
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 24, 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 B_{L} spin Calogero model and three for G_{2} spin Calogero model. They are all related to the
halfloop algebra and its twisted versions. Some of the result are extended to any
finite Coxeter group.
Key words:
Calogero models; symmetry algebra; twisted halfloop algebra.
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References
 Dunkl C.F.,
Differentialdifference operators associated to reflection groups,
Trans. Amer. Math. Soc. 311 (1989), 167183.
 Calogero F.,
Solution of a threebody problem in one dimension,
J. Math. Phys. 10 (1969), 21912196.
Calogero F.,
Ground state of a onedimensional Nbody system,
J. Math. Phys. 10 (1969), 21972200.
Calogero F., Solution of the onedimensional Nbody problems
with quadratic and/or inversely quadratic pair potentials,
J. Math. Phys. 12 (1971), 419436, Erratum, J. Math. Phys. 37 (1996), 3646.
 Sutherland B.,
Quantum manybody problem in one dimension: ground state,
J. Math. Phys. 12 (1971), 246250.
Sutherland B., Quantum manybody problem in one dimension: thermodynamics,
J. Math. Phys. 12 (1971), 251256.
Sutherland B., Exact results for a quantum manybody problem in one dimension,
Phys. Rev. A 4 (1971), 20192021.
 Olshanetsky M.A., Perelomov A.M.,
Quantum integrable systems related to Lie algebras,
Phys. Rep. 94 (1983), 313404.
 Polychronakos A.P.,
Exchange operator formalism for integrable systems of particles,
Phys. Rev. Lett. 69, 703705.
 Minahan J.A., Polychronakos A.P.,
Integrable systems for particles with internal degrees of freedom,
Phys. Lett. B 302 (1993), 265270, hepth/9206046.
 Hikami K., Wadati M.,
Integrability of CalogeroMoser spin system,
J. Phys. Soc. Japan 62 (1993), 469472.
 Bernard D., Gaudin M., Haldane F.D.M., Pasquier V.,
YangBaxter equation in longrange interacting systems,
J. Phys. A: Math. Gen. 26 (1993), 52195236, hepth/9301084.
 Faddeev L.D., Reshetikhin N.Y., Takhtajan L.A.,
Quantization of Lie groups and Lie algebras,
Leningrad Math. J. 1 (1990), 193225.
 Drinfel'd V.G., Hopf algebras and the quantum YangBaxter equation,
Sov. Math. Dokl. 32 (1985), 254258.
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, 798820.
 Takemura K., Uglov D.,
The orthogonal eigenbasis and norms of eigenvectors in the spin CalogeroSutherland model,
J. Phys. A: Math. Gen. 30 (1997), 36853717, solvint/9611006.
 Takemura K.,
The Yangian symmetry in the spin Calogero model and its applications,
J. Phys. A: Math. Gen. 30 (1997), 61856204, solvint/9701015.
 Uglov D.,
Yangian GelfandZetlin bases, gl_{N}Jack polynomials and computation of dynamical correlation functions in the spin CalogeroSutherland model,
Comm. Math. Phys. 191 (1998), 663696, hepth/9702020.
 Ha Z.N.C., Haldane F.D.M.,
On models with inversesquare exchange,
Phys. Rev. B 46 (1992), 93599368, condmat/9204017.
 Haldane F.D.M., Ha Z.N.C., Talstra J.C., Bernard D., Pasquier V.,
Yangian symmetry of integrable quantum chains with longrange interactions and a new description of states in conformal field theory,
Phys. Rev. Lett. 69 (1992), 20212025.
 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,
hepth/9311013.
 Hikami K.,
Yangian symmetry and Virasoro character in a lattice spin system with longrange interactions,
Nuclear Phys. B 441 (1995), 530548.
 Murakami S., Wadati M.,
Connection between Yangian symmetry and the quantum inverse scattering method,
J. Phys. A: Math. Gen. 29 (1996), 79037915.
 Mintchev M., Ragoucy E., Sorba P., Zaugg Ph.,
Yangian symmetry in the nonlinear Schrödinger hierarchy,
J. Phys. A: Math. Gen. 32 (1999), 58855900, hepth/9905105.
 Uglov D.B., Korepin V.E.,
The Yangian symmetry of the Hubbard model,
Phys. Lett. A 190 (1994), 238242, hepth/9310158.
 Sklyanin E.K.,
Boundary conditions for integrable quantum systems,
J. Phys. A: Math. Gen. 21 (1988), 23752389.
 Caudrelier V., Crampé N.,
Integrable Nparticle Hamiltonians with Yangian or reflection algebra symmetry,
J. Phys. A: Math. Gen. 7 (2004), 62856298, mathph/0310028.
 Humphreys J.H., Reflection groups and Coxeter groups,
Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.
 Belavin A.A., Drinfel'd V.G., Solutions of the classical YangBaxter equation for simple Lie algebras,
Funct. Anal. Appl. 16 (1982), 159180.
Belavin A.A., Drinfel'd V.G., Classical YangBaxter equation for simple Lie algebras,
Funct. Anal. Appl. 17 (1983), 220221.
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, 93165.
 Crampé N., Young C.A.S.,
Integrable models from twisted halfloop algebras,
J. Phys. A: Math. Theor. 40 (2007), 54915509, mathph/0609057.
 Yamamoto T.,
Multicomponent Calogero model of B_{N}type confined in a harmonic potential,
Phys. Lett. A 208 (1995), 293302, condmat/9508012.
 Inozemtsev V.I., Sasaki R.,
Universal Lax pairs for spin CalogeroMoser models and spin exchange models,
J. Phys. A: Math. Gen. 34 (2001), 76217632, hepth/0106164.
 Quesne C.,
An exactly solvable threeparticle problem with threebody interaction,
Phys. Rev. A 55 (1997), 39313934, hepth/9612173.
 Nazarov M., Tarasov V.,
Yangians and GelfandZetlin bases,
Publ. Res. Inst. Math. Sci. 30 (1994), 459478, hepth/9302102.
 Talalaev D.,
Quantization of the Gaudin system, hepth/0404153.

