Skip to main content
SpringerLink
Log in

Poisson–Lie Generalization of the Kazhdan–Kostant–Sternberg Reduction

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

Abstract

The trigonometric Ruijsenaars–Schneider model is derived by symplectic reduction of Poisson–Lie symmetric free motion on the group U(n). The commuting flows of the model are effortlessly obtained by reducing canonical free flows on the Heisenberg double of U(n). The free flows are associated with a very simple Lax matrix, which is shown to yield the Ruijsenaars–Schneider Lax matrix upon reduction.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alekseev A.Yu., Malkin A.Z.: Symplectic structures associated to Lie–Poisson groups. Commun. Math. Phys. 162, 147–174 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Arutyunov G.E., Frolov S.A., Medvedev P.B.: Elliptic Ruijsenaars–Schneider model via the Poisson reduction of the affine Heisenberg double. J. Phys. A Math. Gen. 30, 5051–5063 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Arutyunov G.E., Frolov S.A., Medvedev P.B.: Elliptic Ruijsenaars–Schneider model from the cotangent bundle over the two-dimensional current group. J. Math. Phys. 38, 5682–5689 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Calogero F.: Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials. J. Math. Phys. 12, 419–436 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  5. Etingof P.I., Kirillov A.A. Jr: Macdonald’s polynomials and representations of quantum groups. Math. Res. Lett. 1, 279–294 (1994)

    MATH  MathSciNet  Google Scholar 

  6. Fock V., Gorsky A., Nekrasov N., Rubtsov V.: Duality in integrable systems and gauge theories. JHEP 0007, 028 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  7. Fock, V.V., Rosly, A.A.: Poisson structure on moduli of flat connections on Riemann surfaces and the r-matrix. In: Moscow Seminar in Mathematical Physics. AMS Transl. Ser. 2, vol. 191, pp. 67–86 (1999)

  8. Gorsky A., Nekrasov N.: Relativistic Calogero–Moser model as gauged WZW theory. Nucl. Phys. B436, 582–608 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  9. Kazhdan D., Kostant B., Sternberg S.: Hamiltonian group actions and dynamical systems of Calogero type. Commun. Pure Appl. Math. XXXI, 481–507 (1978)

    Article  MathSciNet  Google Scholar 

  10. Klimčík C.: On moment maps associated to a twisted Heisenberg double. Rev. Math. Phys. 18, 781–821 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lu, J.-H.: Moment maps and reduction of Poisson actions. In: Proc. of Sem. Sud- Rodanian de Geometrie à Berkeley (1989). Springer, MSRI Publ., vol. 20, pp. 209–226 (1991)

  12. Moser J.: Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16, 197–220 (1975)

    Article  MATH  Google Scholar 

  13. Nekrasov, N.: Infinite-dimensional algebras, many-body systems and gauge theories. In: Moscow Seminar in Mathematical Physics. AMS Transl. Ser. 2, vol. 191, pp. 263–299 (1999)

  14. Noumi M.: Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces. Adv. Math. 123, 16–77 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Olshanetsky M.A., Perelomov A.M.: Explicit solution of the Calogero model in the classical case and geodesic flows on symmetric spaces of zero curvature. Lett. Nouvo Cim. 16, 333–339 (1976)

    Article  MathSciNet  Google Scholar 

  16. Olshanetsky M.A., Perelomov A.M.: Explicit solutions of some completely integrable systems. Lett. Nouvo Cim. 17, 97–101 (1976)

    Article  MathSciNet  Google Scholar 

  17. Semenov-Tian-Shansky M.A.: Dressing transformations and Poisson groups actions. Publ. RIMS 21, 1237–1260 (1985)

    Article  MathSciNet  Google Scholar 

  18. Sutherland B.: Exact results for a quantum many body problem in one dimension. Phys. Rev. A4, 2019–2021 (1971)

    ADS  Google Scholar 

  19. Ruijsenaars S.N.M.: Action-angle maps and scattering theory for some finite-dimensional integrable systems I. The pure soliton case. Commun. Math. Phys. 115, 127–165 (1988)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Ruijsenaars S.N.M.: Action-angle maps and scattering theory for some finite- dimensional integrable systems III. Sutherland type systems and their duals. Publ. RIMS 31, 247–353 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ruijsenaars, S.N.M.: Systems of Calogero–Moser type. In: Proceedings of the 1994 CRM—Banff Summer School ‘Particles and Fields’. Springer, pp. 251–352 (1999)

  22. Ruijsenaars S.N.M., Schneider H.: A new class of integrable models and their relation to solitons. Ann. Phys. (N.Y.) 170, 370–405 (1986)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. van Diejen J.F., Vinet L.: The quantum dynamics of the compactified trigonometric Ruijsenaars–Schneider model. Commun. Math. Phys. 197, 33–74 (1998)

    Article  MATH  ADS  Google Scholar 

  24. Zakrzewski S.: Free motion on the Poisson SU(N) group. J. Phys. A Math. Gen. 30, 6535–6543 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Theoretical Physics, MTA KFKI RMKI, P.O.B. 49, 1525, Budapest, Hungary

    László Fehér

  2. Department of Theoretical Physics, University of Szeged, Tisza Lajos krt 84-86, 6720, Szeged, Hungary

    László Fehér

  3. Institute de mathématiques de Luminy, 163, Avenue de Luminy, 13288, Marseille, France

    Ctirad Klimčík

Authors
  1. László Fehér
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ctirad Klimčík
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to László Fehér.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fehér, L., Klimčík, C. Poisson–Lie Generalization of the Kazhdan–Kostant–Sternberg Reduction. Lett Math Phys 87, 125–138 (2009). https://doi.org/10.1007/s11005-009-0298-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-009-0298-3

Mathematics Subject Classification (2000)

Keywords

Search

Navigation