Categorical relations between Langlands dual quantum affine algebras: doubly laced types

Masaki Kashiwara, Se jin Oh

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We prove that the Grothendieck rings of category CQ(t) over quantum affine algebras Uq′(g(t))(t= 1 , 2) associated with each Dynkin quiver Q of finite type A2n-1 (resp. Dn+1) are isomorphic to one of the categories CQ over the Langlands dual Uq′(Lg(2)) of Uq′(g(2)) associated with any twisted adapted class [Q] of A2n-1 (resp. Dn+1). This results provide simplicity-preserving correspondences on Langlands duality for finite-dimensional representation of quantum affine algebras, suggested by Frenkel–Hernandez.

Original languageEnglish
Pages (from-to)401-435
Number of pages35
JournalJournal of Algebraic Combinatorics
Issue number4
StatePublished - 15 Jun 2019

Bibliographical note

Funding Information:
M. Kashiwara: The research was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science. S. Oh: This work was supported by NRF Grant # 2016R1C1B2013135.

Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.


  • Combinatorial Auslander–Reiten quivers
  • Langlands duality
  • Longest element
  • Schur–Weyl diagram
  • r-Cluster point


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