Monoidal categories of modules over quantum affine algebras of type A and B

Masaki Kashiwara; Myungho Kim; Se jin Oh

doi:10.1112/plms.12160

Monoidal categories of modules over quantum affine algebras of type A and B

Masaki Kashiwara, Myungho Kim, Se jin Oh

Department of Mathematics

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

9 Scopus citations

Abstract

We construct an exact tensor functor from the category A of finite-dimensional graded modules over the quiver Hecke algebra of type A∞ to the category C_Bn(1) of finite-dimensional integrable modules over the quantum affine algebra of type B_n ¹. It factors through the category T_2n, which is a localization of A. As a result, this functor induces a ring isomorphism from the Grothendieck ring of T_2n (ignoring the gradings) to the Grothendieck ring of a subcategory C⁰ _Bn(1) of C_Bn(1). Moreover, it induces a bijection between the classes of simple objects. Because the category T_2n is related to categories (Formula presented.) (t = 1, 2) of the quantum affine algebras of type A⁽¹⁾ _2n-1, we obtain an interesting connection between those categories of modules over quantum affine algebras of type A and type B. Namely, for each t = 1, 2, there exists an isomorphism between the Grothendieck ring of C⁰ _Bn(1) and the Grothendieck ring of (Formula presented.), which induces a bijection between the classes of simple modules.

Original language	English
Pages (from-to)	43-77
Number of pages	35
Journal	Proceedings of the London Mathematical Society
Volume	118
Issue number	1
DOIs	https://doi.org/10.1112/plms.12160
State	Published - Jan 2019

Bibliographical note

Funding Information:
Received 21 October 2017; revised 3 April 2018; published online 19 June 2018. 2010 Mathematics Subject Classification 81R50, 16G99, 16T25,17B37 (primary). 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). The research of Oh was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korea government(MSIP) (NRF-2016R1C1B2013135).

Publisher Copyright:
© 2018 London Mathematical Society

Keywords

16G99
16T25,17B37 (primary)
81R50

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10.1112/plms.12160

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title = "Monoidal categories of modules over quantum affine algebras of type A and B",

abstract = "We construct an exact tensor functor from the category A of finite-dimensional graded modules over the quiver Hecke algebra of type A∞ to the category CBn(1) of finite-dimensional integrable modules over the quantum affine algebra of type Bn 1. It factors through the category T2n, which is a localization of A. As a result, this functor induces a ring isomorphism from the Grothendieck ring of T2n (ignoring the gradings) to the Grothendieck ring of a subcategory C0 Bn(1) of CBn(1). Moreover, it induces a bijection between the classes of simple objects. Because the category T2n is related to categories (Formula presented.) (t = 1, 2) of the quantum affine algebras of type A(1) 2n-1, we obtain an interesting connection between those categories of modules over quantum affine algebras of type A and type B. Namely, for each t = 1, 2, there exists an isomorphism between the Grothendieck ring of C0 Bn(1) and the Grothendieck ring of (Formula presented.), which induces a bijection between the classes of simple modules.",

keywords = "16G99, 16T25,17B37 (primary), 81R50",

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

note = "Funding Information: Received 21 October 2017; revised 3 April 2018; published online 19 June 2018. 2010 Mathematics Subject Classification 81R50, 16G99, 16T25,17B37 (primary). 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). The research of Oh was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korea government(MSIP) (NRF-2016R1C1B2013135). Publisher Copyright: {\textcopyright} 2018 London Mathematical Society",

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N2 - We construct an exact tensor functor from the category A of finite-dimensional graded modules over the quiver Hecke algebra of type A∞ to the category CBn(1) of finite-dimensional integrable modules over the quantum affine algebra of type Bn 1. It factors through the category T2n, which is a localization of A. As a result, this functor induces a ring isomorphism from the Grothendieck ring of T2n (ignoring the gradings) to the Grothendieck ring of a subcategory C0 Bn(1) of CBn(1). Moreover, it induces a bijection between the classes of simple objects. Because the category T2n is related to categories (Formula presented.) (t = 1, 2) of the quantum affine algebras of type A(1) 2n-1, we obtain an interesting connection between those categories of modules over quantum affine algebras of type A and type B. Namely, for each t = 1, 2, there exists an isomorphism between the Grothendieck ring of C0 Bn(1) and the Grothendieck ring of (Formula presented.), which induces a bijection between the classes of simple modules.

AB - We construct an exact tensor functor from the category A of finite-dimensional graded modules over the quiver Hecke algebra of type A∞ to the category CBn(1) of finite-dimensional integrable modules over the quantum affine algebra of type Bn 1. It factors through the category T2n, which is a localization of A. As a result, this functor induces a ring isomorphism from the Grothendieck ring of T2n (ignoring the gradings) to the Grothendieck ring of a subcategory C0 Bn(1) of CBn(1). Moreover, it induces a bijection between the classes of simple objects. Because the category T2n is related to categories (Formula presented.) (t = 1, 2) of the quantum affine algebras of type A(1) 2n-1, we obtain an interesting connection between those categories of modules over quantum affine algebras of type A and type B. Namely, for each t = 1, 2, there exists an isomorphism between the Grothendieck ring of C0 Bn(1) and the Grothendieck ring of (Formula presented.), which induces a bijection between the classes of simple modules.

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Monoidal categories of modules over quantum affine algebras of type A and B

Abstract

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