOI: https://doi.org/10.1007/s10468-019-09937-w