Quantum groups and noncommutative spacetimes with cosmological constant

Ballesteros, Ángel; Gutiérrez-Sagredo, Iván; Herranz, Francisco J.; Meusburger, Catherine; Naranjo, Pedro

doi:10.1088/1742-6596/880/1/012023

Cited by 13 publications

(37 citation statements)

References 52 publications

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“…At this point it could seem surprising that the noncommutative (A)dS spacetimes (64) and (66) do not depend on the cosmological constant and, therefore, coincide with the corresponding noncommutative Minkowski spacetimes. Indeed, this is true only at first order, and higher order contributions depending on are expected to appear when the full quantum coproduct z is constructed and the all-orders noncommutative spacetime is obtained by applying the full Hopf algebra duality or, alternatively, by quantizing the allorders Poisson-Lie group whose linearization corresponds to the extended noncommutative spacetimes here presented (see [50][51][52] for explicit examples, including the noncommutative κ-(A)dS spacetime in (2 + 1) dimensions, which turns out to be a nonlinear deformation of the κ-Minkowski spacetime).…”

Section: C2 Whenmentioning

confidence: 99%

Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries

Ballesteros

Mercati

2018

Eur. Phys. J. C

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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 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 admit the construction of a noncommutative principal gauge bundle on a quantum homogeneous spacetime. It is shown that several different types of hybrid noncommutativity between the spacetime and gauge coordinates are allowed. 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.

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Section: C2 Whenmentioning

confidence: 99%

Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries

Ballesteros

Mercati

2018

Eur. Phys. J. C

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“…Note that the basis {J, K 1 , K 2 , P 0 , P 1 , P 2 } is related to the basis {J 0 , J 1 , J 2 , P 0 , P 1 , P 2 } considered in [50][51][52][53] via a very simple change of basis [53] …”

Section: Lorentzian 3d Constant Curvature Spacetimes As Poisson Homogmentioning

confidence: 99%

“…An important example that arises from a classical r-matrix that gives the Lie algebra g Λ the structure of a classical double is the twisted ('space-like') κ-AdS Poisson Lie group from [53] given by r =…”

Section: Lorentzian 3d Constant Curvature Spacetimes As Poisson Homogmentioning

confidence: 99%

“…In terms of the local coordinates a + , a − , χ, the left and right invariant vector fields of SL(2, R) associated with the basis {J 3 , J ± } of sl(2, R) are given by [53] …”

Section: A Coordinate Functions and Vector Fields For The 2d Cayley-kmentioning

confidence: 99%

“…We consider two representative Lie bialgebra structures on so(2, 2) that are canonical in the sense that they are the two possible realisations of so(2, 2) as a classical double so(2, 2) = D(a) [50][51][52][53]. This choice is motivated by their applications in 3d gravity, where Poisson-Lie structures arise naturally from the description of 3d gravity as a Chern-Simons gauge theory [54,55] whose symplectic structure can be described in terms of Poisson-Lie structures [56,57].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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Towards (3+1) gravity through Drinfel'd doubles with cosmological constant

2015

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We present the generalisation to (3 + 1) dimensions of a quantum deformation of the (2 + 1) (Anti)-de Sitter and Poincaré Lie algebras that is compatible with the conditions imposed by the Chern-Simons formulation of (2 + 1) gravity. Since such compatibility is automatically fulfilled by deformations coming from Drinfel'd double structures, we believe said structures are worth being analysed also in the (3 + 1) scenario as a possible guiding principle towards the description of (3 + 1) gravity. To this aim, a canonical classical r-matrix arising from a Drinfel'd double structure for the three (3 + 1) Lorentzian algebras is obtained. This r-matrix turns out to be a twisted version of the one corresponding to the (3 + 1) κ-deformation, and the main properties of its associated noncommutative spacetime are analysed. In particular, it is shown that this new quantum spacetime is not isomorphic to the κ-Minkowski one, and that the isotropy of the quantum space coordinates can be preserved through a suitable change of basis of the quantum algebra generators. Throughout the paper the cosmological constant appears as an explicit parameter, thus allowing the (flat) Poincaré limit to be straightforwardly obtained.

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Quantum groups and noncommutative spacetimes with cosmological constant

Cited by 13 publications

References 52 publications

Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries

Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries

AdS Poisson homogeneous spaces and Drinfel’d doubles

Towards (3+1) gravity through Drinfel'd doubles with cosmological constant

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