Duke Mathematical Journal

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

Seok-Jin Kang, Masaki Kashiwara, and Myungho Kim

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Let g be an untwisted affine Kac–Moody algebra of type A n ( 1 ) ( n 1 ) or D n ( 1 ) ( n 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 introduced a tensor subcategory C Q of the category of finite-dimensional integrable U ' q ( g ) -modules and proved that the Grothendieck ring of C 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 to construct an exact functor F from the category of finite-dimensional graded R -modules to the category C 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.

Article information

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

Received: 2 August 2013
Revised: 11 July 2014
First available in Project Euclid: 28 May 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37]
Secondary: 16G 16T25: Yang-Baxter equations 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]

quantum affine algebra quiver Hecke algebra quantum group


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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