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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.
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Let $U_q(mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $mathcal{C}^0_{mathfrak{g}}$ be Hernandez-Leclercs category. For a duality datum $mathcal{D}$ in $mathcal{C}^0_{mathfrak{g}}$, we denote by $mathcal{F}_{mathcal{D}}$ the quantum affine Weyl-Schur duality functor. We give sufficient conditions for a duality datum $mathcal{D}$ to provide the functor $mathcal{F}_{mathcal{D}}$ sending simple modules to simple modules. Then we introduce the notion of cuspidal modules in $mathcal{C}^0_{mathfrak{g}}$, and show that all simple modules in $mathcal{C}^0_{mathfrak{g}}$ can be constructed as the heads of ordered tensor products of cuspidal modules.
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.
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.
We construct a basis for a modified quantum group of finite type, extending the PBW bases of positive and negative halves of a quantum group. Generalizing Lusztigs classic results on PBW bases, we show that this basis is orthogonal with respect to its natural bilinear form (and hence called a PBW basis), and moreover, the matrix for the PBW-expansion of the canonical basis is unital triangular. All these follow by a new construction of the modified quantum group of arbitrary type, which is built on limits of sequences of elements in tensor products of lowest and highest weight modules. Explicitly formulas are worked out in the rank one case.
Let $U_q(mathfrak{g})$ be a twisted affine quantum group of type $A_{N}^{(2)}$ or $D_{N}^{(2)}$ and let $mathfrak{g}_{0}$ be the finite-dimensional simple Lie algebra of type $A_{N}$ or $D_{N}$. For a Dynkin quiver of type $mathfrak{g}_{0}$, we define a full subcategory ${mathcal C}_{Q}^{(2)}$ of the category of finite-dimensional integrable $U_q(mathfrak{g})$-modules, a twisted version of the category ${mathcal C}_{Q}$ introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur-Weyl duality, we construct an exact faithful KLR-type duality functor ${mathcal F}_{Q}^{(2)}: Rep(R) rightarrow {mathcal C}_{Q}^{(2)}$, where $Rep(R)$ is the category of finite-dimensional modules over the quiver Hecke algebra $R$ of type $mathfrak{g}_{0}$ with nilpotent actions of the generators $x_k$. We show that ${mathcal F}_{Q}^{(2)}$ sends any simple object to a simple object and induces a ring isomorphism $K(Rep(R)) simeq K({mathcal C}_{Q}^{(2)})$.
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