Isomorphisms among quantum Grothendieck rings and propagation of positivity

Let (g, g) be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with g being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories C-g and C-g of finite-dimensional representations over the quantum loop algebras of g and g, respectively. As a consequence, we solve long-standing problems: the positivity of the analogs of Kazhdan-Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced g. In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur-Weyl dualities, we prove the analog of Kazhdan-Lusztig conjecture (formulated in [D. Hernandez, Algebraic approach to q, t-characters, Adv. Math. 187 (2004), no. 1, 1-52]) for simple modules in remarkable monoidal subcategories of C-g for any non-simply-laced g, and for any simple finite-dimensional modules in C-g for g of type B-n. In the course of the proof we obtain and combine several new ingredients. In particular, we establish a quantum analog of T-systems, and also we generalize the isomorphisms of [D. Hernandez and B. Leclerc, Quantum Grothendieck rings and derived Hall algebras, J. reine angew. Math. 701 (2015), 77-126, D. Hernandez and H. Oya, Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm, Adv. Math. 347 (2019), 192-272] to all g in a unified way, that is, isomorphisms between subalgebras of the quantum group of g and subalgebras of the quantum Grothendieck ring of C-g.