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Monoidal categorification and quantum affine algebras

Part of: Lie algebras and Lie superalgebras Arithmetic rings and other special rings Categories with structure

Published online by Cambridge University Press:  27 April 2020

Masaki Kashiwara ,
Myungho Kim ,
Se-jin Oh  and
Euiyong Park
Masaki Kashiwara
Affiliation:
Kyoto University Institute for Advanced Study, Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan Korea Institute for Advanced Study, Seoul 02455, Korea email masaki@kurims.kyoto-u.ac.jp
Myungho Kim
Affiliation:
Department of Mathematics, Kyung Hee University, Seoul 02447, Korea email mkim@khu.ac.kr
Se-jin Oh
Affiliation:
Department of Mathematics, Ewha Womans University, Seoul 03760, Korea email sejin092@gmail.com
Euiyong Park
Affiliation:
Department of Mathematics, University of Seoul, Seoul 02504, Korea email epark@uos.ac.kr
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Abstract

We introduce and investigate new invariants of pairs of modules $M$ and $N$ over quantum affine algebras $U_{q}^{\prime }(\mathfrak{g})$ by analyzing their associated $R$-matrices. Using these new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable $U_{q}^{\prime }(\mathfrak{g})$-modules to become a monoidal categorification of a cluster algebra.

Keywords

quantum affine algebramonoidal categorificationR-matricescluster algebra

MSC classification

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations 13F60: Cluster algebras 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 5 , May 2020 , pp. 1039 - 1077
Copyright
© The Authors 2020

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Footnotes

The research of M. Kashiwara was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science. The research of M. Kim was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2017R1C1B2007824). The research of S.-j. Oh was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2019R1A2C4069647). The research of E. Park was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIP) (NRF-2017R1A1A1A05001058).

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