Researcher Database

NAOI Katsuyuki

FacultyInstitute of Engineering Division of Mathematical Sciences
PositionAssociate Professor
Last Updated :2023/08/18

Activity information

Name and contact details

  • Name

    ナオイ カツユキ, 直井 克之, NAOI Katsuyuki

Affiliation / Position

  • Institute of Engineering Division of Mathematical Sciences, Associate Professor

Other affiliation

  • Faculty of Engineering Department of Mechanical Systems Engineering
  • Graduate School of Engineering Department of Mechanical Systems Engineering

Research History

  •  
    Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo
    From 01 Apr. 2012, To 31 Jan. 2013
  •  
    東京大学大学院数理科学研究科Global COE研究員
    From 01 Apr. 2011, To 31 Mar. 2012

Education

  • The University of Tokyo
    Faculty of Science
    mathematics
    To Mar. 2006, graduated
  • The University of Tokyo
    Graduate School, Division of Mathematical Sciences
    数理科学専攻
    To Mar. 2008, completed, master course
  • The University of Tokyo
    Graduate School, Division of Mathematical Sciences
    数理科学専攻
    To Mar. 2011, doctor course
  • 愛知県立一宮高校
    To Mar. 2002, graduated

Degree

  • 博士(数理科学)
    東京大学

Research Areas

  • Natural Science, Algebra

Research Interests

  • Representation theory of infinite-dimensional Lie algebras and their quantum analog

Teaching Experience

  • 機械数学特論
    2021, 専門科目等
  • 【1063301016】 機械数学特論
    2021, 専門科目等
  • 【3026901001】 線形代数学Ⅰ
    2021, 専門科目等
  • 【3026901004】 微分積分学Ⅱおよび演習
    2021, 専門科目等

Grants-in-Aid for Scientific Research

  • 基盤研究(C)
    Kirillov-Reshetikhin加群の結晶基底
    From 2020, To 2023
  • 若手研究(B)
    量子アフィン代数の有限次元表現の古典極限に関する研究
    From 2016, To 2019

Papers

  • Equivalence between module categories over quiver Hecke algebras and Hernandez-Leclerc's categories in general types
    Naoi, Katsuyuki
    ADVANCES IN MATHEMATICS
    ACADEMIC PRESS INC ELSEVIER SCIENCE
    We prove in full generality that the generalized quantum affine Schur-Weyl duality functor, introduced by Kang- Kashiwara-Kim, gives an equivalence between the category of finite-dimensional modules over a quiver Hecke algebra and a certain full subcategory of finite-dimensional modules over a quantum affine algebra which is a generalization of the Hernandez-Leclerc's category C-Q. This was previously proved in untwisted ADE types by Fujita using the geometry of quiver varieties, which is not applicable in general. Our proof is purely algebraic, and so can be extended uniformly to general types. (C) 2021 Elsevier Inc. All rights reserved.
    08 Oct. 2021, Research paper (scientific journal), only, 389, 0001-8708, DOI(公開)(r-map)
  • Existence of Kirillov-Reshetikhin crystals for near adjoint nodes in exceptional types
    Naoi, Katsuyuki; Scrimshaw, Travis
    JOURNAL OF PURE AND APPLIED ALGEBRA
    ELSEVIER
    We prove that, in types E-6,7,8,((1)) F-4((1)) and E-6((1)) every Kirillov-Reshetikhin module associated with the node adjacent to the adjoint one (near adjoint node) has a crystal pseudobase, by applying the criterion introduced by Kang et al. In order to apply the criterion, we need to prove some statements concerning values of a bilinear form. We achieve this by using the global bases of extremal weight modules. (C) 2020 Elsevier B.V. All rights reserved.
    May 2021, Research paper (scientific journal), joint, 225, 5, 0022-4049, DOI(公開)(r-map)
  • Kirillov-Reshetikhin加群の結晶擬基底の存在について
    直井克之
    第4回Algebraic Lie Theory and Representation Theory報告集
    Feb. 2019, Research paper (research society, symposium materials, etc.), only, 4, 171, 178
  • Existence of Kirillov-Reshetikhin crystals of type G(2)((1)) and D-4((3))
    Naoi, Katsuyuki
    JOURNAL OF ALGEBRA
    ACADEMIC PRESS INC ELSEVIER SCIENCE
    In this paper we prove that every Kirillov-Reshetikhin module of type G(2)((1)) and D-4((3)) has a crystal pseudobase (crystal base modulo signs), by applying the criterion for the existence of a crystal pseudobase due to Kang et al. (C) 2018 Elsevier Inc. All rights reserved.
    15 Oct. 2018, Research paper (scientific journal), only, 512, 0021-8693, DOI(公開)(r-map), 47, 65
  • Tensor Products of Kirillov-Reshetikhin Modules and Fusion Products
    Naoi, Katsuyuki
    INTERNATIONAL MATHEMATICS RESEARCH NOTICES
    OXFORD UNIV PRESS
    We study the classical limit of a tensor product of Kirillov-Reshetikhin modules over a quantum loop algebra, and show that it is realized from the classical limits of the tensor factors using the notion of fusion products. In the process of the proof, we also give defining relations of the fusion product of the (graded) classical limits of Kirillov-Reshetikhin modules.
    Sep. 2017, Research paper (scientific journal), only, 18, 1073-7928, DOI(公開)(r-map), 5667, 5709
  • 量子ループ代数の加群におけるテンソル積と古典極限を取る操作の非可換性について
    直井克之
    第2回Algebraic Lie Theory and Representation Theory 報告集
    Jan. 2017, Research paper (research society, symposium materials, etc.), only, 2, 218, 227
  • Graded limits of minimal affinizations over the quantum loop algebra of type G_2
    Jian-Rong Li and Katsuyuki Naoi
    Algebras and Representation Theory
    Mar. 2016, Research paper (scientific journal), joint, 19, 4, 957, 973
  • Defining relations of fusion products and Schur positivity
    Katsuyuki Naoi
    Toyama Mathmatical Journal
    Jan. 2016, Research paper (scientific journal), only, 37, 87, 106
  • カレント代数のフュージョン積とSchur正値性
    直井克之
    RIMS講究録1977 「表現論および関連する調和解析と微分方程式」
    Dec. 2015, Research paper (scientific journal), only, 1977, 1, 8
  • minimal affinizationのJacobi-Trudi型指標公式について
    直井克之
    第59回代数学シンポジウム報告集
    Sep. 2014, only, 59, 176, 187
  • Graded limits of minimal affinizations in type D
    Katsuyuki Naoi
    Symmetry, Integrability and Geometry: Methods and Applications
    Apr. 2014, Research paper (scientific journal), only, 10, 1, 20
  • Demazure modules and graded limits of minimal affinizations
    Katsuyuki Naoi
    Representation Theory
    Oct. 2013, Research paper (scientific journal), only, 17, 524, 556
  • Demazure crystals and tensor products of perfect Kirillov-Reshetikhin crystals with various levels
    Katsuyuki Naoi
    Journal of Algebra
    Jan. 2013, Research paper (scientific journal), only, 374, 1, 26
  • 量子アファイン代数の極小アファイン化について
    直井克之
    第15回代数群と量子群の表現論研究集会報告集
    Dec. 2012, only, 15, 101, 108
  • KR クリスタルのテンソル積と Demazure クリスタル
    直井克之
    RIMS講究録1795「組合せ論的表現論の拡がり」
    Oct. 2012, only, 1795, 59, 69
  • Fusion products of Kirillov-Reshetikhin modules and the X=M conjecture
    Katsuyuki Naoi
    Advances in Mathematics
    Sep. 2012, Research paper (scientific journal), only, 231, 3-4, 1546, 1571
  • Loewy series of Weyl modules and the Poincaré polynomials of quiver varieties
    Ryosuke Kodera, Katsuyuki Naoi
    Publications of RIMS
    May 2012, Research paper (scientific journal), only, 48, 3, 477, 500
  • Weyl modules, Demazure modules and finite crystals for non-simply laced type
    Katsuyuki Naoi
    Advances in Mathematics
    Jan. 2012, Research paper (scientific journal), only, 229, 2, 875, 934
  • Demazure加群、DemazureクリスタルとX=M予想
    直井克之
    第14回代数群と量子群の表現論研究集会報告集
    Jun. 2011, only, 14, 132, 141
  • Weyl moduleおよびDemazure moduleと有限crystalとの関係について
    直井克之
    2010年度表現論シンポジウム講演集
    Nov. 2010, only, 19, 12
  • 有限位数自己同型を用いたextended affine Lie algebra の構成について
    直井克之
    RIMS講究録別冊
    Aug. 2010, Research paper (bulletin of university, research institution), only, B20, 21, 30
  • Multiloop Lie algebras and the construction of extended affine Lie algebras
    Katsuyuki Naoi
    Journal of Algebra
    Apr. 2010, Research paper (scientific journal), only, 323, 8, 2103, 2129
  • Construction of extended affine Lie algebras from multiloop Lie algebras
    直井克之
    第11回代数群と量子群の表現論研究集会報告集
    Jun. 2008, only, 11, 85, 93

Presentations

  • Strong duality data of type A and extended T-systems
    東工大表現論セミナー
    30 Jun. 2023, Public discourse, seminar, tutorial, course, lecture and others
  • 一般化量子アフィンSchur-Weyl双対性と圏同値
    第67回代数学シンポジウム
    31 Aug. 2022, Oral presentation(invited, special)
  • Equivalence between module categories over quiver Hecke algebras and Hernandez-Leclerc's subcategories
    Quantum Groups and Cluster Algebras
    15 Feb. 2022, Oral presentation(invited, special)
  • Equivalence between module categories over quiver Hecke algebras and Hernandez-Leclerc's categories
    Conference on Algebraic Representation Theory 2021
    29 Nov. 2021, Oral presentation(invited, special)
  • Equivalence via generalized quantum affine Schur-Weyl duality
    代数的リー理論と表現論2021
    28 Jun. 2021, Oral presentation(general)
  • Equivalence via generalized quantum affine Schur-Weyl duality
    南大阪代数セミナー
    19 Mar. 2021, Public discourse, seminar, tutorial, course, lecture and others
  • Existence of Kirillov-Reshetikhin crystals for the near adjoint nodes in exceptional types
    Representation Theory of Algebraic Groups and Quantum Groups -in honor of Professor Ariki's 60th birthday-
    21 Oct. 2019, Oral presentation(invited, special)
  • Existence of KR crystals in types G_2^{(1)}, D_4^{(3)} and E_6^{(1)}
    Algebraic Representation Theory and Representation Theory 2018
    May 2018, Oral presentation(general)
  • Existence of Kirillov-Reshetikhin crystals of type G_2^{(1)} and D_4^{(3)}
    Infinite Analysis 17 -Algebraic and Combinatorial Aspects in Integrable Systems-
    04 Dec. 2017, Oral presentation(invited, special)
  • U_q(Lg)加群に対するテンソル積と古典極限を取る操作の非可換性について
    日本数学会2017年度年会無限可積分系セッション
    Mar. 2017, Oral presentation(general)
  • 量子ループ代数の有限次元表現と古典極限
    農工大数学セミナー2017
    Mar. 2017, Oral presentation(invited, special)
  • Noncommutativity between the operations of tensor product and taking a classical limit
    Combinatorics of Lie Type
    Oct. 2016, Oral presentation(invited, special)
  • Noncommutativity between operations of tensor product and classical limit on modules over a quantum loop algebra
    Algebraic Lie Theory and Representation Theory
    Jun. 2016, Oral presentation(invited, special)
  • Fusion products of current algebra and Schur positivity
    Representation theory, harmonic analysis and differential equation
    Jun. 2015, Oral presentation(invited, special)
  • Minimal affinizations and their graded limits
    Repsentation theory and Related Topics
    Feb. 2015, Oral presentation(invited, special)
  • Jacobi-Trudi formula and quantum loop algebras
    組み合せ論セミナー
    Dec. 2014, Other
  • Minimal affi nizations and their graded limits
    Shanghai Workshop on Representation Theory
    Dec. 2014, Oral presentation(invited, special)
  • Minimal affinizations and their graded limits
    Conference on Cluster Algebras and Representation Theory
    Nov. 2014, Oral presentation(invited, special)
  • Graded limits of finite-dimensional modules over quantum loop algebras
    Tsukuba Workshop on Infinite Dimensional Lie Theory and Related Topics -- History and Development --
    Oct. 2014, Oral presentation(invited, special)
  • 量子アフィン代数の有限次元既約表現とその次数付き極限
    第59回代数学シンポジウム
    Sep. 2014, Oral presentation(invited, special)
  • Graded limits of minimal affinizations
    Combinatorial representation theory
    Apr. 2014, Oral presentation(invited, special)
  • An approach to the X=M conjecture using modules over a current algebra
    日本数学会秋季総合分科会無限可積分系セッション特別講演
    Sep. 2013, Oral presentation(invited, special)
  • Finite-dimensional representations over a quantum loop algebra and their classical limits
    GT Seminar
    May 2013, Other
  • An approach to the X=M conjecture using current algebras
    Infinite Analysis 13
    Mar. 2013, Oral presentation(invited, special)
  • Graded limits of minimal affinizations over a quantum loop algebra
    日本数学会2013年度年会
    Mar. 2013, Oral presentation(general)
  • Graded limits of minimal affinizations
    Shanghai workshop on Representation Theory
    Dec. 2012, Oral presentation(general)
  • 量子ループ代数のminimal affinizationについて
    信州代数セミナー
    Nov. 2012, Other
  • BC型量子ループ代数のminimal affinizationについて
    RIMS 表現論セミナー
    Nov. 2012, Other
  • The X=M conjecture for a quantum affine algebra
    MS Seminar
    May 2012, Other
  • Classical limits of minimal affinizations and generalized Demazure modules
    第15回代数群と量子群の表現論研究集会
    May 2012, Oral presentation(general)
  • Loewy series of Weyl modules and the Poincare polynomials of quiver varieties
    Lie Theory Seminar
    Feb. 2012, Other
  • Generalized Demazure module and the restricted classical limit of a tensor product of KR modules
    Lie Theory Seminar
    Feb. 2012, Other
  • Demazure加群とX=M予想
    信州代数セミナー
    Oct. 2011, Other
  • Demazure modules, Demazure crystals and the X=M conjecture
    組合せ論的表現論の拡がり
    Oct. 2011, Oral presentation(general)
  • Relations among Weyl modules, Demazure modules and finite crystals
    Infinite Analysis 11
    Jul. 2011, Poster presentation
  • Demazure modules, Demazure crystalsとX=M予想
    第14回代数群と量子群の表現論研究集会
    Jun. 2011, Oral presentation(general)
  • Weyl加群、Demazure加群とfundamental representationのテンソル積のcrystal basisとの関係について
    広域科学専攻相関基礎科学系セミナー
    Apr. 2011, Other
  • Weyl加群、Demazure加群とfundamental representationのテンソル積のcrystal basisとの関係について
    RIMS 表現論セミナー
    Apr. 2011, Other
  • Weyl加群、Demazure加群とfundamental representationのcrystal basisの関係について
    大阪表現論セミナー
    Feb. 2011, Other
  • Weyl加群、Demazure加群とfundamental representationのcrystal basisの関係について
    RAQセミナー
    Jan. 2011, Other
  • Some relations between the Weyl module and the crystal basis of the tensor product of fundamental representations
    Lie Groups and Representation Theory Seminar
    Dec. 2010, Other
  • Weyl moduleおよびDemazure moduleと有限crystalとの関係について
    表現論シンポジウム 2010
    Oct. 2010, Oral presentation(general)
  • Categorical approach for Weyl modules, after Chari-Fourier-Khandai
    MS Seminar
    Aug. 2009, Other
  • loop代数とmultiloop代数の表現論について
    Algebras, Groups, and Geometry 2009 in Tambara
    Aug. 2009, Oral presentation(invited, special)
  • Construction of extended affine Lie algebras from multiloop Lie algebras
    表現論と非可換調和解析における新しい視点
    Sep. 2008, Oral presentation(general)
  • Construction of extended affine Lie algebras from multiloop Lie algebras
    Lie Groups and Representation Theory Seminar
    Jul. 2008, Other
  • Construction of extended affine Lie algebras from multiloop Lie algebras
    第11回代数群と量子群の表現論研究集会
    May 2008, Oral presentation(general)

Professional Memberships

  • Mathematical Society of Japan


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