On homomorphisms from Ringel-Hall algebras to quantum cluster algebras

In \cite{rupel3},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 \cite{DX}. Moreover, if the underlying graph of $Q$ is bipartite and the matrix $B$ associated to the quiver $Q$ is of full rank, we show that the image of the algebra homomorphisms is in the corresponding quantum cluster algebra.