PBW Theoretic Approach to the Module Category of Quantum Affine Algebras

M. J.A. Masaki Kashiwara; Myungho Kim; Se Jin Oh; Euiyong Park

doi:10.3792/PJAA.97.007

PBW Theoretic Approach to the Module Category of Quantum Affine Algebras

M. J.A. Masaki Kashiwara, Myungho Kim, Se Jin Oh, Euiyong Park

Department of Mathematics

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

1 Scopus citations

Abstract

Let (Formula presented) be a quantum affine algebra of untwisted affine ADE type and let (Formula presented) be Hernandez-Leclerc’s category. For a duality datum D in (Formula presented), we denote by (Formula presented) the quantum affine Weyl-Schur duality functor. We give a sufficient condition for a duality datum D to provide the functor (Formula presented) sending simple modules to simple modules. Moreover, under the same condition, the functor (Formula presented) has compatibility with the new invariants introduced by the authors. Then we introduce the notion of cuspidal modules in (Formula presented), and show that all simple modules in (Formula presented) can be constructed as the heads of ordered tensor products of cuspidal modules. We next state that the ordered tensor products of cuspidal modules have the unitriangularity property.

Original language	English
Pages (from-to)	33-37
Number of pages	5
Journal	Proceedings of the Japan Academy Series A: Mathematical Sciences
Volume	97
Issue number	6
DOIs	https://doi.org/10.3792/PJAA.97.007
State	Published - 2021

Bibliographical note

Funding Information:
Acknowledgements. The research of Ka- shiwara was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science, the research of Kim was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2017R1C1B2007824) and the research of Oh was supported by the Ministry of Education of the Republic of Korea and the NRF of Korea (NRF-2019R1A2C4069647).

Publisher Copyright:
© The Japan Academy

Keywords

Cuspidal modules
Hernandez-Leclerc category
quantum affine Weyl-Schur duality
quantum affine algebra
quiver Hecke algebra

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10.3792/PJAA.97.007

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title = "PBW Theoretic Approach to the Module Category of Quantum Affine Algebras",

abstract = "Let (Formula presented) be a quantum affine algebra of untwisted affine ADE type and let (Formula presented) be Hernandez-Leclerc{\textquoteright}s category. For a duality datum D in (Formula presented), we denote by (Formula presented) the quantum affine Weyl-Schur duality functor. We give a sufficient condition for a duality datum D to provide the functor (Formula presented) sending simple modules to simple modules. Moreover, under the same condition, the functor (Formula presented) has compatibility with the new invariants introduced by the authors. Then we introduce the notion of cuspidal modules in (Formula presented), and show that all simple modules in (Formula presented) can be constructed as the heads of ordered tensor products of cuspidal modules. We next state that the ordered tensor products of cuspidal modules have the unitriangularity property.",

keywords = "Cuspidal modules, Hernandez-Leclerc category, quantum affine Weyl-Schur duality, quantum affine algebra, quiver Hecke algebra",

author = "{Masaki Kashiwara}, {M. J.A.} and Myungho Kim and Oh, {Se Jin} and Euiyong Park",

note = "Funding Information: Acknowledgements. The research of Ka- shiwara was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science, the research of Kim was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2017R1C1B2007824) and the research of Oh was supported by the Ministry of Education of the Republic of Korea and the NRF of Korea (NRF-2019R1A2C4069647). Publisher Copyright: {\textcopyright} The Japan Academy",

year = "2021",

doi = "10.3792/PJAA.97.007",

language = "English",

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pages = "33--37",

journal = "Proceedings of the Japan Academy Series A: Mathematical Sciences",

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AU - Masaki Kashiwara, M. J.A.

AU - Kim, Myungho

AU - Oh, Se Jin

AU - Park, Euiyong

N1 - Funding Information: Acknowledgements. The research of Ka- shiwara was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science, the research of Kim was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2017R1C1B2007824) and the research of Oh was supported by the Ministry of Education of the Republic of Korea and the NRF of Korea (NRF-2019R1A2C4069647). Publisher Copyright: © The Japan Academy

PY - 2021

Y1 - 2021

N2 - Let (Formula presented) be a quantum affine algebra of untwisted affine ADE type and let (Formula presented) be Hernandez-Leclerc’s category. For a duality datum D in (Formula presented), we denote by (Formula presented) the quantum affine Weyl-Schur duality functor. We give a sufficient condition for a duality datum D to provide the functor (Formula presented) sending simple modules to simple modules. Moreover, under the same condition, the functor (Formula presented) has compatibility with the new invariants introduced by the authors. Then we introduce the notion of cuspidal modules in (Formula presented), and show that all simple modules in (Formula presented) can be constructed as the heads of ordered tensor products of cuspidal modules. We next state that the ordered tensor products of cuspidal modules have the unitriangularity property.

AB - Let (Formula presented) be a quantum affine algebra of untwisted affine ADE type and let (Formula presented) be Hernandez-Leclerc’s category. For a duality datum D in (Formula presented), we denote by (Formula presented) the quantum affine Weyl-Schur duality functor. We give a sufficient condition for a duality datum D to provide the functor (Formula presented) sending simple modules to simple modules. Moreover, under the same condition, the functor (Formula presented) has compatibility with the new invariants introduced by the authors. Then we introduce the notion of cuspidal modules in (Formula presented), and show that all simple modules in (Formula presented) can be constructed as the heads of ordered tensor products of cuspidal modules. We next state that the ordered tensor products of cuspidal modules have the unitriangularity property.

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KW - Hernandez-Leclerc category

KW - quantum affine Weyl-Schur duality

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KW - quiver Hecke algebra

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JO - Proceedings of the Japan Academy Series A: Mathematical Sciences

JF - Proceedings of the Japan Academy Series A: Mathematical Sciences

IS - 6

ER -

PBW Theoretic Approach to the Module Category of Quantum Affine Algebras

Abstract

Bibliographical note

Keywords

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