Skip to main content
Log in

Deformed Quantum Calogero-Moser Problems and Lie Superalgebras

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

Abstract

The deformed quantum Calogero-Moser-Sutherland problems related to the root systems of the contragredient Lie superalgebras are introduced. The construction is based on the notion of the generalized root systems suggested by V. Serganova. For the classical series a recurrent formula for the quantum integrals is found, which implies the integrability of these problems. The corresponding algebras of the quantum integrals are investigated, the explicit formulas for their Poincare series for generic values of the deformation parameter are presented.

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.

Institutional subscriptions

Similar content being viewed by others

References

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

    MATH  Google Scholar 

  2. Sutherland, B.: Exact results for a quantum many-body problem in one dimension. Phys. Rev. A 4, 2019–2021 (1971)

    Article  Google Scholar 

  3. Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Rep. 94, 313–404 (1983)

    Article  MathSciNet  Google Scholar 

  4. Olshanetsky, M.A., Perelomov, A.M.: Quantum systems related to root systems and radial parts of Laplace operators. Funct. Anal. Appl. 12, 121–128 (1978)

    Google Scholar 

  5. Berezin, F.A., Pokhil, G.P., Finkelberg, V.M.: Schrödinger equation for a system of one-dimensional particles with point interaction. Vestnik MGU 1, 21–28 (1964)

    MATH  Google Scholar 

  6. Helgason, S.: Groups and Geometric Analysis. London-New York: Academic Press, 1984

  7. Heckman, G.J., Opdam, E.M.: Root systems and hypergeometric functions I. Comp. Math. 64, 329–352 (1987)

    MathSciNet  MATH  Google Scholar 

  8. Heckman, G.J.: A remark on the Dunkl differential-difference operators. Progress in Math. 101, 181–191 (1991)

    MATH  Google Scholar 

  9. Veselov, A.P., Feigin, M.V., Chalykh, O.A.: New integrable deformations of quantum Calogero- Moser problem. Russ. Math. Surv. 51(3), 185–186 (1996)

    MATH  Google Scholar 

  10. Chalykh, O.A., Feigin, M.V., Veselov, A.P.: New integrable generalizations of Calogero-Moser quantum problem. J. Math. Phys. 39(2), 695–703 (1998)

    Article  MATH  Google Scholar 

  11. Chalykh, O.A., Feigin, M.V., Veselov, A.P.: Multidimensional Baker-Akhiezer Functions and Huygens’ Principle. Commun. Math. Phys. 206, 533–566 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Veselov, A.P.: Deformations of the root systems and new solutions to generalized WDVV equations. Phys. Lett. A 261, 297–302 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sergeev, A.N.: Superanalogs of the Calogero operators and Jack polynomials. J. Nonlin. Math. Phys. 8(1), 59–64 (2001)

    MATH  Google Scholar 

  14. Sergeev, A.N.: Calogero operator and Lie superalgebras. Theor. Math. Phys. 131(3), 747–764 (2002)

    Article  Google Scholar 

  15. Serganova, V.: On generalization of root system. Commun. in Algebra 24(13), 4281–4299 (1996)

    MATH  Google Scholar 

  16. Bourbaki, N.: Groupes et algèbres de Lie. Chap. VI, Paris: Masson, 1981

  17. Kac, V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)

    MATH  Google Scholar 

  18. Berest, Yu.: Private communication to A.P. Veselov. October 1997

  19. Matsuo, A.: Integrable connections related to zonal spherical functions. Invent. Math. 110, 95–121 (1992)

    MathSciNet  MATH  Google Scholar 

  20. Veselov, A.P.: On generalisations of the Calogero-Moser-Sutherland quantum problem and WDVV equations. J. Math. Phys. 43(11), 5675–5682 (2002)

    Article  Google Scholar 

  21. Chalykh, O.A., Veselov, A.P.: Locus configurations and ∨-systems. Phys. Lett. A 285, 339–349 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chalykh, O.A., Veselov, A.P.: Commutative rings of partial differential operators and Lie algebras. Commun. Math. Phys. 126, 597–611 (1990)

    MathSciNet  MATH  Google Scholar 

  23. Feigin, M., Veselov, A.P.: Quasi-invariants of Coxeter groups and m-harmonic polynomials. Int. Math. Res. Notices 10, 521–545 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Macdonald, I.: Symmetric functions and Hall polynomials. 2nd edition, Oxford: Oxford Univ. Press, 1995

  25. Oshima, T.: Completely integrable systems with a symmmetry in coordinates. Asian J. Math. 2, 935–955 (1998)

    MathSciNet  MATH  Google Scholar 

  26. Berezin, F.A.: Laplace operators on semisimple Lie groups. Proc. Moscow Math. Soc. 6, 371–463 (1971) (Russian)

    MATH  Google Scholar 

  27. Kerov, S., Okounkov, A., Olshanski, G.: The boundary of the Young graph with Jack edge multipliers. Intern. Math. Res. Notices 4, 173–199 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Okounkov, A.: On N-point correlations in the log-gas at rational temperature. hep-th/9702001

  29. Orellana, R.C., Zabrocki, M.: Some remarks on the characters of the general Lie superalgebra. math.CO/0008152

  30. Atiyah, M., Macdonald, I.G.: Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1969

  31. Fulton, W., Pragacz, P.: Schubert Varieties and Degeneracy Loci. Lect. Notes in Math. 1689, Berlin Heidelberg-New York: Springer, 1998

  32. Feigin, M., Veselov, A.P.: Quasi-invariants and quantum integrals of the deformed Calogero-Moser systems. Intern. Math. Research Notices 46, 2487–2511 (2003)

    Article  Google Scholar 

  33. Khodarinova, L.A.: On quantum elliptic Calogero-Moser problem. Vestnik MGU, Ser.I Math. Mech. 5, 16–19 (1998)

    Google Scholar 

  34. Khodarinova, L.A., Prikhodsky, I.A.: On algebraic integrability of the deformed elliptic Calogero-Moser problem. J. Nonlin. Math. Phys. 8(1), 50–53 (2001)

    MATH  Google Scholar 

  35. Chalykh, O., Etingof, P., Oblomkov, A.: Generalized Lamé operators. Commun. Math. Phys. 239, 115–153 (2003)

    MATH  Google Scholar 

  36. Ruijsenaars, S.N.M.: Complete integrability of relativistic Calogero-Moser systems and elliptic functions identities. Commun. Math. Phys. 110, 191–213 (1987)

    MathSciNet  MATH  Google Scholar 

  37. Chalykh, O.: Macdonald polynomials and algebraic integrability. Adv. Math. 166, 193–259 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Sergeev, A.N., Veselov, A.P.: Generalised discriminants, deformed quantum Calogero-Moser system and Jack polynomials. math-ph/ 0307036

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A.P. Veselov.

Additional information

Communicated by L. Takhtajan

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sergeev, A., Veselov, A. Deformed Quantum Calogero-Moser Problems and Lie Superalgebras. Commun. Math. Phys. 245, 249–278 (2004). https://doi.org/10.1007/s00220-003-1012-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-003-1012-4

Keywords

Navigation