Skip to main content
SpringerLink
Log in

Integrability of Characteristic Hamiltonian Systems on Simple Lie Groups with Standard Poisson Lie Structure

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Phase space of a characteristic Hamiltonian system is a symplectic leaf of a factorizable Poisson Lie group. Its Hamiltonian is a restriction to the symplectic leaf of a function on the group which is invariant with respect to conjugations. It is shown in this paper that such a system is always integrable.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Kostant, B.: The solution to a generalized Toda lattice and representation theory. Adv. Math. 34, 195–338 (1979)

    MathSciNet  MATH  Google Scholar 

  2. Adler, M.: On a trace functional for formal pseudo-differential operators and the symplectic structure of the kdv-type equations. Inv. Math. 50, 219–248 (1979)

    MathSciNet  MATH  Google Scholar 

  3. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Second Edition, Berlin-Heidelberg-New York: Springer, 1989

  4. Symes, W.W.: Systems of Toda type, inverse spectral problems, and representation theory. Invent. Math. 50, 13–51 (1980)

    MathSciNet  Google Scholar 

  5. Sklyanin, E.K.: Quantum version of inverse scattering method. Zap. Nauch. Semin. LOMI 95, 55–128 (1980)

    MATH  Google Scholar 

  6. Reyman, A.G., Semenov-Tian-Shansky, M.A.: Reduction of Hamiltonian systems, affine Lie algebras and Lax equations.I. Invent. Math. 54, 81–100 (1979)

    MathSciNet  MATH  Google Scholar 

  7. Drinfeld, V.G.: Quantum groups. In: Proc. Intern. Congress of Math. (Berkeley 1986), Providence, RI: AMS, 1987, pp. 798–820

  8. Semenov-Tian-Shansky, M.A.: Dressing transformations and Poisson group actions. Pub. Res. Inst. Math. Sci. Kyoto Univ. 21, 1237–1260, (1985)

    MATH  Google Scholar 

  9. Deift, P., Li, L.C., Nanda, T., Tomei, C.: The Toda flow on a generic orbit is integrable. Comm. Pure Appl. Math. 39, 183–232, (1986)

    MathSciNet  MATH  Google Scholar 

  10. Ercolani, N.M., Flaschka, H., Singer, S.F.: The geometry of the full Kostant-Toda lattice. Progr. Math. 115, 181–225 (1993)

    MATH  Google Scholar 

  11. Gekhtman, M.I., Shapiro, M.Z.: Non-commutative and commutative integrability of generic Toda flow in simple Lie algberas. Comm. Pure Appl. Math. 52, 53–84 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nekhoroshev, N.N.: Action-angle variables and their generalizations. Trans. Moscow Math. Soc. 26, 180–197 (1972)

    Google Scholar 

  13. Fomenko, A.T.: Symplectic geometry. Advanced Studies in Contemporary Mathematics. 5, New York: Gordon and Breach Science Publishers, 1988

  14. Frish, J., Mandrosov, V., Smorodinsky, Y.A., Uhlir, M., Winternitz, P.: On higher symmetries in quantum mechanics. Phys. Lett. 16, 354–356 (1965)

    Article  Google Scholar 

  15. Pauli, W.: Z. Physik 36, 336 (1935)

    Google Scholar 

  16. Hoffman, T., Kellendonk, J., Kutz, N., Reshetikhin, N.: Factorization dynamics and Coxeter-Toda lattices. Commun. Math. Phys. 212, 297–321 (2000)

    Article  Google Scholar 

  17. Li, L.-C.: The SVD flows on generic symplectic leaves are completely integrable. Adv. Math. 128(1), 82–118 (1997)

    Article  MATH  Google Scholar 

  18. De~Concini, C., Kac, V.G., Procesi, C.: Some quantum analogues of solvable Lie groups. In: Geometry and analysis. Papers presented at the Bombay colloquium, India, January 6–14, 1992, Oxford: Oxford University Press, 1995, pp. 41–65.

  19. Fomin, S., Zelevinsky, A.: Double bruhat cells and total positivity. J. AMS 12, 335–380, (1999)

    MathSciNet  MATH  Google Scholar 

  20. Hodges, T., Levasseur, T.: Primitive ideals of C q [SL(3)]. Commun. Math. Phys. 156, 581–605, (1993)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California-Berkeley, Berkeley, CA, 94720-3840, USA

    N. Reshetikhin

Authors
  1. N. Reshetikhin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Reshetikhin.

Additional information

Communicated by R.H. Dijkgraaf

Rights and permissions

Reprints and permissions

About this article

Cite this article

Reshetikhin, N. Integrability of Characteristic Hamiltonian Systems on Simple Lie Groups with Standard Poisson Lie Structure. Commun. Math. Phys. 242, 1–29 (2003). https://doi.org/10.1007/s00220-003-0916-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-003-0916-3

Keywords

Search

Navigation