2,485 research outputs found
On exact solution of a classical 3D integrable model
We investigate some classical evolution model in the discrete 2+1 space-time.
A map, giving an one-step time evolution, may be derived as the compatibility
condition for some systems of linear equations for a set of auxiliary linear
variables. Dynamical variables for the evolution model are the coefficients of
these systems of linear equations. Determinant of any system of linear
equations is a polynomial of two numerical quasimomenta of the auxiliary linear
variables. For one, this determinant is the generating functions of all
integrals of motion for the evolution, and on the other hand it defines a high
genus algebraic curve. The dependence of the dynamical variables on the
space-time point (exact solution) may be expressed in terms of theta functions
on the jacobian of this curve. This is the main result of our paper
Quantum Calogero-Moser systems: a view from infinity
Various infinite-dimensional versions of Calogero-Moser operator are
discussed in relation with the theory of symmetric functions and representation
theory of basic classical Lie superlagebras. This is a version of invited talk
given by the second author at XVI International Congress on Mathematical
Physics in Prague, August 2009.Comment: 6 pages, to appear in Proceedings of XVI International Congress on
Mathematical Physics, Prague, August 200
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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