A Casimir operator for a Calogero $W$ algebra

TL;DR

This work constructs and analyzes nonlinear $W$ algebras arising from permutation invariant observables of the rational Calogero model, focusing on the three-particle case. After decoupling the center of mass, the relative algebra $W'_3$ is organized into a spin-1 and a spin-3/2 $sl(2)$ representation, with level-3 commutators generating a central term that depends on the Calogero coupling $g$. The authors identify the lowest Casimir ${\cal C}_6$ of $W'_3$ and its quantum deformation ${\cal C}_6^{\rm quant}$ for the full $W_3$, providing explicit Weyl-ordered expressions and showing how ${\cal C}_6^{\rm quant}$ encodes both $\hbar$ and $g$ through the parameter $\lambda=g(g-1)$. They derive these Casimirs using nested Weyl orderings, discuss classical and free-particle limits, and outline extensions to general $N$ and higher Casimirs. The results offer a concrete route to classifying irreducible representations of Calogero induced nonlinear $W$ algebras and demonstrate a scalable computational approach for higher $W_N$ structures.

Abstract

We investigate the nonlinear algebra $W_3$ generated by the 9 functionally independent permutation-symmetric operators in the three-particle rational quantum Calogero model. Decoupling the center of mass, we pass to a smaller algebra $W'_3$ generated by 7 operators, which fall into a spin-$1$ and a spin-$\frac32$ representation of the conformal $sl(2)$ subalgebra. The commutators of the spin-$\frac32$ generators with each other are quadratic in the spin-$1$ generators, with a central term depending on the Calogero coupling. One expects this algebra to feature three Casimir operators, and we construct the lowest one explicitly in terms of Weyl-ordered products of the 7 generators. It is a polynomial of degree 6 in these generators, with coefficients being up to quartic in $\hbar$ and quadratic polynomials in the Calogero coupling $\hbar^2g(g{-}1)$. Putting back the center of mass, our Casimir operator for $W_3$ is a degree-9 polynomial in the 9 generators. The computations require the evaluation of nested Weyl orderings. The classical and free-particle limits are also given. Our scheme can be extended to any finite number $N$ of Calogero particles and the corresponding nonlinear algebras $W_N$ and $W'_N$.