Drinfel’d double structures for Poincaré and Euclidean groups

In this contribution (arXiv:1812.02075) all non-isomorphic three-dimensional Poisson homogeneous Euclidean spaces are constructed and analyzed, based on the classification of coboundary Lie bialgebra structures of the Euclidean group in 3-dimensions, and the only Drinfel’d double structure for this group is explicitly given. The similar construction for the Poincaré case is reviewed and the striking differences between the Lorentzian and Euclidean cases are underlined. Finally, the contraction scheme starting from Drinfel’d double structures of the so(3,1) Lie algebra is presented.

The Poincaré group as a Drinfel’d double

In this new paper (arXiv:1809.09207) the eight nonisomorphic Drinfel’d double (DD) structures for the Poincaré Lie group in (2+1) dimensions are explicitly constructed in the kinematical basis. Also, the two existing DD structures for a non-trivial central extension of the (1+1) Poincaré group are also identified and constructed, while in (3+1) dimensions no Poincaré DD structure does exist. Each of the DD structures here presented has an associated canonical quasitriangular Poincaré r-matrix whose properties are analysed. Some of these r-matrices give rise to coisotropic Poisson homogeneous spaces with respect to the Lorentz subgroup, and their associated Poisson Minkowski spacetimes are constructed. Two of these (2+1) noncommutative DD Minkowski spacetimes turn out to be quotients by a Lorentz Poisson subgroup: the first one corresponds to the double of 𝔰𝔩(2) with trivial Lie bialgebra structure, and the second one gives rise to a quadratic noncommutative Poisson Minkowski spacetime. With these results, the explicit construction of DD structures for all Lorentzian kinematical groups in (1+1) and (2+1) dimensions is completed, and the connection between (anti-)de Sitter and Poincaré r-matrices through the vanishing cosmological constant limit is also analysed.

Global versus local superintegrability of nonlinear oscillators

Liouville (super)integrability of a Hamiltonian system of differential equations is based on the existence of globally well-defined constants of the motion, while Lie point symmetries provide a local approach to conserved integrals. In this new paper (arXiv:1809.02248) we investigate in which sense Lie point symmetries can be used to provide information concerning the superintegrability of a given Hamiltonian system. The two-dimensional oscillator and the central force problem are used as benchmark examples to show that the relationship between standard Lie point symmetries and superintegrability is neither straightforward nor universal. In general, it turns out that superintegrability is not related to either the size or the structure of the algebra of variational dynamical symmetries. Nevertheless, all of the first integrals for a given Hamiltonian system can be obtained through an extension of the standard point symmetry method, which is applied to a superintegrable nonlinear oscillator describing the motion of a particle on a space with non-constant curvature and spherical symmetry.

Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries

In this new paper (arXiv:1805.07099) we study the Lie bialgebra structures that can be built on the one-dimensional central extension of the Poincaré and (A)dS algebras in (1+1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, U(1) or SO(2) on the (1+1) dimensional Minkowski or (A)dS spacetime. We show that this highlights the possibility that the algebra of functions on the gauge bundle becomes noncommutative. This is a new way in which the Coleman-Mandula theorem could be circumvented by noncommutative structures, and it is related to a mixing of spacetime and gauge symmetry generators when they act on tensor-product states. We obtain all Lie bialgebra structures on centrally-extended Poincaré and (A)dS which are coisotropic w.r.t. the Lorentz algebra, and therefore all of them admit the construction of a noncommutative principal gauge bundle on a quantum homogeneous Minkowski spacetime. It is shown that several different types of hybrid noncommutativity between the spacetime and gauge coordinates are allowed by introducing quantum extended Poincaré symmetries. In one of these cases, an alternative interpretation of the central extension leads to a new description of the well-known canonical noncommutative spacetime as the quantum homogeneous space of a quantum Poincaré algebra of symmetries.