Dunkl Operators at Infinity and Calogero–Moser Systems

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.

On spin Calogero–Moser system at infinity

“…Theorem The representation (11) of Yangian Y (gl s ) commutes with the Hamiltonians of the CS system. The algebra of higher Hamiltonians coincides with the center of the Yangian, which is generated by the coefficients of the quantum determinant…”

“…Here we review recent results on the scalar CS system [8,11] mainly following the approach of [8]. The main idea is to regard the equivariant Heckman-Dunkl operators as a quantum L-operator acting on the space of polynomial functions of one variable with coefficients being symmetric polynomials of the remaining N − 1 variables.…”

“…There is a difference between the definitions (15) and (24) for the vertex operatorsṼ (x) and i , see [8] and [11].…”

“…In [10] A.Veselov and A.Sergeev suggested to define a bosonic limit of the CS system as a projective limit of finite models. Precise bosonic constructions of higher Hamiltonians in a Fock space were then presented by M.Nazarov and E.Sklyanin in [8] and by A.Veselov and A.Sergeev in [11]. It was also shown in [8] that the classical limit of the infinite system can be identified with the Benjamin-Ono hierarchy whose Lax presentation can be derived from the suggested description of the quantum system.…”

Fermionic limit of the Calogero-Sutherland system

“…This paper can be regarded as a further development of the latter ideas to the CS systems restricted to the antisymmetric wave functions. The both approaches [13], [16] in the bosonic case regard the space C[x 1 ] ⊗ Λ (+) [x 2 , . .…”