# Acyclic quantum cluster algebras via Hall algebras of morphisms

## TL;DR

This paper constructs a realization of quantum cluster algebras with principal coefficients for acyclic quivers using Hall algebras of morphisms between projective modules, providing a new algebraic framework.

## Contribution

It introduces a novel approach to realize quantum cluster algebras via Hall algebras of morphisms, connecting cluster theory with representation theory of quivers.

## Key findings

- Quantum cluster algebra realized as a sub-quotient of Hall algebra
- Establishes a link between cluster algebras and morphism categories
- Provides algebraic tools for studying quantum cluster structures

## Abstract

Let $A$ be the path algebra of a finite acyclic quiver $Q$ over a finite field. We realize the quantum cluster algebra with principal coefficients associated to $Q$ as a sub-quotient of a certain Hall algebra involving the category of morphisms between projective $A$-modules.