Categorical relations between Langlands dual quantum affine algebras: Exceptional cases

Abstract

We first compute the denominator formulas for quantum affine algebras of all exceptional types. Then we prove the isomorphisms among Grothendieck rings of categories $C_Q^{(t)}$ $(t=1,2,3)$, ${\mathcal{C}}_{\mathcal{Q}}^{(1)}$ and ${\mathcal{C}}_{\mathfrak{Q}}^{(1)}$. These results give Dorey's rule for all exceptional affine types, prove the conjectures of Kashiwara-Kang-Kim and Kashiwara-Oh, and provides the partial answers of Frenkel-Hernandez on Langlands duality for finite dimensional representations of quantum affine algebras of exceptional types.

arXiv comment67 pages, 1 figure; v2 incorporated changes from referee report; v3 incorporated an addendum that removes ambiguities

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Cited by in corpus (12)

• Monoidal categorification and quantum affine algebras
• Q-data and representation theory of untwisted quantum affine algebras
• 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
• 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
• Poles of finite-dimensional representations of Yangians
• The $(q,t)$-Cartan matrix specialized at $q=1$
• Deformed Cartan matrices and generalized preprojective algebras I: Finite type
• Equivalence between module categories over quiver Hecke algebras and Hernandez-Leclerc's categories in general types
• $t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras