Skip to main content
SpringerLink
Log in

A New Model in the Calogero–Ruijsenaars Family

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

Abstract

Hamiltonian reduction is used to project a trivially integrable system on the Heisenberg double of SU(n, n), to obtain a system of Ruijsenaars type on a suitable quotient space. This system possesses BC n symmetry and is shown to be equivalent to the standard three-parameter BC n hyperbolic Sutherland model in the cotangent bundle limit.

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. 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 

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

    Article  ADS  MATH  Google Scholar 

  3. Sutherland B.: Exact results for a quantum many-body problem in one dimension. II. Phys. Rev. A 5, 1372–1376 (1972)

    Article  ADS  Google Scholar 

  4. Olshanetsky M.A., Perelomov A.M.: Classical integrable finite-dimensional systems related to Lie algebras. Phys. Rep. 71, 313–400 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  5. Olshanetsky M.A., Perelomov A.M.: Completely integrable Hamiltonian systems connected with semisimple Lie algebras. Invent. Math. 37, 93–108 (1976)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Inozemtsev V.I., Meshcheryakov D.V.: Extension of the class of integrable dynamical systems connected with semisimple Lie algebras. Lett. Math. Phys. 9, 13–18 (1985)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. 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 

  8. Reyman, A.G., Semenov-Tian-Shansky, M.A.: Group-theoretical methods in the theory of finite-dimensional integrable systems. In: Arnol’d V.I., Novikov, S.P. (eds.) Integrable Systems II. Dynamical Systems VII, Encyclopaedia of Math. Sciences, vol.16. Springer-Verlag, Berlin, pp. 116–220 (1994)

  9. Fehér L., Pusztai B.G.: A class of Calogero type reductions of free motion on a simple Lie group. Lett. Math. Phys. 79, 263–277 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. 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  ADS  MATH  MathSciNet  Google Scholar 

  11. 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)

  12. Gorsky A., Nekrasov N.: Relativistic Calogero–Moser model as gauged WZW theory. Nucl. Phys. B 436, 582–608 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Gorsky A.: Integrable many body systems in field theories. Theor. Math. Phys. 103, 681–700 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

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

    Article  ADS  MathSciNet  Google Scholar 

  17. Feher L., Klimcik C.: Poisson–Lie generalization of the Kazhdan–Kostant–Sternberg reduction. Lett. Math. Phys. 87, 125–138 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Feher L., Klimcik C.: Poisson–Lie interpretation of trigonometric Ruijsenaars duality. Commun. Math. Phys. 301, 55–104 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Feher L., Klimcik C.: Self-duality of the compactified Ruijsenaars–Schneider system from quasi-Hamiltonian reduction. Nucl. Phys. B 860, 464–515 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Pusztai B.G.: The hyperbolic BC(n) Sutherland and the rational BC(n) Ruijsenaars–Schneider–van Diejen models: lax matrices and duality. Nucl. Phys. B 856(2), 528–551 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Feher L., Ayadi V.: Trigonometric Sutherland systems and their Ruijsenaars duals from symplectic reduction. J. Math. Phys. 51, 103511 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  22. Feher L., Gorbe T.F.: Duality between the trigonometric BC(n) Sutherland system and a completed rational Ruijsenaars–Schneider–van Diejen system. J. Math. Phys. 55, 102704 (2014)

    Article  ADS  Google Scholar 

  23. Feher L., Tsutsui I.: Regularization of Toda lattices by Hamiltonian reduction. J. Geom. Phys. 21, 97–135 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. van Diejen J.F.: Deformations of Calogero–Moser systems. Theor. Math. Phys. 99, 549–554 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  25. Schneider H.: Integrable relativistic N-particle systems in an external potential. Physica D 26, 203–209 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Semenov-Tian-Shansky M.A.: Dressing transformations and Poisson group actions. Publ. Res. Inst. Math. Sci. 21(6), 1237–1260 (1985)

    Article  MathSciNet  Google Scholar 

  28. Abraham R., Marsden J.E.: Foundations of Mechanics, 2nd edn. Addison-Wesley, Boston (1978)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Higher School of Economics, Ulitsa Vavilova 7, Moscow, Russia

    Ian Marshall

Authors
  1. Ian Marshall
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ian Marshall.

Additional information

Communicated by N. Reshetikhin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Marshall, I. A New Model in the Calogero–Ruijsenaars Family. Commun. Math. Phys. 338, 563–587 (2015). https://doi.org/10.1007/s00220-015-2388-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-015-2388-7

Keywords

Search

Navigation