Quantum cluster characters of Hall algebras revisited

Abstract

Let Q be a finite acyclic valued quiver. We give a bialgebra structure and an integration map on the Hall algebra associated to the morphism category of projective valued 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 their cluster multiplication formulas.