AdS Poisson homogeneous spaces and Drinfel'd doubles

Abstract

The correspondence between Poisson homogeneous spaces over a Poisson-Lie group $G$ and Lagrangian Lie subalgebras of the classical double $D(\mathfrak{g})$ is revisited and explored in detail for the case in which $\mathfrak{g}=D(\mathfrak{a})$ is a classical double itself. We apply these results to give an explicit description of some coisotropic 2d Poisson homogeneous spaces over the group $\mathrm{SL}(2,R)\cong \mathrm{SO}(2,1)$, 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 ${\mathrm{AdS}}_3$ 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 ${\mathrm{AdS}}_3$ that arise from two Drinfel'd double structures on $\mathrm{SO}(2,2)$. The first one realises ${\mathrm{AdS}}_3$ as a quotient of $\mathrm{SO}(2,2)$ by the Poisson-subgroup $\mathrm{SL}(2,R)$, while the second one, the non-commutative spacetime of the twisted $\kappa$-AdS deformation, realises ${\mathrm{AdS}}_3$ as a coisotropic Poisson homogeneous space.

25 pages, journal version, section 2 fully rewritten, some comments and references added

References in corpus (7)

• Generalised Chern-Simons actions for 3d gravity and kappa-Poincare symmetry
• A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions
• Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra
• Quaternionic and Poisson-Lie structures in 3d gravity: the cosmological constant as deformation parameter
• The kappa-(A)dS quantum algebra in (3+1) dimensions
• Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations
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Cited by in corpus (11)

• Pauli-Jordan Function and Scalar Field Quantization in $\kappa$-Minkowski Noncommutative Spacetime
• The $\kappa$-(A)dS noncommutative spacetime
• Noncommutative spaces of worldlines
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• Extending general covariance: Moyal-type noncommutative manifolds
• Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces
• All noncommutative spaces of $\kappa$-Poincaré geodesics
• Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries
• The noncommutative space of light-like worldlines
• Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups
• Noncommutative Lightcones from Quantum SO(2,1) Conformal Groups