Conformal compactification and cycle-preserving symmetries of spacetimes

The cycle-preserving symmetries for the nine two-dimensional real spaces of constant curvature are collectively obtained within a Cayley-Klein framework. This approach affords a unified and global study of the conformal structure of the three classical Riemannian spaces as well as of the six relativistic and non-relativistic spacetimes (Minkowskian, de Sitter, anti-de Sitter, both Newton-Hooke and Galilean), and gives rise to general expressions holding simultaneously for all of them. Their metric structure and cycles (lines with constant geodesic curvature that include geodesics and circles) are explicitly characterized. The corresponding cyclic (Mobius-like) Lie groups together with the differential realizations of their algebras are then deduced; this derivation is new and much simpler than the usual ones and applies to any homogeneous space in the Cayley-Klein family, whether flat or curved and with any signature. Laplace and wave-type differential equations with conformal algebra symmetry are constructed. Furthermore, the conformal groups are realized as matrix groups acting as globally defined linear transformations in a four-dimensional "conformal ambient space", which in turn leads to an explicit description of the "conformal completion" or compactification of the nine spaces.Comment: 43 pages, LaTe

Maximal superintegrability on N-dimensional curved spaces

A unified algebraic construction of the classical Smorodinsky-Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie groups SO(N+1), ISO(N), and SO(N,1) is presented. Firstly, general expressions for the Hamiltonian and its integrals of motion are given in a linear ambient space $R^{N+1}$, and secondly they are expressed in terms of two geodesic coordinate systems on the ND spaces themselves, with an explicit dependence on the curvature as a parameter. On the sphere, the potential is interpreted as a superposition of N+1 oscillators. Furthermore each Lie algebra generator provides an integral of motion and a set of 2N-1 functionally independent ones are explicitly given. In this way the maximal superintegrability of the ND Euclidean Smorodinsky-Winternitz system is shown for any value of the curvature.Comment: 8 pages, LaTe

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

Integrable potentials on spaces with curvature from quantum groups

A family of classical integrable systems defined on a deformation of the two-dimensional sphere, hyperbolic and (anti-)de Sitter spaces is constructed through Hamiltonians defined on the non-standard quantum deformation of a sl(2) Poisson coalgebra. All these spaces have a non-constant curvature that depends on the deformation parameter z. As particular cases, the analogues of the harmonic oscillator and Kepler--Coulomb potentials on such spaces are proposed. Another deformed Hamiltonian is also shown to provide superintegrable systems on the usual sphere, hyperbolic and (anti-)de Sitter spaces with a constant curvature that exactly coincides with z. According to each specific space, the resulting potential is interpreted as the superposition of a central harmonic oscillator with either two more oscillators or centrifugal barriers. The non-deformed limit z=0 of all these Hamiltonians can then be regarded as the zero-curvature limit (contraction) which leads to the corresponding (super)integrable systems on the flat Euclidean and Minkowskian spaces.Comment: 19 pages, 1 figure. Two references adde

Superintegrability on sl(2)-coalgebra spaces

We review a recently introduced set of N-dimensional quasi-maximally superintegrable Hamiltonian systems describing geodesic motions, that can be used to generate "dynamically" a large family of curved spaces. From an algebraic viewpoint, such spaces are obtained through kinetic energy Hamiltonians defined on either the sl(2) Poisson coalgebra or a quantum deformation of it. Certain potentials on these spaces and endowed with the same underlying coalgebra symmetry have been also introduced in such a way that the superintegrability properties of the full system are preserved. Several new N=2 examples of this construction are explicitly given, and specific Hamiltonians leading to spaces of non-constant curvature are emphasized.Comment: 12 pages. Based on the contribution presented at the "XII International Conference on Symmetry Methods in Physics", Yerevan (Armenia), July 2006. To appear in Physics of Atomic Nucle

Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry

A new method to obtain trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method encapsulates trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an absolute trigonometry, and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic homogeneous spacetimes; therefore a complete discussion of trigonometry in the six de Sitter, minkowskian, Newton--Hooke and galilean spacetimes follow as particular instances of the general approach. Any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes; in most cases these equations are new. Distinctive traits of the method are universality and self-duality: every equation is meaningful for the nine spaces at once, and displays explicitly invariance under a duality transformation relating the nine spaces. The derivation of the single basic trigonometric equation at group level, its translation to a set of equations (cosine, sine and dual cosine laws) and the natural apparition of angular and lateral excesses, area and coarea are explicitly discussed in detail. The exposition also aims to introduce the main ideas of this direct group theoretical way to trigonometry, and may well provide a path to systematically study trigonometry for any homogeneous symmetric space.Comment: 51 pages, LaTe

Contractions, deformations and curvature

The role of curvature in relation with Lie algebra contractions of the pseudo-ortogonal algebras so(p,q) is fully described by considering some associated symmetrical homogeneous spaces of constant curvature within a Cayley-Klein framework. We show that a given Lie algebra contraction can be interpreted geometrically as the zero-curvature limit of some underlying homogeneous space with constant curvature. In particular, we study in detail the contraction process for the three classical Riemannian spaces (spherical, Euclidean, hyperbolic), three non-relativistic (Newtonian) spacetimes and three relativistic ((anti-)de Sitter and Minkowskian) spacetimes. Next, from a different perspective, we make use of quantum deformations of Lie algebras in order to construct a family of spaces of non-constant curvature that can be interpreted as deformations of the above nine spaces. In this framework, the quantum deformation parameter is identified as the parameter that controls the curvature of such "quantum" spaces.Comment: 17 pages. Based on the talk given in the Oberwolfach workshop: Deformations and Contractions in Mathematics and Physics (Germany, january 2006) organized by M. de Montigny, A. Fialowski, S. Novikov and M. Schlichenmaie

Superintegrability on N-dimensional spaces of constant curvature from so(N+1) and its contractions

The Lie-Poisson algebra so(N+1) and some of its contractions are used to construct a family of superintegrable Hamiltonians on the ND spherical, Euclidean, hyperbolic, Minkowskian and (anti-)de Sitter spaces. We firstly present a Hamiltonian which is a superposition of an arbitrary central potential with N arbitrary centrifugal terms. Such a system is quasi-maximally superintegrable since this is endowed with 2N-3 functionally independent constants of the motion (plus the Hamiltonian). Secondly, we identify two maximally superintegrable Hamiltonians by choosing a specific central potential and finding at the same time the remaining integral. The former is the generalization of the Smorodinsky-Winternitz system to the above six spaces, while the latter is a generalization of the Kepler-Coulomb potential, for which the Laplace-Runge-Lenz N-vector is also given. All the systems and constants of the motion are explicitly expressed in a unified form in terms of ambient and polar coordinates as they are parametrized by two contraction parameters (curvature and signature of the metric).Comment: 14 pages. Based on the contribution presented at the "XII International Conference on Symmetry Methods in Physics", Yerevan (Armenia), July 2006. To appear in Physics of Atomic Nucle

Casimir invariants for the complete family of quasi-simple orthogonal algebras

A complete choice of generators of the center of the enveloping algebras of real quasi-simple Lie algebras of orthogonal type, for arbitrary dimension, is obtained in a unified setting. The results simultaneously include the well known polynomial invariants of the pseudo-orthogonal algebras $so(p,q)$, as well as the Casimirs for many non-simple algebras such as the inhomogeneous $iso(p,q)$, the Newton-Hooke and Galilei type, etc., which are obtained by contraction(s) starting from the simple algebras $so(p,q)$. The dimension of the center of the enveloping algebra of a quasi-simple orthogonal algebra turns out to be the same as for the simple $so(p,q)$ algebras from which they come by contraction. The structure of the higher order invariants is given in a convenient "pyramidal" manner, in terms of certain sets of "Pauli-Lubanski" elements in the enveloping algebras. As an example showing this approach at work, the scheme is applied to recovering the Casimirs for the (3+1) kinematical algebras. Some prospects on the relevance of these results for the study of expansions are also given.Comment: 19 pages, LaTe