Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras II

Abstract

Let $\g$ be an untwisted affine Kac-Moody algebra of type $A_n^{(1)}$ $(n\ge 1)$ or $D_n^{(1)}$ $(n\ge 4)$ and let $\g_0$ be the underlying finite-dimensional simple Lie subalgebra of $\g$. For each Dynkin quiver $Q$ of type $\g_0$, Hernandez and Leclerc (\cite{HL11}) introduced a tensor subcategory $\CC_Q$ of the category of finite-dimensional integrable $\uqpg$-modules and proved that the Grothendieck ring of $\CC_Q$ is isomorphic to $\C [N]$, the coordinate ring of the unipotent group $N$ associated with $\g_0$. We apply the generalized quantum affine Schur-Weyl duality introduced in \cite{KKK13} to construct an exact functor $\F$ from the category of finite-dimensional graded $R$-modules to the category $\CC_Q$, where $R$ denotes the symmetric quiver Hecke algebra associated to $\g_0$. We prove that the homomorphism induced by the functor $\F$ coincides with the homomorphism of Hernandez and Leclerc and show that the functor $\F$ sends the simple modules to the simple modules.

46 pages

References in corpus (3)

• Graded quiver varieties, quantum cluster algebras and dual canonical basis
• PBW bases and KLR algebras
• Homological properties of finite type Khovanov-Lauda-Rouquier algebras

Cited by in corpus (16)

• Mirkovic-Vilonen polytopes and Khovanov-Lauda-Rouquier algebras
• Monoidal categorification and quantum affine algebras
• Q-data and representation theory of untwisted quantum affine algebras
• Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
• Categorical relations between Langlands dual quantum affine algebras: Exceptional cases
• Isomorphisms among quantum Grothendieck rings and propagation of positivity
• Geometric realization of Dynkin quiver type quantum affine Schur-Weyl duality
• Twisted and folded Auslander-Reiten quivers and applications to the representation theory of quantum affine algebras
• Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras III
• Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types
• Graded quiver varieties and singularities of normalized R-matrices for fundamental modules
• The $(q,t)$-Cartan matrix specialized at $q=1$
• $t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras
• Equivalence between module categories over quiver Hecke algebras and Hernandez-Leclerc's categories in general types
• Affinization of $q$-oscillator representations of $U_q({\mathfrak{gl}}_n)$
• Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras