Maximal Abelian Subgroups of the Isometry and Conformal Groups of Euclidean and Minkowski Spaces

Classical Klein-Gordon solutions, symplectic structures and isometry actions on AdS spacetimes

We study classical, real Klein-Gordon theory on Lorentzian Anti de Sitter (AdS_{1,d}) spacetimes with spatial dimension d. We give a complete list of well defined and bounded Klein-Gordon solutions for three types of regions on AdS: slice (time interval times all of space), rod hypercylinder (all of time times solid ball in space), and tube hypercylinder (all of time times solid shell in space). Hypercylinder regions are of natural interest for AdS since the neighborhood of the AdS-boundary is a tube. For the solution spaces of our regions we find the actions induced by the AdS isometry group …

Chirality in Affine Spaces and in Spacetime

An object is chiral when its symmetry group contains no indirect isometry. It can be difficult to classify isometries as direct or indirect, except in the Euclidean case. We classify them with the help of outer semidirect products of isometry groups, in particular in the case of an affine space defined over a finite-dimensional real quadratic space. We also classify as direct or indirect the isometries of the real Lorentz-Minkowski spacetime and those of the classical spacetime defined by the Newton-Cartan theory.

AdS Geometry, projective embedded coordinates and associated isometry groups

This work is intended to investigate the geometry of anti-de Sitter spacetime (AdS), from the point of view of the Laplacian Comparison Theorem (LCT), and to give another description of the hyperbolical embedding standard formalism of the de Sitter and anti-de Sitter spacetimes in a pseudo-Euclidean spacetime. It is shown how to reproduce some geometrical properties of AdS, from the LCT in AdS, choosing suitable functions that satisfy basic properties of Riemannian geometry. We also introduce and discuss the well-known embedding of a 4-sphere and a 4-hyperboloid in a 5-dimensional pseudo-Eucli…

具有无观测视界时空的等距变换

Isometries of spacetimes without observer horizons

研究了具有良好因果结构的非紧致洛伦兹流形的等距群，证明时间定向保持等距群在时空上作用良好，并推导出不变的因果时间函数及等距群的分裂结构。

Yang-Mills solutions on Minkowski space via non-compact coset spaces

We find a two-parameter family of solutions of the Yang-Mills equations for gauge group SO(1,3) on Minkowski space by foliating different parts of it with non-compact coset spaces with SO(1,3) isometry. The interior of the lightcone is foliated with hyperbolic space $H^3\cong \text{SO}(1,3)/\text{SO}(3)$, while the exterior of the lightcone employs de Sitter space dS$_3\cong \text{SO}(1,3)/\text{SO}(1,2)$. The lightcone itself is parametrized by SO(1,3)/ISO(2) in a nilpotent fashion. Equivariant reduction of the SO(1,3) Yang-Mills system on the first two coset spaces yields a mechanical system…

Noninertial symmetry group with invariant Minkowski line element consistent with Heisenberg quantum commutation relations

The maximal symmetry of a quantum system with Heisenberg commutation relations is given by the projective representations of the automorphism group of the Weyl-Heisenberg algebra. The automorphism group is the central extension of the inhomogeneous symplectic group with a conformal scaling that acts on extended phase space. We determine the subgroup that also leaves invariant a degenerate orthogonal Minkowski line element. This defines noninertial relativistic symmetry transformations that have the expected classical limit as c becomes infinite.

Causal and conformal structures of globally hyperbolic spacetimes

The group of conformal diffeomorphisms and the group of causal automorphisms on two-dimensional globally hyperbolic spacetimes are clarified. It is shown that if spacetimes have non-compact Cauchy surfaces, then the groups are subgroups of that of two-dimensional Minkowski spacetime, and if spacetimes have compact Cauchy surfaces, then the groups are subgroups of that of two-dimensional Einstein's static universe.

Subgroups of the Group of Generalized Lorentz Transformations and Their Geometric Invariants

It is shown that the group of generalized Lorentz transformations serves as relativistic symmetry group of a flat Finslerian event space. Being the generalization of Minkowski space, the Finslerian event space arises from the spontaneous breaking of initial gauge symmetry and from the formation of anisotropic fermion-antifermion condensate. The principle of generalized Lorentz invariance enables exact taking into account the influence of condensate on the dynamics of fundamental fields. In particular, the corresponding generalized Dirac equation turns out to be nonlinear. We have found two non…

Cohomogeneity-one Einstein-Weyl structures : a local approach

We analyse in a systematic way the (non-)compact n-dimensional Einstein Weyl spaces equipped with a cohomogeneity-one metric. With no compactness hypothesis, we prove that, as soon as the (n-1)-dimensional space is an homogeneous reductive Riemannian space with an unimodular group of left-acting isometries G 1)a non-exact Einstein-Weyl stucture may exist only if the (n-1)-dimensional homogeneous space G/H contains a non trivial subgroup H' that commutes with the isotropy subgroup H, 2) the extra isometry due to this Killing vector corresponds to the right-action of one of the generators of the…

The Poincaré group as a Drinfel'd double

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…