Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature

A family of classical superintegrable Hamiltonians, depending on an arbitrary radial function, which are defined on the 3D spherical, Euclidean and hyperbolic spaces as well as on the (2+1)D anti-de Sitter, Minkowskian and de Sitter spacetimes is constructed. Such systems admit three integrals of the motion (besides the Hamiltonian) which are explicitly given in terms of ambient and geodesic polar coordinates. The resulting expressions cover the six spaces in a unified way as these are parametrized by two contraction parameters that govern the curvature and the signature of the metric on each space. Next two maximally superintegrable Hamiltonians are identified within the initial superintegrable family by finding the remaining constant of the motion. The former potential is the superposition of a (curved) central harmonic oscillator with other three oscillators or centrifugal barriers (depending on each specific space), so that this generalizes the Smorodinsky-Winternitz system. The latter one is a superposition of the Kepler-Coulomb potential with another two oscillators or centrifugal barriers. As a byproduct, the Laplace-Runge-Lenz vector for these spaces is deduced. Furthermore both potentials are analysed in detail for each particular space. Some comments on their generalization to arbitrary dimension are also presented.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Humour in the post-war press: short stories of Gloria Fuertes in the falangist magazine Maravillas

The Spanish civil war entailed an impasse in the development of press as a communication platform. In fact, its instrumentalization for propaganda purposes explains its role in the consolidation of the new State. So, in the newspaper network created around the Movimiento Nacional (National Movement), the children’s press occupied a predominant place due to its contribution to the education of future generations. So much so, that the strategical-ideological aims imposed by the regime coexisted in these platforms together with the collaboration of artists of prominent aesthetical level. Such is the case of Gloria Fuertes (1917-1998), who gave life in the magazine Maravillas, to very popular characters such as Coleta and Pelines to disseminate their short stories once more, back in the eighties, in three monographic books: Coleta, la poeta (1982), Coleta, payasa ¿qué pasa? (1983) and Pelines (1986). However, such protagonists, beyond constituting aesthetical projections of the writer as heteronyms, would end up becoming authentic icons of children’s imagery

Towards (3+1) gravity through Drinfel'd doubles with cosmological constant

We present the generalisation to (3+1) dimensions of a quantum deformation of the (2+1) (Anti)-de Sitter and Poincar\'e Lie algebras that is compatible with the conditions imposed by the Chern-Simons formulation of (2+1) gravity. Since such compatibility is automatically fulfilled by deformations coming from Drinfel'd double structures, we believe said structures are worth being analysed also in the (3+1) scenario as a possible guiding principle towards the description of (3+1) gravity. To this aim, a canonical classical $r$-matrix arising from a Drinfel'd double structure for the three (3+1) Lorentzian algebras is obtained. This $r$-matrix turns out to be a twisted version of the one corresponding to the (3+1) $\kappa$-deformation, and the main properties of its associated noncommutative spacetime are analysed. In particular, it is shown that this new quantum spacetime is not isomorphic to the $\kappa$-Minkowski one, and that the isotropy of the quantum space coordinates can be preserved through a suitable change of basis of the quantum algebra generators. Throughout the paper the cosmological constant appears as an explicit parameter, thus allowing the (flat) Poincar\'e limit to be straightforwardly obtained.Comment: 12 pages. References and comments added. One misprint correcte

A (2+1) non-commutative Drinfel'd double spacetime with cosmological constant

We show that the Drinfel'd double associated to the standard quantum deformation $sl_\eta(2,R)$ is isomorphic to the (2+1)-dimensional AdS algebra with the initial deformation parameter $\eta$ related to the cosmological constant $\Lambda=-\eta^2$. This gives rise to a generalisation of a non-commutative Minkowski spacetime that arises as a consequence of the quantum double symmetry of (2+1) gravity to non-vanishing cosmological constant. The properties of the AdS quantum double that generalises this symmetry to the case $\Lambda\neq 0$ are sketched, and it is shown that the new non-commutative AdS spacetime is a nonlinear $\Lambda$-deformation of the Minkowskian one.Comment: 14 pages; some comments and references adde