Capelli type identities and enveloping algebras of Lie algebras

• Project/Area Number: 20740020
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Kagoshima University
• Principal Investigator: ITOH Minoru 鹿児島大学, 大学院・理工学研究科, 准教授 (60381141)
• Project Period (FY): 2008 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: テンソル代数 / dual pair / 外積代数 / 不変式論の第一・第二基本定理 / Cayley-Hamiltonの定理 / 普遍包絡環の中心元 / 量子群 / 不変式論の第二基本定理 / Amitsur-Levitzki定理 / カペリ恒等式 / 普遍包絡環 / 量子展開環 / 不変式論の第一基本定理 / 不変式論の第一基定理

Research Abstract

The first result is the notion of derivations on tensor algebras and its applications to the invariant theory of noncommutative algebras(e. g. tensor algebras or universal enveloping algebras of Lie algebras). I also obtained a q-analogue of these results. The second result is some series of first and second fundamental theorems of invariant theory for polynomial algebras and exterior algebras. We have Cayley-Hamilton type theorems behind these second fundamental theorems. In addition, these second fundamental theorems for exterior algebras are closely related to the theory of polynomial identities.

Research Products

• [Journal Article] Extensions of the tensor algebra and their application (2012)
  • Author(s): Minoru Itoh
  • Journal Title: Commmunications in Algebra Volume: (印刷中)
  • Peer Reviewed
• [Journal Article] Extensions of the tensor algebra and their applications (2012)
  • Author(s): Minoru Itoh
  • Journal Title: Communications in Algebra Volume: 40(In press)
  • Peer Reviewed
• [Journal Article] On extensions of the tensor algebra, Casimir Force, Casimir Operators and the Riemann Hypothesis, Mathematics for Innovation in Industry and Science(edited by Gerrit van Dijk and Masato Wakayama) (2010)
  • Author(s): Minoru Itoh
  • Journal Title: DE GRUYTER Pages: 135-145
  • DOI: 10.1515/9783110226133.135
  • ISBN: 9783110226126
  • Peer Reviewed
• [Journal Article] テンソル代数上の微分のq類似とq Schur-Weyl双対性 (2010)
  • Author(s): 伊藤稔
  • Journal Title: 京都大学数理解析研究所講究録 Volume: Vol.1722 Pages: 90-96
• [Journal Article] テンソル代数上の微分のq類似とqSchur-Weyl双対性 (2010)
  • Author(s): 伊藤稔
  • Journal Title: 京都大学数理解析研究所講究録 Vol .1722 Pages: 90-96
• [Journal Article] On extensions of the tensor algebra, Casimir Force, Casimir Operators and the Riemann Hypothesis. (2010)
  • Author(s): Minoru Itoh
  • Journal Title: Mathematics for Innovation in Industry and Science(edited by Gerrit van Dijk and Masato Wakayama DE GRUYTER ISBN 978-3-11-022612-6 Pages: 135-145
  • Peer Reviewed
• [Journal Article] On Extensions of the Tensor Algebra (2010)
  • Author(s): Minoru Itoh
  • Journal Title: "Casimir Force, Casimir Operators and the Riemann Hypothesis," Walter de Gruyter Publishing Pages: 135-145
  • Peer Reviewed
• [Journal Article] Two permanents in the universal enveloping algebras of the symplectic Lie algebras (2009)
  • Author(s): Minoru Itoh
  • Journal Title: Two permanents in the universal enveloping algebras of the symplectic Lie algebras Volume: Vol.20, No.3 Issue: 03 Pages: 339-368
  • DOI: 10.1142/s0129167x09005327
  • Peer Reviewed
• [Journal Article] Schur type functions associated with polynomial sequences of binomial type (2009)
  • Author(s): Minoru Itoh
  • Journal Title: Selecta Mathematica(N. S.) Volume: Vol.14, No.2 Issue: 2 Pages: 247-274
  • DOI: 10.1007/s00029-008-0061-0
  • Peer Reviewed
• [Journal Article] Two permanents in the universal enveloping algebras of the symplectic Lie algebras (2009)
  • Author(s): Minoru Itoh
  • Journal Title: International Journal of Mathematics Vol.20, No.3 Pages: 339-368
  • Peer Reviewed
• [Journal Article] Schur type functions associated with polynomial sequences of binomial type (2009)
  • Author(s): Minoru Itoh
  • Journal Title: Selecta Mathematica(N.S.) Vol.14, No.2 Pages: 247-274
  • Peer Reviewed
• [Journal Article] Two permanents in the universal enveloping algebras of the symplectic Lie algebras (2009)
  • Author(s): Minoru Itoh
  • Journal Title: International Journal of Mathematics 20 Pages: 339-368
  • Peer Reviewed
• [Journal Article] Schur type functions associated with polynomial sequences of binomial type (2009)
  • Author(s): Minoru Itoh
  • Journal Title: Selecta Mathematica (New Series) 14 Pages: 247-274
  • Peer Reviewed
• [Journal Article] 二項型多項式列に付随するシューア型函数と普遍包絡環の中心元の固有値 (2008)
  • Author(s): 伊藤稔
  • Journal Title: 数理解析研究所講究録別冊 Volume: B7 Pages: 157-176
  • Peer Reviewed
• [Journal Article] 二項型多項式列に付随するシューア型函数と普遍包絡環の中心元の固有値 (2008)
  • Author(s): 伊藤稔
  • Journal Title: 数理解析研究所講究録別冊 B7 Pages: 157-176
  • Peer Reviewed
• [Journal Article] On Extensions of the Tensor Algebra
  • Author(s): Minoru Itoh
  • Journal Title: "Casimir force, Casimir Operators and the Riemann Hypothesis," Walter de Gruyter Publishing (印刷中)
• [Presentation] Invariant Theory for Exterior Algebras (2012)
  • Author(s): Minoru Itoh
  • Organizer: The 7th Kagoshima Algebra. Analysis. Geometry Seminar
  • Place of Presentation: 鹿児島大学
  • Year and Date: 2012-02-16
• [Presentation] Invariant Theory for Exterior Algebras (2012)
  • Author(s): Minoru Itoh
  • Organizer: The 7th Kagoshima Algebra-Analysis-Geometry Seminar
  • Place of Presentation: 鹿児島大学(招待講演)
  • Year and Date: 2012-02-16
• [Presentation] A mixture of the tensor algebra and the infinite symmetric group and its application to the Schur-Weyl duality (2011)
  • Author(s): Minoru Itoh
  • Organizer: MPI-Oberseminar
  • Place of Presentation: Max-Planck-Institute for mathematics, Germany
  • Year and Date: 2011-05-05
• [Presentation] A mixture of the tensor algebra and the infinite symmetric group and its application to the Schur-Weyl duality (2011)
  • Author(s): Minoru Itoh
  • Organizer: MPI-Oberseminar
  • Place of Presentation: Max-Planck-Institute for mathematics, Germany(招待講演)
  • Year and Date: 2011-05-05
• [Presentation] テンソル代数上の微分のq類似とq Schur-Weyl双対性 (2010)
  • Author(s): 伊藤稔
  • Organizer: RIMS研究集会「等質空間と非可換調和解析」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-06-15
• [Presentation] Extensions of the tensor algebra and their applications (2010)
  • Author(s): Minoru Itoh
  • Organizer: Workshop on Invariant Theory and Related Topics
  • Place of Presentation: Inha University, Korea
• [Presentation] Extensions of the Tensor Algebra and Their Applications (2010)
  • Author(s): Minoru Ttoh
  • Organizer: Invariant Theory and Related Topics
  • Place of Presentation: Inha University(韓国)
• [Presentation] On extensions of the tensor algebra (2009)
  • Author(s): Minoru Itoh
  • Organizer: Forum "Math-for-Industry" Casimir Force, Casimir Operators and the Riemann Hypothesis
  • Place of Presentation: Kyushu University
  • Year and Date: 2009-11-11
• [Presentation] On extensions of the tensor algebra,Forum "Math-for-Industry" Casimir Force (2009)
  • Author(s): Minoru Itoh
  • Organizer: Casimir Operators and the Riemann Hypothesis
  • Place of Presentation: Kyushu University
  • Year and Date: 2009-11-11
• [Presentation] On Extensions of the Tensor Algebra (2009)
  • Author(s): Minoru Itoh
  • Organizer: Casimir Force, Casimir Operators and the Riemann Hypothesis
  • Place of Presentation: 明治大学リバティタワー
  • Year and Date: 2009-11-11
• [Presentation] テンソル代数の拡張とその応用 (2009)
  • Author(s): 伊藤稔
  • Organizer: 第54回代数学シンポジウム
  • Place of Presentation: 明治大学
  • Year and Date: 2009-08-06
• [Presentation] テンソル代数の拡張とその応用 (2009)
  • Author(s): 伊藤稔
  • Organizer: 代数学シンポジウム
  • Place of Presentation: 明治大学リバティタワー
  • Year and Date: 2009-08-06
• [Presentation] テンソル代数の拡張とそのimmanantへの応用 (2008)
  • Author(s): 伊藤稔
  • Organizer: 2008年度表現論シンポジウム
  • Place of Presentation: ウェルハートビア熱海(静岡県熱海市)
  • Year and Date: 2008-11-14
• [Presentation] テンソル代数の拡張とその表現論への応用 (2008)
  • Author(s): 伊藤稔
  • Organizer: 第119回日本数学会九州支部例会
  • Place of Presentation: 鹿児島大学理学部
  • Year and Date: 2008-10-18
• [Presentation] Extensions of the tensor algebra and their application to immamants (2008)
  • Author(s): Minoru Itoh
  • Organizer: JSPS-RFBR Workshop "Harmonic Analysis on Homogeneous Spaces and Quantization,"
  • Place of Presentation: Derzhavin Tambov State University, Russia
  • Year and Date: 2008-09-26
• [Presentation] Extensions of the tensor algebra and their application to immamants,JSPS-RFBR Workshop (2008)
  • Author(s): Minoru Itoh
  • Organizer: Harmonic Analysis on Homogeneous Spaces and Quantization
  • Place of Presentation: Derzhavin Tambov State University, Russia
  • Year and Date: 2008-09-26
• [Presentation] Extensions of the tensor algebra and their application to immamants (2008)
  • Author(s): Minoru Itoh
  • Organizer: JSPS-RFBR Workshop, Harmonic Analysis on Homogeneous Spaces and Quantization
  • Place of Presentation: Derzhavin Tambov State University, Russia
  • Year and Date: 2008-09-26
• [Presentation] Extensions of the tensor algebra and their application to Schur-Weyl type dualities (2008)
  • Author(s): Minoru Itoh
  • Organizer: JSPS-RFBR Workshop "Harmonic Analysis on Homogeneous Spaces and Quantization,"
  • Place of Presentation: Tambara Institute of Mathematical Sciences, The University of Tokyo
  • Year and Date: 2008-08-28
• [Presentation] Extensions of the tensor algebra and their application to Schur-Weyl type dualities,JSPS-RFBR Workshop (2008)
  • Author(s): Minoru Itoh
  • Organizer: Harmonic Analysis on Homogeneous Spaces and Quantization
  • Place of Presentation: Tambara Institute of Mathematical Sciences, The University of Tokyo
  • Year and Date: 2008-08-28
• [Presentation] Extensions of the tensor algebra and their application to Schur-Weyl type dualities (2008)
  • Author(s): Minoru Itoh
  • Organizer: JSPS-RFBR Workshop, Harmonic Analysis on Homogeneous Spaces and Quantization
  • Place of Presentation: Tambara Institute of Mathematical Sciences, University of Tokyo
  • Year and Date: 2008-08-28