Abstract
We define the Dunkl and Dunkl-Heckman operators in infinite number of variables and use them to construct the quantum integrals of the Calogero-Moser-Sutherland problems at infinity. As a corollary we have a simple proof of integrability of the deformed quantum CMS systems related to classical Lie superalgebras. We show how this naturally leads to a quantum version of the Moser matrix, which in the deformed case was not known before.
Highlights
The usual Calogero–Moser, or Calogero–Moser–Sutherland (CMS), system describes the interaction of N particles with equal masses on the line with the inverse square potential or, in the trigonometric version, with the inverse sin2 potential [6]
We show that it does not allow to construct the Dunkl operators in the deformed case, it naturally leads to the quantum Moser matrix for the deformed CMS system
This gives an interpretation of the integrals of the deformed CMS systems from [21] in terms of the quantum Lax pair, which, for the usual CMS system was first considered by Ujino et al [31] and Wadati et al [32]
Summary
The usual Calogero–Moser, or Calogero–Moser–Sutherland (CMS), system describes the interaction of N particles with equal masses on the line with the inverse square potential or, in the trigonometric version, with the inverse sin potential [6]. We show that it does not allow to construct the Dunkl operators in the deformed case, it naturally leads to the quantum Moser matrix for the deformed CMS system. This gives an interpretation of the integrals of the deformed CMS systems from [21] in terms of the quantum Lax pair, which, for the usual CMS system was first considered by Ujino et al [31] and Wadati et al [32]. We present similar formulae in the BC case as well Another result of the paper is the new formulae for the quantum CMS integrals at infinity in both rational and trigonometric cases for types A and BC. We comment on the relation with our results in the last section
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