On rooted cluster morphisms and cluster structures in 2-Calabi–Yau triangulated categories

Wen Chang Shaanxi Normal University Bin Zhu Tsinghua University

Representation Theory mathscidoc:1611.30001

Journal of Algebra, 458, 387-421, 2016
We study rooted cluster algebras and rooted cluster mor-phisms which were introduced in [1]recently and cluster struc-tures in 2-Calabi–Yau triangulated categories. An example of rooted cluster morphism which is not ideal is given, this clar-ifyinga doubt in [1]. We introduce the notion of freezing of a seed and show that an injective rooted cluster morphism always arises from a freezing and a subseed. Moreover, it is a section if and only if it arises from a subseed. This answers the Problem 7.7 in [1]. We prove that an inducible rooted clus-ter morphism is ideal if and only if it can be decomposed as a surjective rooted cluster morphism and an injective rooted cluster morphism. For rooted cluster algebras arising from a 2-Calabi–Yau triangulated category Cwith cluster tilting ob-jects, we give an one-to-one correspondence between certain pairs of their rooted cluster subalgebras which we call com-plete pairs (see Definition2.27) and cotorsion pairs in C.
Rooted cluster algebra;(Ideal) Rooted cluster morphism;Rooted cluster subalgebra
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@inproceedings{wen2016on,
  title={On rooted cluster morphisms and cluster structures in 2-Calabi–Yau triangulated categories},
  author={Wen Chang, and Bin Zhu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161123155438152556626},
  booktitle={Journal of Algebra},
  volume={458},
  pages={387-421},
  year={2016},
}
Wen Chang, and Bin Zhu. On rooted cluster morphisms and cluster structures in 2-Calabi–Yau triangulated categories. 2016. Vol. 458. In Journal of Algebra. pp.387-421. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161123155438152556626.
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