D. Hernandez and B. Leclerc [Duke Math. J. 154, No. 2, 265–341 (2010; Zbl 1284.17010)] first realized that the Grothendieck ring of a certain monoidal subcategory \(\mathcal{C}_1\) of the category \(\mathcal{C}\) of finite-dimensional \(U_q(L\mathfrak{g})\)-modules had the structure of a cluster algebra. They thus proved that the Grothendieck ring of a certain monoidal subcategory \(\mathcal{O}^+_\mathbb{Z}\) of the category \(\mathcal{O}\) had a cluster algebra structure of infinite rank, for which one can take as initial seed the classes of the positive prefundamental representations. That is, the category \(\mathcal{O}^+_{\mathbb{Z}}\) contains the finite-dimensional representations and the positive prefundamental representations whose spectral parameter satisfy an integrality condition. Moreover, certain exchange relations, such as the Baxter relation, coming from cluster mutations appear naturally.
In order to construct of quantum Grothendieck ring for the category \(\mathcal{O}\) of representations of the quantum loop algebra introduced by D. Hernandez and M. Jimbo [Compos. Math. 148, No. 5, 1593–1623 (2012; Zbl 1266.17010)], previous approaches were no longer applicable. The geometrical approach of H. Nakajima [Ann. Math. (2) 160, No. 3, 1057–1097 (2005; Zbl 1140.17015)] and M. Varagnolo and E. Vasserot [Prog. Math. 210, 345–365 (2003; Zbl 1162.17307)] (in which the \(t\)-graduation naturally comes from the graduation of cohomological complexes) requires a geometric interpretation of the objects in the category \(\mathcal{O}\), which is yet to be found. The more algebraic approach consisting of realizing the (quantum) Grothendieck ring as an invariant under a sort of Weyl symmetry, which allowed D. Hernandez [Adv. Math. 187, No. 1, 1–52 (2004; Zbl 1098.17009)] to define a quantum Grothendieck ring of finite-dimensional representations in non-simply laced types, is again no longer relevant for the category \(\mathcal{O}\). However, only the cluster algebra approach yields results in this context.
The author constructs a quantum Grothendieck ring for a certain monoidal subcategory of the category \(\mathcal{O}\) (Theorem 5.2.1, page 180). She uses 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 \(A\), she proves that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite-dimensional representations of the associated quantum affine algebra (Theorem 8.1.1, page 193).