Quantum cluster characters of Hall algebras revisited

Let Q be a finite acyclic valued quiver. We define a bialgebra structure and an integration map on the Hall algebra associated to the morphism category of projective representations of Q. As an application, we recover the surjective homomorphism defined in [12], which realizes the principal coefficient quantum cluster algebra A(q) (Q) as a sub-quotient of the Hall algebra of morphisms. Moreover, we also recover the quantum Caldero-Chapoton formula, as well as some multiplication formulas between quantum Caldero-Chapoton characters.