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 RR--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 $R$--matrices.Comment: 19 pages, LaTeX; revised version to appear in J. Phys. A; quantization of bialgebras is complete

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

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 $N$ coupled oscillators are obtained from Poisson realizations of Heisenberg--Weyl, harmonic oscillator and $sl(2,\R)$ 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 $sl(2,\R)$ coalgebra and angular momentum chains is presented, and a non-standard integrable deformation of the hyperbolic Gaudin system is obtained.Comment: 17 pages, LaTe

(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

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 $R$-matrix linked to this new quantum algebra is presented.Comment: 8 pages, LaTeX; communication presented in the XXI ICGTMP, Goslar (Germany) 199

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 $-1/c^2$ 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 $c\to \infty$ 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 $\pm 1/\tau^2$ the spacetime, while keeping a flat space of worldlines.Comment: 14 pages, LaTeX. The expansion method is clarifie

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 $-1/c^2$ 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