Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 090, 15 pages      arXiv:0809.3948      https://doi.org/10.3842/SIGMA.2008.090
Contribution to the Special Issue on Dunkl Operators and Related Topics

Symmetries of Spin Calogero Models

Vincent Caudrelier a and Nicolas Crampé b
a) Centre for Mathematical Science, City University, Northampton Square, London, EC1V 0HB, United Kingdom
b) International School for Advanced Studies, Via Beirut 2-4, 34014 Trieste, Italy

Received September 24, 2008, in final form December 17, 2008; Published online December 23, 2008

Abstract
We investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra to a given Coxeter group W is wrong. More precisely, the symmetry algebra heavily depends on the representation of W on the spins. We prove this by identifying two different symmetry algebras for a BL spin Calogero model and three for G2 spin Calogero model. They are all related to the half-loop algebra and its twisted versions. Some of the result are extended to any finite Coxeter group.

Key words: Calogero models; symmetry algebra; twisted half-loop algebra.

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References

  1. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  2. Calogero F., Solution of a three-body problem in one dimension, J. Math. Phys. 10 (1969), 2191-2196.
    Calogero F., Ground state of a one-dimensional N-body system, J. Math. Phys. 10 (1969), 2197-2200.
    Calogero F., Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436, Erratum, J. Math. Phys. 37 (1996), 3646.
  3. Sutherland B., Quantum many-body problem in one dimension: ground state, J. Math. Phys. 12 (1971), 246-250.
    Sutherland B., Quantum many-body problem in one dimension: thermodynamics, J. Math. Phys. 12 (1971), 251-256.
    Sutherland B., Exact results for a quantum many-body problem in one dimension, Phys. Rev. A 4 (1971), 2019-2021.
  4. Olshanetsky M.A., Perelomov A.M., Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.
  5. Polychronakos A.P., Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69, 703-705.
  6. Minahan J.A., Polychronakos A.P., Integrable systems for particles with internal degrees of freedom, Phys. Lett. B 302 (1993), 265-270, hep-th/9206046.
  7. Hikami K., Wadati M., Integrability of Calogero-Moser spin system, J. Phys. Soc. Japan 62 (1993), 469-472.
  8. Bernard D., Gaudin M., Haldane F.D.M., Pasquier V., Yang-Baxter equation in long-range interacting systems, J. Phys. A: Math. Gen. 26 (1993), 5219-5236, hep-th/9301084.
  9. Faddeev L.D., Reshetikhin N.Y., Takhtajan L.A., Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225.
  10. Drinfel'd V.G., Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985), 254-258.
    Drinfel'd V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
  11. Takemura K., Uglov D., The orthogonal eigenbasis and norms of eigenvectors in the spin Calogero-Sutherland model, J. Phys. A: Math. Gen. 30 (1997), 3685-3717, solv-int/9611006.
  12. Takemura K., The Yangian symmetry in the spin Calogero model and its applications, J. Phys. A: Math. Gen. 30 (1997), 6185-6204, solv-int/9701015.
  13. Uglov D., Yangian Gelfand-Zetlin bases, glN-Jack polynomials and computation of dynamical correlation functions in the spin Calogero-Sutherland model, Comm. Math. Phys. 191 (1998), 663-696, hep-th/9702020.
  14. Ha Z.N.C., Haldane F.D.M., On models with inverse-square exchange, Phys. Rev. B 46 (1992), 9359-9368, cond-mat/9204017.
  15. Haldane F.D.M., Ha Z.N.C., Talstra J.C., Bernard D., Pasquier V., Yangian symmetry of integrable quantum chains with long-range interactions and a new description of states in conformal field theory, Phys. Rev. Lett. 69 (1992), 2021-2025.
  16. Bernard D., Pasquier V., Serban D., A one dimensional ideal gas of spinons, or some exact results on the XXX spin chain with long range interaction, hep-th/9311013.
  17. Hikami K., Yangian symmetry and Virasoro character in a lattice spin system with long-range interactions, Nuclear Phys. B 441 (1995), 530-548.
  18. Murakami S., Wadati M., Connection between Yangian symmetry and the quantum inverse scattering method, J. Phys. A: Math. Gen. 29 (1996), 7903-7915.
  19. Mintchev M., Ragoucy E., Sorba P., Zaugg Ph., Yangian symmetry in the nonlinear Schrödinger hierarchy, J. Phys. A: Math. Gen. 32 (1999), 5885-5900, hep-th/9905105.
  20. Uglov D.B., Korepin V.E., The Yangian symmetry of the Hubbard model, Phys. Lett. A 190 (1994), 238-242, hep-th/9310158.
  21. Sklyanin E.K., Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen. 21 (1988), 2375-2389.
  22. Caudrelier V., Crampé N., Integrable N-particle Hamiltonians with Yangian or reflection algebra symmetry, J. Phys. A: Math. Gen. 7 (2004), 6285-6298, math-ph/0310028.
  23. Humphreys J.H., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.
  24. Belavin A.A., Drinfel'd V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl. 16 (1982), 159-180.
    Belavin A.A., Drinfel'd V.G., Classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl. 17 (1983), 220-221.
    Belavin A.A., Drinfel'd V.G., Triangle equation and simple Lie algebras, Soviet Sci. Rev. Sect. C Math. Phys. Rev., Mathematical Physics Reviews, Vol. 4, Harwood Academic Publ., Chur, 1984, 93-165.
  25. Crampé N., Young C.A.S., Integrable models from twisted half-loop algebras, J. Phys. A: Math. Theor. 40 (2007), 5491-5509, math-ph/0609057.
  26. Yamamoto T., Multicomponent Calogero model of BN-type confined in a harmonic potential, Phys. Lett. A 208 (1995), 293-302, cond-mat/9508012.
  27. Inozemtsev V.I., Sasaki R., Universal Lax pairs for spin Calogero-Moser models and spin exchange models, J. Phys. A: Math. Gen. 34 (2001), 7621-7632, hep-th/0106164.
  28. Quesne C., An exactly solvable three-particle problem with three-body interaction, Phys. Rev. A 55 (1997), 3931-3934, hep-th/9612173.
  29. Nazarov M., Tarasov V., Yangians and Gelfand-Zetlin bases, Publ. Res. Inst. Math. Sci. 30 (1994), 459-478, hep-th/9302102.
  30. Talalaev D., Quantization of the Gaudin system, hep-th/0404153.


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