Poisson structure and symmetry in the Chern–Simons formulation of (2   1)-dimensional gravity

Meusburger, Catherine; Schroers, Bernd J.

doi:10.1088/0264-9381/20/11/318

Cited by 71 publications

(170 citation statements)

References 33 publications

(90 reference statements)

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“…Due to the isomorphism o (2,2) o(2, 1) ⊕ o(2, 1) our results can be also applied to the description of D = 3 AdS symmetries [19]. We recall that o(2, 2) symmetry has been employed in Chern-Simons formulation of D = 3 gravity [20][21][22], with Lorentzian signature and nonvanishing negative cosmological constant. Subsequently, the quantum deformations of D = 3 Chern-Simons theory have been used for the description of D = 3 quantum gravity as deformed D = 3 topological QFT [23,24].…”

Section: Introductionmentioning

confidence: 87%

Quantizations of $$D=3$$ D = 3 Lorentz symmetry

Lukierski

Tolstoy

2017

Eur. Phys. J. C

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Using the isomorphism o(3; C) sl(2; C) we develop a new simple algebraic technique for complete classification of quantum deformations (the classical r -matrices) for real forms o(3) and o(2, 1) of the complex Lie algebra o(3; C) in terms of real forms of sl(2; C): su(2), su(1, 1) and sl(2; R). We prove that the D = 3 Lorentz symmetry o(2, 1) su(1, 1) sl(2; R) has three different Hopfalgebraic quantum deformations, which are expressed in the simplest way by two standard su(1, 1) and sl(2; R) qanalogs and by simple Jordanian sl(2; R) twist deformation. These quantizations are presented in terms of the quantum Cartan-Weyl generators for the quantized algebras su(1, 1) and sl(2; R) as well as in terms of quantum Cartesian generators for the quantized algebra o(2, 1). Finally, some applications of the deformed D = 3 Lorentz symmetry are mentioned.

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Section: Introductionmentioning

confidence: 87%

Quantizations of $$D=3$$ D = 3 Lorentz symmetry

Lukierski

Tolstoy

2017

Eur. Phys. J. C

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“…This aspect represents a crucial departure from the four dimensional case and it allows for a different approach to the quantization of the system. In order to clarify this point, let us first recall some basic elements of the inclusion of particles in 3D gravity (see, for instance, [14,[30][31][32][33][34][35] and references therein).…”

Section: Coupling To Massive Point Particlesmentioning

confidence: 99%

Symmetries of quantum spacetime in three dimensions

et al. 2016

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By applying loop quantum gravity techniques to 3D gravity with a positive cosmological constant Λ, we show how the local gauge symmetry of the theory, encoded in the constraint algebra, acquires the quantum group structure of so q (4), with q = exp (i ̵ h √ Λ 2κ). By means of an Inonu-Wigner contraction of the quantum group bi-algebra, keeping κ finite, we obtain the kappa-Poincaré algebra of the flat quantum space-time symmetries.

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“…A description of a moving defect can be obtained by boosting the conical metric, in this case the three momentum of the particle will be a general element of SL(2, R) [3,51]. Various treatments exist for the description of the phase space of point particles coupled to gravity in three dimensions [3,52,53] and its symmetries [54,55].…”

Section: Deforming Momentum Space To the Group Sl(2 R)mentioning

confidence: 99%

Deformed phase spaces with group valued momenta

Arzano

Nettel

2016

Phys. Rev. D

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We introduce a general framework for describing deformed phase spaces with group valued momenta. Using techniques from the theory of Poisson-Lie groups and Lie bi-algebras we develop tools for constructing Poisson structures on the deformed phase space starting from the minimal input of the algebraic structure of the generators of the momentum Lie group. The tools developed are used to derive Poisson structures on examples of group momentum space much studied in the literature such as the n-dimensional generalization of the κ-deformed momentum space and the SL(2, Ê) momentum space in three space-time dimensions. We discuss classical momentum observables associated to multi-particle systems and argue that these combine according the usual four-vector addition despite the non-abelian group structure of momentum space.

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Poisson structure and symmetry in the Chern–Simons formulation of (2 1)-dimensional gravity

Cited by 71 publications

References 33 publications

Quantizations of $$D=3$$ D = 3 Lorentz symmetry

Quantizations of $$D=3$$ D = 3 Lorentz symmetry

Symmetries of quantum spacetime in three dimensions

Deformed phase spaces with group valued momenta

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