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Dunkl operators at infinity and Calogero-Moser systems
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.Comment: 22 pages. Corrected version with minor change
Deformed quantum Calogero-Moser problems and Lie superalgebras
The deformed quantum Calogero-Moser-Sutherland problems related to the root
systems of the contragredient Lie superalgebras are introduced. The
construction is based on the notion of the generalized root systems suggested
by V. Serganova. For the classical series a recurrent formula for the quantum
integrals is found, which implies the integrability of these problems. The
corresponding algebras of the quantum integrals are investigated, the explicit
formulas for their Poincare series for generic values of the deformation
parameter are presented.Comment: 30 pages, 1 figur
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