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Jordan algebraic and differential geometric study of homogeneous complex manifolds

Research Project

Project/Area Number 15540066
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionUniversity of Fukui

Principal Investigator

YASUKURA Osami  University of Fukui, Faculty of Engineering, Professor, 工学部, 教授 (00191122)

Co-Investigator(Kenkyū-buntansha) ASANO Hiroshi  Kanagawa University, Faculty of Engineering, Professor, 工学部, 教授 (00046012)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥300,000 (Direct Cost: ¥300,000)
Fiscal Year 2003: ¥500,000 (Direct Cost: ¥500,000)
Keywordscomplex sinple Lie algebras / adjoint varieties / Freudenthal's varieties / symplectic triple systems / complex contact type gradation / reductive Lie groups / homogeneity / complex projective varieties / 接触型階数別分解 / 階数1部分空間 / adjoint variety / Freudenthal's variety / symplectic triple systems / projective algebraic variety / complex homogeneous manifolds / フロイデンタール多様体 / シンプレクテイック三項系 / 冪零軌道 / 例外型複素単純リー群-E_7 / D型複素単純リー代数
Research Abstract

For any complex simple Lie algebra, the Freudenthal variety is defined as the linear section of the corresponding adjoint variety by the 1-st graded subspace with respect to the complex contact type gradation of the Lie algebra. In this research [KY], the homogeneity of the Freudenthal varieties is proved in a priori way. And a unified description is given for the orbit structure of the first graded subspace by the Lie subgroup of the adjoint group corresponding to the 0-th graded Lie subalgebra. Then several properties of the Freidenthal varieties as complex projective varieties are derived only from the axioms of symplectic triple systems which are enjoyed by the 1-st graded subspace. Moreover, it provides a short cutting alternating proof for the 1-st graded subspace to enjoy the axioms of symplectic triple systems. These results are based on the construction of all complex simple Lie algebra of rank non less than two from simple complex symplectic triple systems by K.Yamaguti-H.Asano (1975), and vice varsa by H.Asano.(1975). Note that these results avoid case-by-case arguments according to the classification of all complex simple Lie algebras, and that these results are drived by considering substructure of symplectic triple systems. Algebraically, a structure theory of symplectic triple systems is studied by H.Asano (unpublished). These results give a geometric light on this algebraic structure theory of symplectic triple systems.
In this research [Y], typical examples are found on two non-isomorphic reductive Lie groups with one dimensional center such that their Lie algebras are isomorphic. In the literature, no proof was published on these examples. This result is firstly observed by considering the result of the orbit decomposition of the 1-st graded subspace by the Lie subgroup of the adjoint group corresponding to the 0-th graded Lie subalgebra of complex simple Lie algebras of type B and D.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (5 results)

All 2004 2003 Other

All Journal Article (4 results) Publications (1 results)

  • [Journal Article] Projective geometry of Freudenthal's varieties of certain type2004

    • Author(s)
      H.Kaji, O.Yasukura
    • Journal Title

      Michigan Mathematical Journal 52

      Pages: 515-542

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Projective geometry of Freudenthal's varieties of certain type2004

    • Author(s)
      H.Kaji, O.Yasukura
    • Journal Title

      Michigan Math.J. 52

      Pages: 515-542

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] 簡約リー群の同型問題2003

    • Author(s)
      保倉 理美
    • Journal Title

      数学(日本数学会,岩波書店) 55-2

      Pages: 199-202

    • NAID

      10011796881

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Isomorphic problem on reductive Lie groups2003

    • Author(s)
      O.Yasukura
    • Journal Title

      Sugaku 55-2

      Pages: 199-202

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] 保倉理美: "簡約リー群の同型問題"数学(日本数学会,岩波書店). 55・2. 199-202 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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