From quantum groups to quantum cluster algebras

We provide a homomorphism of algebras from the quantum group ${\mathbf{U}}_v^{+}(\mathfrak{g})$ to the corresponding quantum cluster algebra ${\mathcal{A}}_q$ with principal coefficients. As a by-product, we show that the quantum cluster variables arising from one-step mutations from the initial cluster variables satisfy the (high order) quantum Serre relations in ${\mathcal{A}}_q$.