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We introduce and investigate new invariants on the pair of modules $M$ and $N$ over quantum affine algebras $U_q(mathfrak{g})$ by analyzing their associated R-matrices. From new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable $U_q(mathfrak{g})$-modules to become a monoidal categorification of a cluster algebra.
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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 provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra.
Let $U_q(mathfrak{g})$ be a quantum affine algebra of untwisted affine $ADE$ type, and $mathcal{C}_{mathfrak{g}}^0$ the Hernandez-Leclerc category of finite-dimensional $U_q(mathfrak{g})$-modules. For a suitable infinite sequence $widehat{w}_0= cdots s_{i_{-1}}s_{i_0}s_{i_1} cdots$ of simple reflections, we introduce subcategories $mathcal{C}_{mathfrak{g}}^{[a,b]}$ of $mathcal{C}_{mathfrak{g}}^0$ for all $a le b in mathbb{Z}sqcup{ pm infty }$. Associated with a certain chain $mathfrak{C}$ of intervals in $[a,b]$, we construct a real simple commuting family $M(mathfrak{C})$ in $mathcal{C}_{mathfrak{g}}^{[a,b]}$, which consists of Kirillov-Reshetikhin modules. The category $mathcal{C}_{mathfrak{g}}^{[a,b]}$ provides a monoidal categorification of the cluster algebra $K(mathcal{C}_{mathfrak{g}}^{[a,b]})$, whose set of initial cluster variables is $[M(mathfrak{C})]$. In particular, this result gives an affirmative answer to the monoidal categorification conjecture on $mathcal{C}_{mathfrak{g}}^-$ by Hernandez-Leclerc since it is $mathcal{C}_{mathfrak{g}}^{[-infty,0]}$, and is also applicable to $mathcal{C}_{mathfrak{g}}^0$ since it is $mathcal{C}_{mathfrak{g}}^{[-infty,infty]}$.
We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ring $A_q(mathfrak{n}(w))$, associated with a symmetric Kac-Moody algebra and its Weyl group element $w$, admits a monoidal categorification via the representations of symmetric Khovanov-Lauda- Rouquier algebras. In order to achieve this goal, we give a formulation of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded $R$-modules to become a monoidal categorification, where $R$ is a symmetric Khovanov-Lauda-Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions, once the first-step mutations are possible. Then, we show the existence of a quantum monoidal seed of $A_q(mathfrak{n}(w))$ which admits the first-step mutations in all the directions. As a consequence, we prove the conjecture that any cluster monomial is a member of the upper global basis up to a power of $q^{1/2}$. In the course of our investigation, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.
Let $U_q(mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $mathcal{C}_{mathfrak{g}}$ be Hernandez-Leclercs category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $mathcal{C}_{mathfrak{g}}$. We introduce the notion of a strong (complete) duality datum $D$ and prove that, when $D$ is strong, the induced duality functor $F_D$ sends simple modules to simple modules and preserves the invariants $Lambda$ and $Lambda^infty$ introduced by the authors. We next define the reflections $mathcal{S}_k$ and $mathcal{S}^{-1}_k$ acting on strong duality data $D$. We prove that if $D$ is a strong (resp. complete) duality datum, then $mathcal{S}_k(D)$ and $mathcal{S}_k^{-1}(D)$ are also strong (resp. complete ) duality data. We finally introduce the notion of affine cuspidal modules in $mathcal{C}_{mathfrak{g}}$ by using the duality functor $F_D$, and develop the cuspidal module theory for quantum affine algebras similarly to the quiver Hecke algebra case.
Let $textbf{U}^+$ be the positive part of the quantum group $textbf{U}$ associated with a generalized Cartan matrix. In the case of finite type, Lusztig constructed the canonical basis $textbf{B}$ of $textbf{U}^+$ via two approaches. The first one is an elementary algebraic construction via Ringel-Hall algebra realization of $textbf{U}^+$ and the second one is a geometric construction. The geometric construction of canonical basis can be generalized to the cases of all types. The generalization of the elementary algebraic construction to affine type is an important problem. We give several main results of algebraic constructions to the affine canonical basis in this ariticle. These results are given by Beck-Nakajima, Lin-Xiao-Zhang, Xiao-Xu-Zhao, respectively.
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