This article is cited in 4 papers

Symmetries of Spin Calogero Models

Vincent Caudrelier^a, Nicolas Crampé<sup>b

a</sup> 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

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 half-loop algebra and its twisted versions. Some of the result are extended to any finite Coxeter group.

Keywords: Calogero models; symmetry algebra; twisted half-loop algebra.

MSC: 70H06; 81R12; 81R50

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

Language: English

DOI: 10.3842/SIGMA.2008.090

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ArXiv: 0809.3948