Dunkl symplectic algebra in generalized Calogero models

TL;DR

This paper explores the deformed symplectic algebra using Dunkl operators in generalized Calogero models, revealing new symmetry structures and classifying wavefunctions through conformal multiplets.

Contribution

It introduces a detailed analysis of Dunkl symplectic algebra in Calogero models, including explicit relations and classification of wavefunctions via conformal multiplets.

Findings

Deformed symplectic algebra contains a subalgebra with deformed unitary and nondeformed sl(2,R) symmetries.

Wavefunctions are classified by infinite-dimensional sl(2,R) multiplets.

Polynomial integrals generate finite-dimensional conformal multiplets.

Abstract

We study the properties of the symplectic sp(2N) algebra deformed using Dunkl operators, which describe the dynamical symmetry of the generalized N-particle quantum Calogero model. It contains a symmetry subalgebra formed by the deformed unitary generators as well as the (nondeformed) sl(2,R) conformal subalgebra. An explicit relation among the deformed symplectic generators is derived. Based on the matching between the Casimir elements of the conformal spin and Dunkl angular momentum algebras, the independent wavefunctions of the both the standard and generalized Calogero models, expressed in terms of the deformed spherical harmonics, are classified according to infinite-dimensional lowest-state sl(2,R) multiplets. Meanwhile, any polynomial integral of motion of the (generalized) Calogero-Moser model generates a finite-dimensional highest-state conformal multiplet with descendants expressed via the Weyl-ordered product in quantum field theory.