Quantum cluster characters of Hall algebras revisited

TL;DR

This paper revisits quantum cluster characters within Hall algebras, establishing new algebraic structures and recovering key formulas, thereby deepening the understanding of quantum cluster algebras related to acyclic quivers.

Contribution

It introduces a bialgebra structure and an integration map on Hall algebras, connecting quantum cluster algebras with Hall algebra morphisms and formulas.

Findings

Recovered the surjective homomorphism for quantum cluster algebra as a Hall algebra sub-quotient.

Established a bialgebra structure and an integration map on Hall algebras.

Reproved the quantum Caldero--Chapoton formula and related multiplication formulas.

Abstract

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 \cite{DXZ}, 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.