Fundamental relations in quantum cluster algebras

Let ${\mathcal{A}}_q$ be an arbitrary quantum cluster algebra with principal coefficients. We give the fundamental relations between the quantum cluster variables arising from one-step mutations from the initial cluster in ${\mathcal{A}}_q$. Immediately and directly, we obtain an algebra homomorphism from the corresponding (untwisted) quantum group to ${\mathcal{A}}_q$.