For the quantum Calogero-Moser model, we construct a set of conserved operators and another set of operators, named boost operators, from its Lax operator. We prove that each conserved operator satisfies both the Lax equation and a remarkable relation named additional relation. From these knowledge, we show that the conserved operators are involutive. Moreover, the conserved operators and the boost operators constitute the U(1)-current algebra. All the proofs are simplified a great deal due to the Lax equations and additional relations.
For any root system ∆ and an irreducible representation R of the reflection (Weyl) group G ∆ generated by ∆, a spin Calogero-Moser model can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member µ of R, to be called a “site”, we associate a vector space Vµ whose element is called a “spin”. Its dynamical variables are the canonical coordinates {qj,pj} of a particle in R r, (r = rank of ∆), and spin exchange operators { ˆ Pρ} (ρ ∈ ∆) which exchange the spins at the sites µ and sρ(µ). Here sρ is the reflection generated by ρ. For each ∆ and R a spin exchange model can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by “freezing ” the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For ∆ = Ar and R = vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials.
For any root system
Δ and an irreducible representation
R of the reflection (Weyl) group
GΔ generated by
Δ, a {\em spin Calogero-Moser model} can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member
μ of
R, to be called a "site", we associate a vector space
Vμ whose element is called a "spin". Its dynamical variables are the canonical coordinates
{qj,pj} of a particle in
Rr, (
r= rank of
Δ), and spin exchange operators
{P^ρ} (
ρ∈Δ) which exchange the spins at the sites
μ and
sρ(μ). Here
sρ is the reflection generated by
ρ. For each
Δ and
R a {\em spin exchange model} can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by "freezing" the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For
Δ=Ar and
R= vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for {\em degenerate} potentials.
An algebraic construction of the eigenstates for the quantum Calogero model is investigated. Extending the method of Lapointe and Vinet, we construct the eigenstates for the second conserved operator of the quantum Calogero model. All the eigenstates can be factorized into symmetric polynomials which we call “Hi-Jack symmetric polynomials” and the ground state wave function. The conjectured formula for the eigenvalue of the second conserved operator is confirmed. The Hi-Jack polynomials are strong candidates for the orthogonal basis of the quantum Calogero model.
An algebraic construction more general and intimately connected with that of Faddeev1, along with its application for generating different classes of quantum integrable models are summarised to complement the recent results of ref. 1 ( L.D. Faddeev, Int. J. Mod. Phys. A10, 1845 (1995)). 1 A significant portion of an excellent recent review by Faddeev 1 is devoted to an important result showing deep relation between the Yang–Baxter equation and the quantum group. We are delighted to note that a scheme for systematic construction of the Lax operators through Yang–Baxterisation of the Faddeev–Reshetikhin–Takhtajan (FRT) algebra persued by us over the last few years 2,3 has an intimate connection with this approach. In fact our formulation gives a more general framework allowing construction of a wide range of Lax operators of lattice models and recovers the corresponding result of ref 1 related to Uqsl(2) as a particular case. Another convincing demonstration of the usefulness of our approach is that, the well known Lax operator of the lattice Liouville model 4 as well as the recently discovered 5 nontrivial spectral parameter dependent variant of it can be constructed directly and easily from our general form,