We study rooted cluster algebras and rooted cluster morphisms which were introduced in \cite{ 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 \cite{ 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 \cite{ 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 $\mathcal{ C}$ 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 \ref{ def of complete pairs} for precise meaning) and cotorsion pairs in $\mathcal{ C}$.
Categories: math.RT