Skip to main content
Log in

Invariant Foliations of Nondegenerate Bi-Hamiltonian Structures

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this paper, we describe all invariant distributions of nondegenerate bi-Hamiltonian structures and investigate their integrability in the neighborhood of a generic point.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. A. V. Bolsinov, “A completeness criterion for a family of function in involution constructed by the argument shift method,” Sov. Math. Dokl., 38, No. 1, 161–165 (1989).

    MATH  Google Scholar 

  2. A. V. Bolsinov, “Compatible Poisson brackets on Lie algebras and completeness of families of functions in involution,” Math. USSR Izv., 38, No. 1, 69–89 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  3. A. V. Bolsinov, A. M. Izosimov, A. Y. Konyaev, and A. A. Oshemkov, “Algebra and topology of integrable systems: research problems,” Tr. Semin. Vekt. Tenz. Anal., 28, 119–191 (2012).

    Google Scholar 

  4. F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).

    MATH  Google Scholar 

  5. G. B. Gurevich, “Canonization of a pair of bivectors,” Tr. Semin. Vekt. Tenz. Anal., 8, 355–363 (1950).

    MathSciNet  MATH  Google Scholar 

  6. I. K. Kozlov, “An elementary proof of the Jordan–Kronecker theorem,” Mat. Zametki, 94, No. 6, 857–870 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. S. Mishchenko and A. T. Fomenko, “Euler equations on finite–dimensional Lie groups,” Izv. Akad. Nauk SSSR, Ser. Mat., 42, No. 2, 396–415 (1978).

    MathSciNet  MATH  Google Scholar 

  8. A. S. Mishchenko and A. T. Fomenko, “Generalized Liouville method of integration of Hamiltonian systems,” Funkts. Anal. Prilozh., 12, No. 2, 46–56 (1978).

    MathSciNet  MATH  Google Scholar 

  9. A. Panasyuk, “Veronese webs for bihamiltonian structures of higher corank,” Banach Center Publ., 51, 251–261 (2000).

    MathSciNet  MATH  Google Scholar 

  10. R. C. Thompson, “Pencils of complex and real symmetric and skew matrices,” Linear Algebra Appl., 147, 323–371 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  11. V. V. Trofimov and A. T. Fomenko, Algebra and Geometry of Integrable Hamiltonian Differential Equations [in Russian], Faktorial, Moscow (1995).

    MATH  Google Scholar 

  12. F. J. Turiel, “Classification locale simultan´ee de deux formes symplectiques compatibles,” Manuscripta Math., 82, No. 1, 349–362 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  13. F. J. Turiel, On the Local Theory of Veronese Webs, arXiv:1001.3098v1.

  14. F. J. Turiel, The Local Product Theorem for Bihamiltonian Structures, arXiv:1107.2243v1.

  15. I. S. Zakharevich, Kronecker Webs, Bihamiltonian Structures, and the Method of Argument Translation, arXiv:math/9908034v3.

  16. P. Zhang, Algebraic Properties of Compatible Poisson Structures, Preprint Loughborough Univ., No. 10-02 (2010).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to I. K. Kozlov.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 20, No. 3, pp. 91–111, 2015.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kozlov, I.K. Invariant Foliations of Nondegenerate Bi-Hamiltonian Structures. J Math Sci 225, 596–610 (2017). https://doi.org/10.1007/s10958-017-3481-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-017-3481-6

Navigation