Quantum Grothendieck rings as quantum cluster algebras

We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category [Formula: see text] of representations of the quantum loop algebra introduced by Hernandez–Jimbo. We use the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type [Formula: see text] , we prove that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite‐dimensional representations of the associated quantum affine algebra. In type [Formula: see text] , we identify remarkable relations in this quantum Grothendieck ring.