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We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category O 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 A , we prove that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite-dimensional representations of the associated q
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