A quantum duality principle for coisotropic subgroups and Poisson quotients

Ciccoli, Nicola; Gavarini, Fabio

doi:10.1016/j.aim.2005.01.009

Cited by 28 publications

(69 citation statements)

References 16 publications

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“…Now, by making use of the Poisson version of the quantum duality principle [38][39][40], the group multiplication law (7) can be immediately rewritten in algebraic terms as a comultiplication map Δ z through the identification of the two copies of the dual group coordinates as…”

Section: A Dual Poisson-lie Group and Curved Momentum Spacementioning

confidence: 99%

Curved momentum spaces from quantum (anti–)de Sitter groups in ( 3+1 ) dimensions

et al. 2018

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Curved momentum spaces associated to the κ-deformation of the (3 þ 1) de Sitter and anti-de Sitter algebras are constructed as orbits of suitable actions of the dual Poisson-Lie group associated to the κ-deformation with nonvanishing cosmological constant. The κ-de Sitter and κ-anti-de Sitter curved momentum spaces are separately analyzed, and they turn out to be, respectively, half of the (6 þ 1)-dimensional de Sitter space and half of a space with SOð4; 4Þ invariance. Such spaces are made of the momenta associated to spacetime translations and the "hyperbolic" momenta associated to boost transformations. The known κ-Poincaré curved momentum space is smoothly recovered as the vanishing cosmological constant limit from both of the constructions.

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Section: A Dual Poisson-lie Group and Curved Momentum Spacementioning

confidence: 99%

Curved momentum spaces from quantum (anti–)de Sitter groups in ( 3+1 ) dimensions

et al. 2018

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“…On the other hand, the constant curvature spacetimes of 3d gravity, their isometry groups and many associated structures can be obtained from associated structures for SL(2, R) and two-dimensional hyperbolic space by analytic continuation techniques, see for instance [63,64]. Although there may be many more noncoisotropic Poisson homogeneous structures on SL(2, R)/H, it can be expected that the coisotropic ones are the simplest to quantise, see [65] and also [24][25][26][27][28][29][30][31][32][33][34][35], and most natural for applications in 3d quantum gravity and noncommutative geometry. It is therefore sensible to focus first on the coisotropic case.…”

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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“…The Poisson homogeneous space G * /H ⊥ is called the complementary dual of G/H in [4] where it is shown that it fits into a quantum duality scheme. This map is also surjective if every g ∈ G can be written as g = kh , with h ∈ K and h ∈ H so that the last statement follows as well.…”

Section: Lemma 33 Let G Be a Poisson-lie Group H A Closed Connectementioning

confidence: 99%

“…One reason of interest lies in the fact that every coisotropic subgroup of a Poisson-Lie group can be quantized in such a way as to fit in a nice duality diagram [4]. Furthermore, coisotropic submanifolds have recently raised a lot of attention in the context of deformation quantization [3,2] and played a role in the analysis of Poisson sigma-models over group manifolds [1].…”

Section: Introductionmentioning

confidence: 99%