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.

In Berenstein and Rupel (2015), the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category \mathcal{A} to an appropriate q-polynomial algebra. In the case that \mathcal{A} is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in (Ding and Xu, Sci. China Math. 55(10) 2045–2066, 2012). Moreover, if the underlying graph of Q associated with \mathcal{A} is bipartite and the matrix B associated to the quiver Q is of full rank, we show that the image of the algebra homomorphism is in the corresponding quantum cluster algebra.

In this paper, we generalize the categorifical construction of a quantum group and its canonical basis by Lusztig (\cite{Lusztig,Lusztig2}) to the generic form of whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula in \cite{Green} and the restriction functor in \cite{Lusztig2}. By a geometric way to prove Green formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework.