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A study on duality of Hopf modules

Research Project

Project/Area Number 15540054
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNiihama National College of Technology

Principal Investigator

YANAI Tadashi  Niihama National College of Technology, Engineering Science, Assistant Professor, 数理科, 助教授 (50220174)

Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2004: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2003: ¥500,000 (Direct Cost: ¥500,000)
KeywordsHopf algebras / Hopf modules / Duality / Integrals / Galois correspondence / クロスK双代数 / ガロア型対応
Research Abstract

Assume that H is a Hopf algebra over a field k, A a right H-comodule algebra, D the subalgebra of all coinvariants of A, and M the (A,D)-bimodule of all left D-linear maps from A to D. If H is a pointed Hopf algebra, and A is simple as an (A,H)-Hopf module and finite-dimensional as a D-module, then, by twisting the D-module structure of A and the H-comodule structure of M suitably, A is isomorphic to M as an (A,H)-Hopf module and as an (A,D)-bimodule. This result can be considered as a generalization of duality of finite-dimensional Hopf algebras to Hopf modules. Besides, suppose that a finite-dimensional Hopf algebra H acts on a division algebra D and A is a right H-comodule subalgebra of D#H including D. Then this duality implies some properties of integrals in A. Moreover, let R be a prime algebra, K its extended centreoid, and H a finite-dimensional pointed Hopf algebra acting on R by an X-outer action. Then, from those properties of integrals, we have a one to one Galois-type correspondence in arbitrary characteristic between the set of all rationally complete subalgebras of R including the subalgebra of invariants and the set of all right H-comodule subalgebras of K#H including K. For a further study, we have a problem whether these results can be generalized to Hopf algebroids, which are objects including Hopf algebras. We also have a problem whether it is possible to give a Galois correspondence theorem for Hopf algebra actions under a condition which is weaker than that of the result above.

Report

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

    (5 results)

All 2003 Other

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

  • [Journal Article] Hopf module duality applied to X-outer Galois theory2003

    • Author(s)
      Akira Masuoka
    • Journal Title

      Journal of Algebra 265

      Pages: 229-246

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Galois correspondence theorem for Hopf algebra actions

    • Author(s)
      Tadashi Yanai
    • Journal Title

      Contemporary Mathematics (掲載予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Galois correspondence theorem for Hopf algebra actions

    • Author(s)
      Tadashi Yanai
    • Journal Title

      Contemporary Mathematics (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Galois correspondence theorem for Hopf algebra actions

    • Author(s)
      Tadashi Yanai
    • Journal Title

      Contemporary Mathematics (掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Publications] Akira Masuoka, Tadashi Yanai: "Hopf module duality applied to X-outer Galois theory"Journal of Algebra. 265. 229-246 (2003)

    • Related Report
      2003 Annual Research Report

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

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