126 research outputs found
Algebraic extensions of Gaudin models
We perform a In\"on\"u--Wigner contraction on Gaudin models, showing how the
integrability property is preserved by this algebraic procedure. Starting from
Gaudin models we obtain new integrable chains, that we call Lagrange chains,
associated to the same linear -matrix structure. We give a general
construction involving rational, trigonometric and elliptic solutions of the
classical Yang-Baxter equation. Two particular examples are explicitly
considered: the rational Lagrange chain and the trigonometric one. In both
cases local variables of the models are the generators of the direct sum of
interacting tops.Comment: 15 pages, corrected formula
B\"acklund transformations for the rational Lagrange chain
We consider a long--range homogeneous chain where the local variables are the
generators of the direct sum of interacting Lagrange
tops. We call this classical integrable model rational ``Lagrange chain''
showing how one can obtain it starting from rational Gaudin
models. Moreover we construct one- and two--point integrable maps (B\"acklund
transformations).Comment: 12 page
Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature
An infinite family of quasi-maximally superintegrable Hamiltonians with a
common set of (2N-3) integrals of the motion is introduced. The integrability
properties of all these Hamiltonians are shown to be a consequence of a hidden
non-standard quantum sl(2,R) Poisson coalgebra symmetry. As a concrete
application, one of this Hamiltonians is shown to generate the geodesic motion
on certain manifolds with a non-constant curvature that turns out to be a
function of the deformation parameter z. Moreover, another Hamiltonian in this
family is shown to generate geodesic motions on Riemannian and relativistic
spaces all of whose sectional curvatures are constant and equal to the
deformation parameter z. This approach can be generalized to arbitrary
dimension by making use of coalgebra symmetry.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium
on Non-Perturbative and Symmetry Methods in Field Theory (June 2006,
Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Superintegrable Deformations of the Smorodinsky-Winternitz Hamiltonian
A constructive procedure to obtain superintegrable deformations of the
classical Smorodinsky-Winternitz Hamiltonian by using quantum deformations of
its underlying Poisson sl(2) coalgebra symmetry is introduced. Through this
example, the general connection between coalgebra symmetry and quasi-maximal
superintegrability is analysed. The notion of comodule algebra symmetry is also
shown to be applicable in order to construct new integrable deformations of
certain Smorodinsky-Winternitz systems.Comment: 17 pages. Published in "Superintegrability in Classical and Quantum
Systems", edited by P.Tempesta, P.Winternitz, J.Harnad, W.Miller Jr.,
G.Pogosyan and M.A.Rodriguez, CRM Proceedings & Lecture Notes, vol.37,
American Mathematical Society, 200
Exact Solution of the Quantum Calogero-Gaudin System and of its q-Deformation
A complete set of commuting observables for the Calogero-Gaudin system is
diagonalized, and the explicit form of the corresponding eigenvalues and
eigenfunctions is derived. We use a purely algebraic procedure exploiting the
co-algebra invariance of the model; with the proper technical modifications
this procedure can be applied to the deformed version of the model, which
is then also exactly solved.Comment: 20 pages Late
Classical Dynamical Systems from q-algebras:"cluster" variables and explicit solutions
A general procedure to get the explicit solution of the equations of motion
for N-body classical Hamiltonian systems equipped with coalgebra symmetry is
introduced by defining a set of appropriate collective variables which are
based on the iterations of the coproduct map on the generators of the algebra.
In this way several examples of N-body dynamical systems obtained from
q-Poisson algebras are explicitly solved: the q-deformed version of the sl(2)
Calogero-Gaudin system (q-CG), a q-Poincare' Gaudin system and a system of
Ruijsenaars type arising from the same (non co-boundary) q-deformation of the
(1+1) Poincare' algebra. Also, a unified interpretation of all these systems as
different Poisson-Lie dynamics on the same three dimensional solvable Lie group
is given.Comment: 19 Latex pages, No figure
Gaudin models with {\CU}_q(\mathfrak{osp}(1 | 2)) symmetry
We consider a Gaudin model related to the q-deformed superalgebra
{\CU}_q(\mathfrak{osp}(1 | 2)). We present an exact solution to that system
diagonalizing a complete set of commuting observables, and providing the
corresponding eigenvectors and eigenvalues. The approach used in this paper is
based on the coalgebra supersymmetry of the model.Comment: 10 page
The spin 1/2 Calogero-Gaudin System and its q-Deformation
The spin 1/2 Calogero-Gaudin system and its q-deformation are exactly solved:
a complete set of commuting observables is diagonalized, and the corresponding
eigenvectors and eigenvalues are explicitly calculated. The method of solution
is purely algebraic and relies on the co-algebra simmetry of the model.Comment: 15 page
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