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Algebras and Representation Theory

, Volume 22, Issue 5, pp 1051–1081 | Cite as

Cluster Subalgebras and Cotorsion Pairs in Frobenius Extriangulated Categories

  • Wen Chang
  • Panyue ZhouEmail author
  • Bin Zhu
Article
  • 104 Downloads

Abstract

Nakaoka and Palu introduced the notion of extriangulated categories by extracting the similarities between exact categories and triangulated categories. In this paper, we study cotorsion pairs in a Frobenius extriangulated category \(\mathcal {C}\). Especially, for a 2-Calabi-Yau extriangulated category \(\mathcal {C}\) with a cluster structure, we describe the cluster substructure in the cotorsion pairs. For rooted cluster algebras arising from \(\mathcal {C}\) with cluster tilting objects, we give a one-to-one correspondence between cotorsion pairs in \(\mathcal {C}\) and certain pairs of their rooted cluster subalgebras which we call complete pairs. Finally, we explain this correspondence by an example relating to a Grassmannian cluster algebra.

Keywords

Frobenius extriangulated categories 2-Calabi-Yau extriangulated (or triangulated) categories Cotorsion pairs Cluster algebras 

Mathematics Subject Classification (2010)

16S99 16S70 18E30 

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Notes

Acknowledgements

The correspondence in Corollary 5.6 was asked by Osamu Iyama in a conference at Seoul Korea in 2014. We are grateful to him for this. The authors also would like to thank the referees for reading the paper carefully and for many suggestions on mathematics and English expressions. The third author thanks Tiwei Zhao for his comments on the paper!

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China
  2. 2.College of MathematicsHunan Institute of Science and TechnologyYueyangPeople’s Republic of China
  3. 3.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China

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