Symmetries of Spin Calogero Models

TL;DR

This paper explores the symmetry algebras of integrable spin Calogero models, revealing that these algebras depend on the spin representation and are related to half-loop algebras, challenging previous assumptions.

Contribution

It demonstrates that symmetry algebras of spin Calogero models vary with spin representations, providing explicit examples and extending results to all finite Coxeter groups.

Findings

Different symmetry algebras for B_L and G_2 models

Symmetry algebras related to half-loop and twisted algebras

Extension of results to all finite Coxeter groups

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.