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

Masaki Kashiwara; Se jin Oh

doi:10.1007/s10801-018-0829-z

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

Masaki Kashiwara, Se jin Oh

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

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 A₂_n_-₁ (resp. D_n₊₁) are isomorphic to one of the categories C_Q over the Langlands dual Uq′(Lg(2)) of Uq′(g(2)) associated with any twisted adapted class [Q] of A₂_n_-₁ (resp. D_n₊₁). This results provide simplicity-preserving correspondences on Langlands duality for finite-dimensional representation of quantum affine algebras, suggested by Frenkel–Hernandez.

Original language	English
Pages (from-to)	401-435
Number of pages	35
Journal	Journal of Algebraic Combinatorics
Volume	49
Issue number	4
DOIs	https://doi.org/10.1007/s10801-018-0829-z
State	Published - 15 Jun 2019

Keywords

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

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@article{1109f06dffad44939388ef6dc0b4034b,

title = "Categorical relations between Langlands dual quantum affine algebras: doubly laced types",

abstract = "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.",

keywords = "Combinatorial Auslander–Reiten quivers, Langlands duality, Longest element, r-Cluster point, Schur–Weyl diagram",

author = "Masaki Kashiwara and Oh, {Se jin}",

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: {\textcopyright} 2018, Springer Science+Business Media, LLC, part of Springer Nature.",

year = "2019",

month = jun,

day = "15",

doi = "10.1007/s10801-018-0829-z",

language = "English",

volume = "49",

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journal = "Journal of Algebraic Combinatorics",

issn = "0925-9899",

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TY - JOUR

T1 - Categorical relations between Langlands dual quantum affine algebras

T2 - doubly laced types

AU - Kashiwara, Masaki

AU - Oh, Se jin

N1 - 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.

PY - 2019/6/15

Y1 - 2019/6/15

N2 - 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.

AB - 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.

KW - Combinatorial Auslander–Reiten quivers

KW - Langlands duality

KW - Longest element

KW - r-Cluster point

KW - Schur–Weyl diagram

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U2 - 10.1007/s10801-018-0829-z

DO - 10.1007/s10801-018-0829-z

M3 - Article

AN - SCOPUS:85049066498

SN - 0925-9899

VL - 49

SP - 401

EP - 435

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

IS - 4

ER -

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

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Keywords

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