## Duke Mathematical Journal

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

#### Abstract

Let ${\mathfrak {g}}$ be an untwisted affine Kac–Moody algebra of type $A^{(1)}_{n}$ ( $n\ge1$) or $D^{(1)}_{n}$ ( $n\ge4$), and let ${\mathfrak {g}}_{0}$ be the underlying finite-dimensional simple Lie subalgebra of ${\mathfrak {g}}$. For each Dynkin quiver $Q$ of type ${\mathfrak {g}}_{0}$, Hernandez and Leclerc introduced a tensor subcategory ${\mathscr {C}}_{Q}$ of the category of finite-dimensional integrable $U'_{q}(\mathfrak{g})$-modules and proved that the Grothendieck ring of ${\mathscr {C}}_{Q}$ is isomorphic to $\mathbb{C}[N]$, the coordinate ring of the unipotent group $N$ associated with ${\mathfrak {g}}_{0}$. We apply the generalized quantum affine Schur–Weyl duality to construct an exact functor $\mathcal {F}$ from the category of finite-dimensional graded $R$-modules to the category ${\mathscr {C}}_{Q}$, where $R$ denotes the symmetric quiver Hecke algebra associated to ${\mathfrak {g}}_{0}$. We prove that the homomorphism induced by the functor $\mathcal {F}$ coincides with the homomorphism of Hernandez and Leclerc and show that the functor $\mathcal {F}$ sends the simple modules to the simple modules.

#### Article information

Source
Duke Math. J., Volume 164, Number 8 (2015), 1549-1602.

Dates
Revised: 11 July 2014
First available in Project Euclid: 28 May 2015

https://projecteuclid.org/euclid.dmj/1432817758

Digital Object Identifier
doi:10.1215/00127094-3119632

Mathematical Reviews number (MathSciNet)
MR3352041

Zentralblatt MATH identifier
1323.81046

#### Citation

Kang, Seok-Jin; Kashiwara, Masaki; Kim, Myungho. Symmetric quiver Hecke algebras and $R$ -matrices of quantum affine algebras, II. Duke Math. J. 164 (2015), no. 8, 1549--1602. doi:10.1215/00127094-3119632. https://projecteuclid.org/euclid.dmj/1432817758

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