432 research outputs found
On the classical r-matrix structure of the rational BC(n) Ruijsenaars-Schneider-van Diejen system
In this paper, we construct a quadratic r-matrix structure for the classical
rational BC(n) Ruijsenaars-Schneider-van Diejen system with the maximal number
of three independent coupling parameters. As a byproduct, we provide a Lax
representation of the dynamics as well.Comment: 36 page
Generalized spin Sutherland systems revisited
We present generalizations of the spin Sutherland systems obtained earlier by
Blom and Langmann and by Polychronakos in two different ways: from SU(n)
Yang--Mills theory on the cylinder and by constraining geodesic motion on the
N-fold direct product of SU(n) with itself, for any N>1. Our systems are in
correspondence with the Dynkin diagram automorphisms of arbitrary connected and
simply connected compact simple Lie groups. We give a finite-dimensional as
well as an infinite-dimensional derivation and shed light on the mechanism
whereby they lead to the same classical integrable systems. The
infinite-dimensional approach, based on twisted current algebras (alias
Yang--Mills with twisted boundary conditions), was inspired by the derivation
of the spinless Sutherland model due to Gorsky and Nekrasov. The
finite-dimensional method relies on Hamiltonian reduction under twisted
conjugations of N-fold direct product groups, linking the quantum mechanics of
the reduced systems to representation theory similarly as was explored
previously in the N=1 case.Comment: 21 page
Derivations of the trigonometric BC(n) Sutherland model by quantum Hamiltonian reduction
The BC(n) Sutherland Hamiltonian with coupling constants parametrized by
three arbitrary integers is derived by reductions of the Laplace operator of
the group U(N). The reductions are obtained by applying the Laplace operator on
spaces of certain vector valued functions equivariant under suitable symmetric
subgroups of U(N)\times U(N). Three different reduction schemes are considered,
the simplest one being the compact real form of the reduction of the Laplacian
of GL(2n,C) to the complex BC(n) Sutherland Hamiltonian previously studied by
Oblomkov.Comment: 30 pages, LateX; v2: final version with minor stylistic modification
On the r-matrix structure of the hyperbolic BC(n) Sutherland model
Working in a symplectic reduction framework, we construct a dynamical
r-matrix for the classical hyperbolic BC(n) Sutherland model with three
independent coupling constants. We also examine the Lax representation of the
dynamics and its equivalence with the Hamiltonian equation of motion.Comment: 20 page
A note on a canonical dynamical r-matrix
It is well known that a classical dynamical -matrix can be associated with
every finite-dimensional self-dual Lie algebra \G by the definition
, where \omega\in \G and is the
holomorphic function given by for
z\in \C\setminus 2\pi i \Z^*. We present a new, direct proof of the statement
that this canonical -matrix satisfies the modified classical dynamical
Yang-Baxter equation on \G.Comment: 17 pages, LaTeX2
- …