A Casimir operator for a Calogero \(W\) algebra

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 \(W'_N\).