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.