Isomorphisms among quantum Grothendieck rings and propagation of positivity

Abstract

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