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Simply laced root systems arising from quantum affine algebras

Part of: Modules, bimodules and ideals Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  08 February 2022

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

Let $U_q'({\mathfrak {g}})$ be a quantum affine algebra with an indeterminate $q$, and let $\mathscr {C}_{\mathfrak {g}}$ be the category of finite-dimensional integrable $U_q'({\mathfrak {g}})$-modules. We write $\mathscr {C}_{\mathfrak {g}}^0$ for the monoidal subcategory of $\mathscr {C}_{\mathfrak {g}}$ introduced by Hernandez and Leclerc. In this paper, we associate a simply laced finite-type root system to each quantum affine algebra $U_q'({\mathfrak {g}})$ in a natural way and show that the block decompositions of $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$ are parameterized by the lattices associated with the root system. We first define a certain abelian group $\mathcal {W}$ (respectively $\mathcal {W} _0$) arising from simple modules of $\mathscr {C}_{\mathfrak {g}}$ (respectively $\mathscr {C}_{\mathfrak {g}}^0$) by using the invariant $\Lambda ^\infty$ introduced in previous work by the authors. The groups $\mathcal {W}$ and $\mathcal {W} _0$ have subsets $\Delta$ and $\Delta _0$ determined by the fundamental representations in $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$, respectively. We prove that the pair $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ is an irreducible simply laced root system of finite type and that the pair $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} , \Delta )$ is isomorphic to the direct sum of infinite copies of $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ as a root system.

Keywords

block decompositionquantum affine algebrasR-matricesroot systems

MSC classification

Primary: 16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 1 , January 2022 , pp. 168 - 210
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

The research of M. Kashiwara was supported by Grant-in-Aid for Scientific Research (B) 20H01795 from the Japan Society for the Promotion of Science.

The research of M. Kim was supported by a National Research Foundation (NRF) grant funded by the government of Korea (MSIP) (NRF-2017R1C1B2007824 and NRF-2020R1A5A1016126).

The research of S.-J.O. 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.P. was supported by a National Research Foundation (NRF) grant funded by the government of Korea (MSIP)(NRF-2020R1F1A1A01065992 and NRF-2020R1A5A1016126).

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