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

JOURNAL OF ALGEBRA

卷:458

Article

Chang, Wen^1,2  Zhu, Bin² 

[1] Shaanxi Normal Univ, Sch Math & Informat Sci, Xian 710062, Peoples R China

[2] Tsinghua Univ, Dept Math Sci, Beijing 10084, Peoples R China

关键词: Rooted cluster algebra;    (Ideal) Rooted cluster morphism;    Rooted cluster subalgebra;    Cotorsion pair;    Cluster structure;   

DOI  :  10.1016/j.jalgebra.2016.03.042

来源: Elsevier

【 摘 要 】

We study rooted cluster algebras and rooted cluster morphisms which were introduced in [1] recently and cluster structures in 2-Calabi-Yau triangulated categories. An example of rooted cluster morphism which is not ideal is given, this clarifying a 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 cluster 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 C with cluster tilting objects, we give an one-to-one correspondence between certain pairs of their rooted cluster subalgebras which we call complete pairs (see Definition 2.27) and cotorsion pairs in C. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free