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

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We study rooted cluster algebras and rooted cluster morphisms which were introduced in ADS13 recently and cluster structures in 2-Calabi-Yau triangulated categories. An example of rooted cluster morphism which is not ideal is given, this clarifies a doubt in ADS13. We introduce the notion of frozenization of a seed and show that an injective rooted cluster morphism always arises from a frozenization and a subseed. Moreover, it is a section if and only if it arises from a subseed. This answers the Problem 7.7 in ADS13. 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. We also introduce the tensor decompositions of a rooted cluster algebra and of a 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 def of complete pairs for precise meaning) and cotorsion pairs in C.