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 is revisited and explored in detail for the case in which is a classical double itself. We apply these results to give an explicit description of some coisotropic 2d Poisson homogeneous spaces over the group , 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 and as a coisotropic one for the others.We then construct families of coisotropic Poisson homogeneous structures for 3d anti de Sitter space and show that the ones that are quotients by a Poisson subgroup are determined by a three-parameter family of classical r-matrices for , while the non Poisson-subgroup cases are much more numerous. In particular, we present the two Poisson homogeneous structures on that arise from two Drinfel’d double structures on . The first one realises as a quotient of by the Poisson-subgroup , while the second one, the non-commutative spacetime of the twisted κ-AdS deformation, realises as a coisotropic Poisson homogeneous space.