Isomorphisms among quantum Grothendieck rings and propagation of positivity

Let g{\mathfrak{g},\mathsf{g})} be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with - {\mathsf{g}} being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories cg\mathscr{C}_{\mathfrak{g}}} and c - {\mathscr{C}_{\mathsf{g}}} of finite-dimensional representations over the quantum loop algebras of g{\mathfrak{g}} and - {\mathsf{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{\mathfrak{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,tq,t-characters, Adv. Math. 187 2004, 1, 1-52]) for simple modules in remarkable monoidal subcategories of cg{\mathscr{C}_{\mathfrak{g}}} for any non-simply-laced g{\mathfrak{g}}, and for any simple finite-dimensional modules in cg{\mathscr{C}_{\mathfrak{g}}} for g{\mathfrak{g}} of type Bn{\mathrm{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{\mathfrak{g}} in a unified way, that is, isomorphisms between subalgebras of the quantum group of - {\mathsf{g}} and subalgebras of the quantum Grothendieck ring of cg{\mathscr{C}_{\mathfrak{g}}}.