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


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 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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