
SIGMA 4 (2008), 090, 15 pages arXiv:0809.3948
http://dx.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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