• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

The study of integrable systems and Lie algebras associated with the period maps for primitive forms

Research Project

Project/Area Number 16340016
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKyoto University

Principal Investigator

SAITO Kyoji  Kyoto University, Mathematical SciResearch Institute for Mathematical Sciences, Professor (20012445)

Co-Investigator(Kenkyū-buntansha) MUKAI Shigeru  Kyoto University, Research Institute for Mathematical Sciences, Professor (80115641)
SAITO Morihiko  Kyoto University, Research Institute for Mathematical Sciences, Associate Professor (10186968)
TERAO Hiroaki  Hokkaido University, Department of Mathematics, Professor (90119058)
SUWA Tatsuo  Niigata University, Faculty of Engineering, Professor (40109418)
AOKI Hiroki  Tokyo University of Science, Faculty of Science and Technology, Lecturer (10333189)
岡 睦雄  東京理科大学, 理学部, 教授 (40011697)
HELMKE Stefan  京都大学, 数理解析研究所, 助手 (40293972)
高橋 篤史  京都大学, 数理解析研究所, 助手 (50314290)
松尾 厚  東京大学, 大学院・数理科学研究科, 助教授 (20238968)
Project Period (FY) 2004 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥17,320,000 (Direct Cost: ¥16,300,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2007: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2006: ¥4,600,000 (Direct Cost: ¥4,600,000)
Fiscal Year 2005: ¥4,000,000 (Direct Cost: ¥4,000,000)
Fiscal Year 2004: ¥4,300,000 (Direct Cost: ¥4,300,000)
Keywordsprimitive forms / Lie algebra / matrix factorization / derived category / reflection group / discontinuous group / partition function / 極限集合 / 正規ウェイト系 / 三角圏 / 無限次元リー環 / 対数的自由因子 / 基本群 / 正規ウエイト系 / エータ積 / 対数型微分形式 / エータ関数 / コクセター変換 / Hook length formula / 組み紐群 / ルート系
Research Abstract

The subject of the present research program is the Lie theoretic study of primitive forms and their period maps. We obtain several results, which are briefly summarized as follows.
1. In a series of joint works with H. Kajiura and A. Takahashi, we developed a new method to approach to the construction of Lie algebras by a use of the category of matrix factorization of singularities associated with a regular system of weights. This has unified not only the construction of the semi-simple and elliptic Lie algebras, but also several concepts such as primitive forms, vanishing cycles, derived category of coherent sheaves, infinite Lie algebras, etc. In particular, we determine certain strongly exceptional collections for the category of ε=-1. This gives an essentially new (wild type) Lie algebras beyond Kac-Moody Lie algebras. The study of primitive forms associated with the Lie algebra should add new contents to mathematics. (3 published papers)
2. By a use of real algebraic geometry, we give a (part of) new proof of the classical fact that the complement of the discriminant loci of a finite reflection group is an Eilenberg-Mclane space with respect to an Artin group. As an application of the proof we obtain a solution to a combinatorial problem: the maximal number of linearization of an oriented tree. (2 published papers)
3. Even it was not stated explicitly in the program of the research, we obtain a remarkable progress, supported by the present fund, in the study of partition functions on non-commutative lattice. Namely, we develop a general theory of the high temperature development of the partition functions on non-commutative lattice. We introduce the compact set Ω (〓,G) of the limit functions, and obtain its comparison with limit set Ω (Pr,〓) associated with the growth function Pr.〓 and a trace formula. (Submitted)

Report

(5 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (45 results)

All 2008 2007 2006 2005 2004

All Journal Article (38 results) (of which Peer Reviewed: 15 results) Presentation (7 results)

  • [Journal Article] Towards a categorical construction of Lie algebras.2008

    • Author(s)
      K. Saito
    • Journal Title

      Advanced Studies in Pure Mathematics 50

      Pages: 101-175

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Geometry of plane curves via taylor expansions2008

    • Author(s)
      M. Oka
    • Journal Title

      Proceeding of Conference on Geometry and Topology (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Residue Theoretical Approach to Intersection Theory2008

    • Author(s)
      T. Suwa
    • Journal Title

      the Proceedings of IX Workshop on Real and Complex Singularities (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Towards a categorical construction of Lie algebras2008

    • Author(s)
      K., Saito
    • Journal Title

      Advanced Studies in Pure Mathematics 50

      Pages: 101-175

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Geometry of plane curves via taylor expansions2008

    • Author(s)
      M., Oka
    • Journal Title

      to appear in Proceeding of Conference on Geometry and Topology, Cuernavaca, Mexico

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Residue Theoretical Approach to Intersection Theory2008

    • Author(s)
      T., Suwa
    • Journal Title

      to appear in the Proceedings of IX Workshop on Real and Complex Singularities, Sao Carlos, Brazil, Contemporary Math., Amer. Math. Soc

      Pages: 1-55

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Towards a categorical construction of Lie algebras2008

    • Author(s)
      K. Saito
    • Journal Title

      Advanced Studies in Pure Mathematics 50

      Pages: 101-175

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Principal Γ-cone for a tree2007

    • Author(s)
      K. Saito
    • Journal Title

      Advances in Mathematics 212

      Pages: 645-668

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Matrix factorizations and representations of quivers II: Type ADE case2007

    • Author(s)
      K. Saito, H. Kajiura and A. Takahashi
    • Journal Title

      Advances in Mathematics 211

      Pages: 327-362

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Crystals and affine Hecke algebras of type D2007

    • Author(s)
      M. Kashiwra and V. Miemietz
    • Journal Title

      Proc. Japan Acad. 83 Ser.A

      Pages: 135-139

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Multiplier ideals, b-function, and spectrum of a hypersurface singularity2007

    • Author(s)
      M. Saito
    • Journal Title

      Compos. Math. 143

      Pages: 1050-1068

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Bases of the contact-order filtration of derivations of coxeter arrangements2007

    • Author(s)
      H. Terao
    • Journal Title

      Proceedings of the American Mathematical Society 133

      Pages: 2029-2034

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Principal P-cone for a tree2007

    • Author(s)
      K., Saito
    • Journal Title

      Advances in Mathematics 212

      Pages: 645-668

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Takahashi, Matrix factorizations and representations of quivers II : Type ADE case2007

    • Author(s)
      K., Saito, H., Kajiura, A
    • Journal Title

      Advances in Mathematics 211

      Pages: 327-362

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Miemietz, Crystals and affine Hecke algebras of type2007

    • Author(s)
      M., Kashiwra, V
    • Journal Title

      Proc. Japan Acad 83

      Pages: 135-139

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Multiplier ideals, b-function, and spectrum of a hypersurface singularity2007

    • Author(s)
      M., Saito
    • Journal Title

      Compos. Math 143

      Pages: 1050-1068

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Bases of the contact-order filtration of derivations of coxeter arrangements2007

    • Author(s)
      H., Terao
    • Journal Title

      Proceedings of the American Mathematical Society 133, No.7

      Pages: 2029-2034

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Crystals and affine Hecke algebras of type D2007

    • Author(s)
      M. Kashiwara and V. Miemietz
    • Journal Title

      Proc. Japan Acad. 83 Ser.A

      Pages: 135-139

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Chambers of arrangements of hyperplanes and Arrow's impossibility theorem2007

    • Author(s)
      H. Terao
    • Journal Title

      Advances in Mathematics 214

      Pages: 366-378

    • NAID

      120000955425

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Geometry of plane curves via taylor expansions2007

    • Author(s)
      M. Oka
    • Journal Title

      Proceeding of Conference on Geometry and Topology (to appear)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Matrix factorizations and representations of quivers II : Type ADE Case.2007

    • Author(s)
      H.Kajiura, K.Saito, A.Takahashi
    • Journal Title

      Advances in Mathematics 211

      Pages: 327-362

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Principal Γ-cone for a tree2007

    • Author(s)
      Kyoji Saito
    • Journal Title

      Advances in Mathematics (to appear)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Tangential Alexander Polynomials and non-reduced degeneration.2007

    • Author(s)
      Mutsuo Oka
    • Journal Title

      Proceedings of 'Singularities in Geometry and Topology' World Scientific, (donated to Prof. K. Saito)

      Pages: 669-704

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Proportionality of 1-forms on singular varieties, Singularities in Geometry and Topology.2007

    • Author(s)
      T.Suwa, J.-P.Brasselet, J.Seade
    • Journal Title

      Advanced Studies in Pure Mathematics 46

      Pages: 49-65

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Polarized K3 surfaces of genus thirteen2006

    • Author(s)
      Shigeru Mukai
    • Journal Title

      Advanced Studies in Pure Mathematics 45

      Pages: 315-326

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A generalization of Griffiths's theorem on rational integrals.2006

    • Author(s)
      Alexandru Dimca, Morihiko Saito
    • Journal Title

      Duke Math. J. 135

      Pages: 303-326

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Eta-product η(7τ)^7/η(τ)2006

    • Author(s)
      齋藤恭司
    • Journal Title

      RIMS Preprint 1532

      Pages: 1-8

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Simple graded rings of siegel modular forms, differential operators and borcherds products2005

    • Author(s)
      H. Aoki and T. Ibukiyama
    • Journal Title

      International Journal of Mathematics 16

      Pages: 249-279

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Ibukiyama, Simple graded rings of siegel modular forms, differential operators and borcherds products2005

    • Author(s)
      H., Aoki, T
    • Journal Title

      International Journal of Mathematics 16, No.3

      Pages: 249-279

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Principal Γ-cones for a tree2005

    • Author(s)
      齋藤恭司
    • Journal Title

      RIMS Preprint 1507

      Pages: 1-25

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Matrix Factorizations and Representations of Quivers2005

    • Author(s)
      齋藤恭司(joint with H.Kajiura, A.Takahashi)
    • Journal Title

      RIMS Preprint 1521

      Pages: 1-36

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Semialgebraic geometry of Braid groups2005

    • Author(s)
      K.Saito
    • Journal Title

      代数幾何学シンポジユーム記録(2004年10月25日-10月29日於兵庫県立城崎大会議館)

      Pages: 1-14

    • NAID

      120005763649

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Polyhedra dual to the Weyl Chamber decomposition: A precis2004

    • Author(s)
      K. Saito
    • Journal Title

      Publ. RIMS, Kyoto Univ. 40

      Pages: 1337-1384

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Geometric realization of T-shaped root systems and counterexamples to Hilbert's fourteenth problem2004

    • Author(s)
      S. Mukai
    • Journal Title

      Algebraic transformation groups and algebraic varieties

      Pages: 123-129

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Polyhedra dual to the Weyl Chamber decomposition : A precis2004

    • Author(s)
      K., Saito
    • Journal Title

      Publ. RIMS, Kyoto Univ 40

      Pages: 1337-1384

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Geometric realization of Tshaped root systems and counterexamples to Hilbert's fourteenth problem2004

    • Author(s)
      S., Mukai
    • Journal Title

      Algebraic transformation groups and algebraic varieties, (ed) V. L. Popov, Springer-Verlag

      Pages: 123-129

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Polyhedra Dual to the Weyl Chamber Decomposition : a Precis2004

    • Author(s)
      K.Saito
    • Journal Title

      Publ.RIMS, Kyoto Univ. 40

      Pages: 1337-1384

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Uniformization of the orbifold of a finite reflection group, Frobenius Manifolds2004

    • Author(s)
      K.Saito
    • Journal Title

      Aspects of Mathematics(C.Hertling and M.Marcolli(eds.)) E36

      Pages: 265-320

    • Related Report
      2004 Annual Research Report
  • [Presentation] Towards a categorical construction of Lie algebras2007

    • Author(s)
      K. Saito
    • Organizer
      The fourth intenational Congress of Chinese Mathematicians (ICCM)
    • Place of Presentation
      杭州大学
    • Year and Date
      2007-12-20
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Towards a categorical construction of Lie algebras2007

    • Author(s)
      K., Saito
    • Organizer
      The fourth intenational Congress of Chinese Mathematicians (ICCM)
    • Place of Presentation
      Zhejiang University
    • Year and Date
      2007-12-20
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Towards a categorical construction of Lie algebras2007

    • Author(s)
      K. Saito
    • Organizer
      The fourth international Congress of Chinese Mathematicians (ICCM)
    • Place of Presentation
      杭州大学
    • Year and Date
      2007-12-20
    • Related Report
      2007 Annual Research Report
  • [Presentation] Towards a categorical construction of Lie algebras2007

    • Author(s)
      K. Saito
    • Organizer
      日仏 特異点シンポジウム
    • Place of Presentation
      立山国際ホテル
    • Year and Date
      2007-08-29
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Towards a categorical construction of Lie algebras2007

    • Author(s)
      K., Saito
    • Organizer
      Singularites, Franco-Japonaises
    • Place of Presentation
      Tateyama Kokusai Hotel
    • Year and Date
      2007-08-29
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Towards a categorical construction of Lie algebras2007

    • Author(s)
      K. Saito
    • Organizer
      Modular Forms and Moduli Spaces
    • Place of Presentation
      St Petersburg, Euler Institute
    • Year and Date
      2007-07-05
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Towards a categorical construction of Lie algebras2007

    • Author(s)
      K. Saito
    • Organizer
      Modular Forms and Moduli Spaces
    • Place of Presentation
      St. Petersburg, Euler Institute
    • Year and Date
      2007-07-05
    • Related Report
      2007 Annual Research Report

URL: 

Published: 2004-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi