Braid Group Action On The Module Category Of Quantum Affine Algebras

Masaki Kashiwara; Myungho Kim; Se Jin Oh; Euiyong Park

doi:10.3792/pjaa.97.003

Braid Group Action On The Module Category Of Quantum Affine Algebras

Masaki Kashiwara, Myungho Kim, Se Jin Oh, Euiyong Park

Department of Mathematics

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

2 Scopus citations

Abstract

Let g₀ be a simple Lie algebra of type ADE and let U^'_q(g) be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group B(g₀) on the quantum Grothendieck ring K_t(g) of Hernandez-Leclerc’s category (Formula Presented). Focused on the case of type A_N-1, we construct a family of monoidal autofunctors (Formula Presented) on a localization T_N of the category of finite-dimensional graded modules over the quiver Hecke algebra of type A_∞. Under an isomorphism between the Grothendieck ring K(T_N) of T_N and the quantum Grothendieck ring (Formula Presented), the functors (Formula Presented) recover the action of the braid group B(A_N-1). We investigate further properties of these functors.

Original language	English
Pages (from-to)	13-18
Number of pages	6
Journal	Proceedings of the Japan Academy Series A: Mathematical Sciences
Volume	97
Issue number	3
DOIs	https://doi.org/10.3792/pjaa.97.003
State	Published - 2021

Bibliographical note

Funding Information:
(c) Let Q be a Q-data, and L := FQ(L(i)). Then the automorphism of Kt(g) induced by SL coincides with σi, i.e., the following diagram commutes: Acknowledgement. The research of Kashiwara 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:
© 2021 The Japan Academy

Keywords

Quantum affine algebra
R-matrix.
braid group action
quantum Grothendieck ring
quiver Hecke algebra

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10.3792/pjaa.97.003

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title = "Braid Group Action On The Module Category Of Quantum Affine Algebras",

abstract = "Let g0 be a simple Lie algebra of type ADE and let U'q(g) be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group B(g0) on the quantum Grothendieck ring Kt(g) of Hernandez-Leclerc{\textquoteright}s category (Formula Presented). Focused on the case of type AN-1, we construct a family of monoidal autofunctors (Formula Presented) on a localization TN of the category of finite-dimensional graded modules over the quiver Hecke algebra of type A∞. Under an isomorphism between the Grothendieck ring K(TN) of TN and the quantum Grothendieck ring (Formula Presented), the functors (Formula Presented) recover the action of the braid group B(AN-1). We investigate further properties of these functors.",

keywords = "Quantum affine algebra, R-matrix., braid group action, quantum Grothendieck ring, quiver Hecke algebra",

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

note = "Funding Information: (c) Let Q be a Q-data, and L := FQ(L(i)). Then the automorphism of Kt(g) induced by SL coincides with σi, i.e., the following diagram commutes: Acknowledgement. The research of Kashiwara 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} 2021 The Japan Academy",

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AU - Kashiwara, Masaki

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AU - Park, Euiyong

N1 - Funding Information: (c) Let Q be a Q-data, and L := FQ(L(i)). Then the automorphism of Kt(g) induced by SL coincides with σi, i.e., the following diagram commutes: Acknowledgement. The research of Kashiwara 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: © 2021 The Japan Academy

PY - 2021

Y1 - 2021

N2 - Let g0 be a simple Lie algebra of type ADE and let U'q(g) be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group B(g0) on the quantum Grothendieck ring Kt(g) of Hernandez-Leclerc’s category (Formula Presented). Focused on the case of type AN-1, we construct a family of monoidal autofunctors (Formula Presented) on a localization TN of the category of finite-dimensional graded modules over the quiver Hecke algebra of type A∞. Under an isomorphism between the Grothendieck ring K(TN) of TN and the quantum Grothendieck ring (Formula Presented), the functors (Formula Presented) recover the action of the braid group B(AN-1). We investigate further properties of these functors.

AB - Let g0 be a simple Lie algebra of type ADE and let U'q(g) be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group B(g0) on the quantum Grothendieck ring Kt(g) of Hernandez-Leclerc’s category (Formula Presented). Focused on the case of type AN-1, we construct a family of monoidal autofunctors (Formula Presented) on a localization TN of the category of finite-dimensional graded modules over the quiver Hecke algebra of type A∞. Under an isomorphism between the Grothendieck ring K(TN) of TN and the quantum Grothendieck ring (Formula Presented), the functors (Formula Presented) recover the action of the braid group B(AN-1). We investigate further properties of these functors.

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KW - R-matrix.

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KW - quiver Hecke algebra

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Braid Group Action On The Module Category Of Quantum Affine Algebras

Abstract

Bibliographical note

Keywords

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