296 research outputs found
New quantum (anti)de Sitter algebras and discrete symmetries
Two new quantum anti-de Sitter so(4,2) and de Sitter so(5,1) algebras are
presented. These deformations are called either time-type or space-type
according to the dimensional properties of the deformation parameter. Their
Hopf structure, universal R matrix and differential-difference realization are
obtained in a unified setting by considering a contraction parameter related to
the speed of light, which ensures a well defined non-relativistic limit. Such
quantum algebras are shown to be symmetry algebras of either time or space
discretizations of wave/Laplace equations on uniform lattices. These results
lead to a proposal fortime and space discrete Maxwell equations with quantum
algebra symmetry.Comment: 10 pages, LaTe
Lie bialgebra quantizations of the oscillator algebra and their universal --matrices
All coboundary Lie bialgebras and their corresponding Poisson--Lie structures
are constructed for the oscillator algebra generated by \{\aa,\ap,\am,\bb\}.
Quantum oscillator algebras are derived from these bialgebras by using the
Lyakhovsky and Mudrov formalism and, for some cases, quantizations at both
algebra and group levels are obtained, including their universal --matrices.Comment: 19 pages, LaTeX; revised version to appear in J. Phys. A;
quantization of bialgebras is complete
Integrable deformations of Hamiltonian systems and q-symmetries
The complete integrability of the hyperbolic Gaudin Hamiltonian and other
related integrable systems is shown to be easily derived by taking into account
their sl(2,R) coalgebra symmetry. By using the properties induced by such a
coalgebra structure, it can be proven that the introduction of any quantum
deformation of the sl(2,R) algebra will provide an integrable deformation for
such systems. In particular, the Gaudin Hamiltonian arising from the
non-standard quantum deformation of the sl(2,R) Poisson algebra is presented,
including the explicit expressions for its integrals of motion. A completely
integrable system of nonlinearly coupled oscillators derived from this
deformation is also introduced.Comment: 11 pages, LaTeX. Contribution to the III Classical and Quantum
Integrable Systems. Edited by L.G. Mardoyan, G.S. Pogosyan and A.N.
Sissakian. JINR, Dubna, pp. 15--25, (1998
The Kepler problem on 3D spaces of variable and constant curvature from quantum algebras
A quantum sl(2,R) coalgebra (with deformation parameter z) is shown to
underly the construction of superintegrable Kepler potentials on 3D spaces of
variable and constant curvature, that include the classical spherical,
hyperbolic and (anti-)de Sitter spaces as well as their non-constant curvature
analogues. In this context, the non-deformed limit z = 0 is identified with the
flat contraction leading to the proper Euclidean and Minkowskian
spaces/potentials. The corresponding Hamiltonians admit three constants of the
motion coming from the coalgebra structure. Furthermore, maximal
superintegrability of the Kepler potential on the spaces of constant curvature
is explicitly shown by finding an additional constant of the motion coming from
an additional symmetry that cannot be deduced from the quantum algebra. In this
way, the Laplace-Runge-Lenz vector for such spaces is deduced and its algebraic
properties are analysed.Comment: 12 pages. Communication presented at the Workshop in honour of Prof.
Jose F. Carinena, "Groups, Geometry and Physics", December 9-10, 2005,
Zaragoza (Spain
Long range integrable oscillator chains from quantum algebras
Completely integrable Hamiltonians defining classical mechanical systems of
coupled oscillators are obtained from Poisson realizations of
Heisenberg--Weyl, harmonic oscillator and coalgebras. Various
completely integrable deformations of such systems are constructed by
considering quantum deformations of these algebras. Explicit expressions for
all the deformed Hamiltonians and constants of motion are given, and the
long-range nature of the interactions is shown to be linked to the underlying
coalgebra structure. The relationship between oscillator systems induced from
the coalgebra and angular momentum chains is presented, and a
non-standard integrable deformation of the hyperbolic Gaudin system is
obtained.Comment: 17 pages, LaTe
Homogeneous phase spaces: the Cayley-Klein framework
The metric structure of homogeneous spaces of rank-one and rank-two
associated to the real pseudo-orthogonal groups SO(p,q) and some of their
contractions (e.g., ISO(p,q), Newton-Hooke type groups...) is studied. All
these spaces are described from a unified setting following a Cayley-Klein
scheme allowing to simultaneously study the main features of their Riemannian,
pesudoRiemannian and semiRiemannian metrics, as well as of their curvatures.
Some of the rank-one spaces are naturally interpreted as spacetime models.
Likewise, the same natural interpretation for rank-two spaces is as spaces of
lines in rank-one spaces; through this relation these rank-two spaces give rise
to homogeneous phase space models. The main features of the phase spaces for
homogeneous spacetimes are analysed.Comment: 20 pages, LaTeX; F.J.H. contribution to WOGDA'9
(Anti)de Sitter/Poincare symmetries and representations from Poincare/Galilei through a classical deformation approach
A classical deformation procedure, based on universal enveloping algebras,
Casimirs and curvatures of symmetrical homogeneous spaces, is applied to
several cases of physical relevance. Starting from the (3+1)D Galilei algebra,
we describe at the level of representations the process leading to its two
physically meaningful deformed neighbours. The Poincare algebra is obtained by
introducing a negative curvature in the flat Galilean phase space (or space of
worldlines), while keeping a flat spacetime. To be precise, starting from a
representation of the Galilei algebra with both Casimirs different from zero,
we obtain a representation of the Poincare algebra with both Casimirs
necessarily equal to zero. The Poincare angular momentum, Pauli-Lubanski
components, position and velocity operators, etc. are expressed in terms of
"Galilean" operators through some expressions deforming the proper Galilean
ones. Similarly, the Newton-Hooke algebras appear by endowing spacetime with a
non-zero curvature, while keeping a flat phase space. The same approach,
starting from the (3+1)D Poincare algebra provides representations of the
(anti)de Sitter as Poincare deformations.Comment: 19 pages, LaTeX.7. Comments and references adde
A new Lie algebra expansion method: Galilei expansions to Poincare and Newton-Hooke
We modify a Lie algebra expansion method recently introduced for the
(2+1)-dimensional kinematical algebras so as to work for higher dimensions.
This new improved and geometrical procedure is applied to expanding the
(3+1)-dimensional Galilei algebra and leads to its physically meaningful
`expanded' neighbours. One expansion gives rise to the Poincare algebra,
introducing a curvature in the flat Galilean space of worldlines,
while keeping a flat spacetime which changes from absolute to relative time in
the process. This formally reverses, at a Lie algebra level, the well known
non-relativistic contraction that goes from the Poincare group to
the Galilei one; this expansion is done in an explicit constructive way. The
other possible expansion leads to the Newton-Hooke algebras, endowing with a
non-zero spacetime curvature the spacetime, while keeping a flat
space of worldlines.Comment: 14 pages, LaTeX. The expansion method is clarifie
Harmonic Oscillator Lie Bialgebras and their Quantization
All possible Lie bialgebra structures on the harmonic oscillator algebra are
explicitly derived and it is shown that all of them are of the coboundary type.
A non-standard quantum oscillator is introduced as a quantization of a
triangular Lie bialgebra, and a universal -matrix linked to this new quantum
algebra is presented.Comment: 8 pages, LaTeX; communication presented in the XXI ICGTMP, Goslar
(Germany) 199
2+1 Kinematical expansions: from Galilei to de Sitter algebras
Expansions of Lie algebras are the opposite process of contractions. Starting
from a Lie algebra, the expansion process goes to another one, non-isomorphic
and less abelian. We propose an expansion method based in the Casimir
invariants of the initial and expanded algebras and where the free parameters
involved in the expansion are the curvatures of their associated homogeneous
spaces. This method is applied for expansions within the family of Lie algebras
of 3d spaces and (2+1)d kinematical algebras. We show that these expansions are
classed in two types. The first type makes different from zero the curvature of
space or space-time (i.e., it introduces a space or universe radius), while the
other has a similar interpretation for the curvature of the space of
worldlines, which is non-positive and equal to in the kinematical
algebras. We get expansions which go from Galilei to either Newton--Hooke or
Poincar\'e algebras, and from these ones to de Sitter algebras, as well as some
other examples.Comment: 15 pages, LaTe
- …