On representation theory of generalized quantum groups and algebras related to them

• Project/Area Number: 25400040
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: University of Toyama
• Principal Investigator: Yamane Hiroyuki 富山大学, 大学院理工学研究部(理学), 教授 (10230517)
• Co-Investigator(Renkei-kenkyūsha): YOSHII Yoji 岩手大学, 教育学部, 教授 (90462126) ; ITO Ken 愛知工業大学, 工学部, 教授 (70410587) ; OSHIMA Kazuyuki 愛知工業大学, 工学部, 准教授 (30547980)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 量子群 / スーパーリー代数 / コクセター群 / ワイル亜群 / 一般化された量子群 / 一般化されたルート系 / ホップ代数 / ルート系 / コクセター亜群 / 国際情報交換 / リー代数

Outline of Final Research Achievements

The article giving the classification of the finite dimensional irreducible representaions of the generalized root system was published. New proof of the Kac's list of the classification the finite dimensional irreducible representaions of the A-G type simmple Lie superalgebras was given. It was proved that any finite dimensional irreducible representaion has a highest weight vector. New and easier definition of the generalizeed root systems was given. Then it became easily seen that the root system of any A-G type simmple Lie superalgebra is a generalized root system.

Research Products

• [Int'l Joint Research] IPM-Branch Isfahan, Irａｎ(Ｉｒａｎ)
• [Int'l Joint Research] Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ Ｒｅｓｅａｒｃｈ Ｉｎｓｔｉｔｕｔｅ(Ｉｎｄｉａ )
• [Journal Article] Generalized root systems and the affine Lie superalgebra ${\mathrm{G}}^{(1)}(3)$ (2016)
  • Author(s): Hiroyuki Yamane
  • Journal Title: São Paulo Journal of Mathematical Sciences Volume: 10 Issue: 1 Pages: 9-19
  • DOI: 10.1007/s40863-015-0021-5
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Skew centers of rank-one generalized quantum groups (2015)
  • Author(s): Punita Batra, Hiroyuki Yamane
  • Journal Title: Toyama Mathematical Journal Volume: 37 Pages: 189-202
  • NAID: 120005750760
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Classification of Finite-Dimensional Irreducible Representations of Generalized Quantum Groups via Weyl Groupoids (2015)
  • Author(s): Saeid Azam, Hiroyuki Yamane and Malihe Yousofzadeh
  • Journal Title: Publications of the Research Institute for Mathematical Sciences Volume: 51 Issue: 1 Pages: 59-130
  • DOI: 10.4171/prims/149
  • NAID: 110009911142
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] The R-matrix of quantum doubles of Nichols algebras of diagonal type (2015)
  • Author(s): Ivan Angiono and Hiroyuki Yamane
  • Journal Title: Journal of Mathematical Physics Volume: 56 Pages: 021702-021702
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Exposition on classification of finite dimensional irreducible representations of the Lie superalgebra B(m,n) (2014)
  • Author(s): Hiroyuki Yamane
  • Journal Title: Journal of Physics: Conference Series Volume: 563 Pages: 012035-012035
  • DOI: 10.1088/1742-6596/563/1/012035
  • Peer Reviewed / Open Access
• [Presentation] Weyl groupoids and representation theory of generalized quantum groups (2016)
  • Author(s): 山根宏之
  • Organizer: 日本数学会2016年度年会 特別講演 無限可積分系
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-19
  • Invited
• [Presentation] Coxeter groups and Weyl groupoids (2015)
  • Author(s): Hiroyuki Yamane
  • Organizer: IPM-Isfahan workshop on Coxeter groups and Weyl groupoids
  • Place of Presentation: IPM-Isfahan Branch, Isfahan, Iran
  • Year and Date: 2015-06-13
  • Invited
• [Presentation] Generalized quantum groups and their representation theory (2015)
  • Author(s): 山根宏之
  • Organizer: Algebraic Lie Theory and Representation Theory 2015
  • Place of Presentation: 岡山いこいの村
  • Year and Date: 2015-06-05
  • Invited
• [Presentation] Application of Weyl groupoids to Representation theory (2014)
  • Author(s): Hiroyuki Yamane
  • Organizer: Mini-workshop on Hopf algebras and representation theory
  • Place of Presentation: West Building 108, Department of Mathematics, Nanjing University, China
  • Year and Date: 2014-12-28
  • Invited
• [Presentation] Representation theory of generalized quantum groups (2014)
  • Author(s): Hiroyuki Yamane
  • Organizer: Conference on “Infinite Dimensional Lie Theory and its Applications” (In Honor of S. E. Rao's 60th Birthday)
  • Place of Presentation: Harish-Chandra Research Institute, Allahabad,India
  • Year and Date: 2014-12-17
  • Invited
• [Presentation] Representation theory of generalized quantum groups via Weyl groupoids (2014)
  • Author(s): Hiroyuki Yamane
  • Organizer: MINI WORKSHOP 2014 : Search for Classical Analysis and Quantum Integrable Systems
  • Place of Presentation: Graduate School of Informatics, Kyoto University
  • Year and Date: 2014-11-16
  • Invited
• [Presentation] Centers of generalized quantum groups (2014)
  • Author(s): Hiroyuki Yamane
  • Organizer: TSUKUBA WORKSHOP ON INFINITE-DIMENSIONAL LIE THEORY
  • Place of Presentation: 筑波大学数学教室
  • Year and Date: 2014-10-22
  • Invited
• [Presentation] Finite dimensional irreducible representations of generalized quantum groups and Weyl groupoids (2014)
  • Author(s): Hiroyuki Yamane, Saeid.Azam and Malihe Yousofzadeh)
  • Organizer: International Congress of Mathematicians, Short communications(Oral Presentation)
  • Place of Presentation: Coex , Seoul , Korea
  • Year and Date: 2014-08-16
  • Invited
• [Presentation] Generalized Quantum Groups and Weyl groupoids (2014)
  • Author(s): Hiroyuki Yamane
  • Organizer: International Conference on Integrable Systems and Quantum symmetries (ISQS)
  • Place of Presentation: Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University Prague Czech
  • Year and Date: 2014-06-25
  • Invited
• [Presentation] Harish-Chandra-type theorem of generalized quantum groups (2014)
  • Author(s): Hiroyuki Yamane
  • Organizer: 第10回駒場幾何学的表現論と量子可積分系のセミナー
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2014-05-10
  • Invited
• [Presentation] Universal R-matrix of a generalized quantum group
  • Author(s): 山根宏之
  • Organizer: 第16回 代数群と量子群の表現論（RAQ2013)
  • Place of Presentation: 強羅青雲荘
• [Presentation] コクセター亜群の松本の定理と一般化された量子群のシャポバロフ行列式
  • Author(s): 山根宏之
  • Organizer: 慶應大学・微分幾何・トポロジーセミナー
  • Place of Presentation: 慶應義塾大学理工学部 14棟631A
• [Presentation] Skew center of a generalized quantized algebra
  • Author(s): 山根宏之
  • Organizer: アルゼンチン・コルドバ大学リー理論セミナー
  • Place of Presentation: アルゼンチン・コルドバ大学数学教室
• [Presentation] 一般化された量子群の中心
  • Author(s): 山根宏之
  • Organizer: 大阪市立大学数学研究所代数セミナー
  • Place of Presentation: 大阪市立大学数学研究所
• [Presentation] コクセター亜群と表現論
  • Author(s): 山根宏之
  • Organizer: 岐阜大学工学部数学教室セミナー
  • Place of Presentation: 岐阜大学工学部
• [Presentation] Irreducible representaions of generalized quantized algebras
  • Author(s): 山根宏之
  • Organizer: ２０１４年度日本数学会春季年会
  • Place of Presentation: 学習院大学
• [Remarks] Ｐｕｂｌｉｃａｔｉｏｎ ｌｉｓｔｓ
  • URL: http://www3.u-toyama.ac.jp/hiroyuki/
• [Remarks] http://hiroyukipersonal.web.fc2.com/
• [Remarks] http://hiroyukipersonal.web.fc2.com/