3 research outputs found
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
(Super)integrability from coalgebra symmetry: formalism and applications
The coalgebra approach to the construction of classical integrable systems
from Poisson coalgebras is reviewed, and the essential role played by
symplectic realizations in this framework is emphasized. Many examples of
Hamiltonians with either undeformed or q-deformed coalgebra symmetry are given,
and their Liouville superintegrability is discussed. Among them,
(quasi-maximally) superintegrable systems on N-dimensional curved spaces of
nonconstant curvature are analysed in detail. Further generalizations of the
coalgebra approach that make use of comodule and loop algebras are presented.
The generalization of such a coalgebra symmetry framework to quantum mechanical
systems is straightforward.Comment: 33 pages. Review-contribution to the "Workshop on higher symmetries
in Physics", 6-8 November 2008, Madrid, Spai