Quantum structure of the motion groups of the two-dimensional Cayley-Klein geometries

Ballesteros, Ángel; Herranz, Francisco J.; Olmo, M. A. del; Santander, Mariano

doi:10.1088/0305-4470/26/21/019

Cited by 72 publications

(97 citation statements)

References 32 publications

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“…2 Working within these safe boundaries guarantees that not only the algebra but also the extra structures required for a relativistic theory (such as conservations laws) can be built in a consistent way. We focus on a quantum deformation of the de Sitter algebra (the algebra of isometries of the de Sitter spacetime), known as q-de Sitter [42][43][44][45]. The dimensionless quantum deformation parameter of the q-de Sitter algebra can be fixed as a function of the Planck length l and the de Sitter radius H −1 , and we choose it so that when the Planck length vanishes, the algebra reduces to the de Sitter algebra, while when the de Sitter radius is sent to infinity one recovers the κ-Poincaré Hopf algebra.…”

Section: Introductionmentioning

confidence: 99%

Kinematics of particles with quantum-de Sitter-inspired symmetries

Barcaroli¹,

Gubitosi

2016

Phys. Rev. D

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We present the first detailed study of the kinematics of free relativistic particles whose symmetries are compatible with the ones described by a quantum deformation of the de Sitter algebra, known as q-de Sitter Hopf algebra. In such algebra, the quantum deformation parameter is a function of the Planck length l and the de Sitter radius H −1 , such that when the Planck length vanishes, the algebra reduces to the de Sitter algebra, while when the de Sitter radius is sent to infinity, one recovers the κ-Poincaré Hopf algebra. In the first limit, the picture is that of a particle with trivial momentum space geometry moving on de Sitter spacetime; in the second one, the picture is that of a particle with de Sitter momentum space geometry moving on Minkowski spacetime. When both the Planck length and the inverse of the de Sitter radius are nonzero, effects due to spacetime curvature and nontrivial momentum space geometry are both present and affect each other. The particles' motion is then described in a full phase-space picture. We find that redshift effects that are usually associated with spacetime curvature become energy dependent. Also, the energy dependence of the particles' travel times that is usually associated with momentum space nontrivial properties is modified in a curvature-dependent way.

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Section: Introductionmentioning

confidence: 99%

Kinematics of particles with quantum-de Sitter-inspired symmetries

Barcaroli¹,

Gubitosi

2016

Phys. Rev. D

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“…The properties of the nine (1+1)-dimensional geometries corresponding to different signs of κ 1 and κ 2 were investigated in detail in [66] and their (2+1)-dimensional counterparts in [67]. The Lie algebra sl(2, R) is the Cayley-Klein algebra (29) for κ 1 = 1 and κ 2 = −1 with the relation between the bases {P 1 , P 2 , J 12 } and {J ± , J 3 } given by…”

Section: Coisotropic Poisson Homogeneous Spaces For Sl(2 R) ≃ So(2 1)mentioning

confidence: 99%

AdS Poisson homogeneous spaces and Drinfel’d doubles

Ballesteros¹,

Meusburger²,

Naranjo³

2017

J. Phys. A: Math. Theor.

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The correspondence between Poisson homogeneous spaces over a Poisson-Lie group G and Lagrangian Lie subalgebras of the classical double D(g) is revisited and explored in detail for the case in which g = D(a) is a classical double itself. We apply these results to give an explicit description of some coisotropic 2d Poisson homogeneous spaces over the group SL(2, R) ∼ = SO(2, 1), namely 2d anti de Sitter space, 2d hyperbolic space and the lightcone in 3d Minkowski space. We show how each of these spaces is obtained as a quotient with respect to a Poisson-subgroup for one of the three inequivalent Lie bialgebra structures on sl(2, R) and as a coisotropic one for the others.We then construct families of coisotropic Poisson homogeneous structures for 3d anti de Sitter space AdS 3 and show that the ones that are quotients by a Poisson subgroup are determined by a three-parameter family of classical r-matrices for so(2, 2), while the non Poisson-subgroup cases are much more numerous. In particular, we present the two Poisson homogeneous structures on AdS 3 that arise from two Drinfel'd double structures on SO(2, 2). The first one realises AdS 3 as a quotient of SO(2, 2) by the Poisson-subgroup SL(2, R), while the second one, the non-commutative spacetime of the twisted κ-AdS deformation, realises AdS 3 as a coisotropic Poisson homogeneous space.

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“…(1) The fundamental LBC's defined by (g (κ 1 ,κ 2 ) , δ (1,1) 02 ) leads to the choice of the transformation law for the deformation parameter (z = √ κ 1 κ 2 w) taken originally for the CK scheme in [18]. In this case, the quantum algebra U w g (κ 1 ,κ 2 ) has coproduct ∆ 02 (3.1) and the following commutation rules…”

Section: Hopf Algebra Contractions Of U Z So(3)mentioning

confidence: 99%

Lie bialgebra contractions and quantum deformations of quasi-orthogonal algebras

Ballesteros

Gromov

Herranz

et al. 1995

Journal of Mathematical Physics

112

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Lie bialgebra contractions are introduced and classified. A non-degenerate coboundary bialgebra structure is implemented into all pseudo-orthogonal algebras so(p, q) starting from the one corresponding to so(N + 1). It allows to introduce a set of Lie bialgebra contractions which leads to Lie bialgebras of quasi-orthogonal algebras. This construction is explicitly given for the cases N = 2, 3, 4. All Lie bialgebra contractions studied in this paper define Hopf algebra contractions for the Drinfel'd-Jimbo deformations U z so(p, q). They are explicitly used to generate new non-semisimple quantum algebras as it is the case for the Euclidean, Poincaré and Galilean algebras.

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Quantum structure of the motion groups of the two-dimensional Cayley-Klein geometries

Cited by 72 publications

References 32 publications

Kinematics of particles with quantum-de Sitter-inspired symmetries

Kinematics of particles with quantum-de Sitter-inspired symmetries

AdS Poisson homogeneous spaces and Drinfel’d doubles

Lie bialgebra contractions and quantum deformations of quasi-orthogonal algebras

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