Monoidal categorification and quantum affine algebras II

We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of \(i\)-boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we prove that various module categories over the quantum affine algebras provide monoidal categorifications of cluster algebras. As special cases, Hernandez-Leclerc categories \(\mathscr{C}_{{\mathfrak{g}}}^{0}\) and \(\mathscr{C}_{{\mathfrak{g}}}^{-}\) provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra.