4,761 research outputs found
(1+1) Schrodinger Lie bialgebras and their Poisson-Lie groups
All Lie bialgebra structures for the (1+1)-dimensional centrally extended
Schrodinger algebra are explicitly derived and proved to be of the coboundary
type. Therefore, since all of them come from a classical r-matrix, the complete
family of Schrodinger Poisson-Lie groups can be deduced by means of the
Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended
Galilei and gl(2) Lie bialgebras within the Schrodinger classification are
studied. As an application, new quantum (Hopf algebra) deformations of the
Schrodinger algebra, including their corresponding quantum universal
R-matrices, are constructed.Comment: 25 pages, LaTeX. Possible applications in relation with integrable
systems are pointed; new references adde
Quantum (1+1) extended Galilei algebras: from Lie bialgebras to quantum R-matrices and integrable systems
The Lie bialgebras of the (1+1) extended Galilei algebra are obtained and
classified into four multiparametric families. Their quantum deformations are
obtained, together with the corresponding deformed Casimir operators. For the
coboundary cases quantum universal R-matrices are also given. Applications of
the quantum extended Galilei algebras to classical integrable systems are
explicitly developed.Comment: 16 pages, LaTeX. A detailed description of the construction of
integrable systems is carried ou
Integrable geodesic motion on 3D curved spaces from non-standard quantum deformations
The link between 3D spaces with (in general, non-constant) curvature and
quantum deformations is presented. It is shown how the non-standard deformation
of a sl(2) Poisson coalgebra generates a family of integrable Hamiltonians that
represent geodesic motions on 3D manifolds with a non-constant curvature that
turns out to be a function of the deformation parameter z. A different
Hamiltonian defined on the same deformed coalgebra is also shown to generate a
maximally superintegrable geodesic motion on 3D Riemannian and (2+1)D
relativistic spaces whose sectional curvatures are all constant and equal to z.
This approach can be generalized to arbitrary dimension.Comment: 7 pages. Communication presented at the 14th Int. Colloquium on
Integrable Systems 14-16 June 2005, Prague, Czech Republi
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
Non-coboundary Poisson-Lie structures on the book group
All possible Poisson-Lie (PL) structures on the 3D real Lie group generated
by a dilation and two commuting translations are obtained. Its classification
is fully performed by relating these PL groups with the corresponding Lie
bialgebra structures on the corresponding "book" Lie algebra. By construction,
all these Poisson structures are quadratic Poisson-Hopf algebras for which the
group multiplication is a Poisson map. In contrast to the case of simple Lie
groups, it turns out that most of the PL structures on the book group are
non-coboundary ones. Moreover, from the viewpoint of Poisson dynamics, the most
interesting PL book structures are just some of these non-coboundaries, which
are explicitly analysed. In particular, we show that the two different
q-deformed Poisson versions of the sl(2,R) algebra appear as two distinguished
cases in this classification, as well as the quadratic Poisson structure that
underlies the integrability of a large class of 3D Lotka-Volterra equations.
Finally, the quantization problem for these PL groups is sketched.Comment: 15 pages, revised version, some references adde
Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature
An infinite family of classical superintegrable Hamiltonians defined on the
N-dimensional spherical, Euclidean and hyperbolic spaces are shown to have a
common set of (2N-3) functionally independent constants of the motion. Among
them, two different subsets of N integrals in involution (including the
Hamiltonian) can always be explicitly identified. As particular cases, we
recover in a straightforward way most of the superintegrability properties of
the Smorodinsky-Winternitz and generalized Kepler-Coulomb systems on spaces of
constant curvature and we introduce as well new classes of (quasi-maximally)
superintegrable potentials on these spaces. Results here presented are a
consequence of the sl(2) Poisson coalgebra symmetry of all the Hamiltonians,
together with an appropriate use of the phase spaces associated to Poincare and
Beltrami coordinates.Comment: 12 page
Finite Size Scaling and ``perfect'' actions: the three dimensional Ising model
Using Finite-Size Scaling techniques, we numerically show that the first
irrelevant operator of the lattice theory in three dimensions
is (within errors) completely decoupled at . This interesting
result also holds in the Thermodynamical Limit, where the renormalized coupling
constant shows an extraordinary reduction of the scaling-corrections when
compared with the Ising model. It is argued that Finite-Size Scaling analysis
can be a competitive method for finding improved actions.Comment: 13 pages, 3 figure
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
- âŠ