Quaternionic and Poisson–Lie structures in three-dimensional gravity: The cosmological constant as deformation parameter

Meusburger, Catherine; Schroers, Bernd J.

doi:10.1063/1.2973040

Cited by 43 publications

(88 citation statements)

References 23 publications

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“…It is worth stressing that the connection between the Poisson-Lie group approach presented here and the role that classical -matrices and Drinfel' d-doubles play in the context of 2 + 1 quantum gravity [76][77][78][79][80][81][82][83][84] has been studied in detail in the works [85][86][87][88][89]. Also, the deformed Casimir operators (57) (or (64)) can be used to provide modified dispersion relations, which should be related to those appearing in several phenomenological approaches to quantum gravity (see [90][91][92][93]).…”

Section: Discussionmentioning

confidence: 99%

Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

Ballesteros

Bruno

Herranz

2017

Advances in High Energy Physics

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The -deformation of the (2 + 1)D anti-de Sitter, Poincaré, and de Sitter groups is presented through a unified approach in which the curvature of the spacetime (or the cosmological constant) is considered as an explicit parameter. The Drinfel' d-double and the Poisson-Lie structure underlying the -deformation are explicitly given, and the three quantum kinematical groups are obtained as quantizations of such Poisson-Lie algebras. As a consequence, the noncommutative (2 + 1)D spacetimes that generalize the -Minkowski space to the (anti-)de Sitter ones are obtained. Moreover, noncommutative 4D spaces of (time-like) geodesics can be defined, and they can be interpreted as a novel possibility to introduce noncommutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related to 2 + 1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related to the Planck length, and the existence of a kind of "duality" between the cosmological constant and the Planck scale is also envisaged.

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Section: Discussionmentioning

confidence: 99%

Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

Ballesteros

Bruno

Herranz

2017

Advances in High Energy Physics

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“…In the 3d case, the resulting q-deformation of the corresponding Lorentz group is very well understood from the canonical point of view in the combinatorial quantization framework for Chern-Simons theory [8][9][10] and from the path integral perspective both from the Chern-Simons quantization [11] and the Turaev-Viro spinfoam model [12] with the asymptotics of the q-deformed {6j}-symbols [13]. From the canonical perspective of loop quantum gravity, which is subtlety different from the Chern-Simons theory, this issue is not entirely settled despite some recent interesting efforts [14,15] and [16][17][18][19].…”

Section: Figmentioning

confidence: 99%

Closure constraints for hyperbolic tetrahedra

Charles

Livine

2015

Class. Quantum Grav.

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We investigate the generalization of loop gravity's twisted geometries to a q-deformed gauge group. In the standard undeformed case, loop gravity is a formulation of general relativity as a diffeomorphism-invariant SU(2) gauge theory. Its classical states are graphs provided with algebraic data. In particular closure constraints at every node of the graph ensure their interpretation as twisted geometries. Dual to each node, one has a polyhedron embedded in flat space R 3 . One then glues them allowing for both curvature and torsion. It was recently conjectured that q-deforming the gauge group SU(2) would allow to account for a non-vanishing cosmological constant Λ = 0, and in particular that deforming the loop gravity phase space with real parameter q ∈ R+ would lead to a generalization of twisted geometries to a hyperbolic curvature. Following this insight, we look for generalization of the closure constraints to the hyperbolic case. In particular, we introduce two new closure constraints for hyperbolic tetrahedra. One is compact and expressed in terms of normal rotations (group elements in SU(2) associated to the triangles) and the second is non-compact and expressed in terms of triangular matrices (group elements in SB(2, C)). We show that these closure constraints both define a unique dual tetrahedron (up to global translations on the three-dimensional one-sheet hyperboloid) and are thus ultimately equivalent.

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“…In particular, this raises the question if the unified description of the isometry groups in 3d gravity in terms of (pseudo) quaternions from [75] can be used to give unified coordinates for the associated homogeneous spaces in which the cosmological constant appears as a deformation parameter. In this context, it would also be worthwhile to explore the connections between Poisson homogeneous spaces and the dynamical Yang-Baxter equation [41,42].…”

Section: Discussionmentioning

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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Quaternionic and Poisson–Lie structures in three-dimensional gravity: The cosmological constant as deformation parameter

Cited by 43 publications

References 23 publications

Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

Closure constraints for hyperbolic tetrahedra

AdS Poisson homogeneous spaces and Drinfel’d doubles

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