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(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.

Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

“…It is worth stressing that the connection between the Poisson-Lie group approach presented here and the role that classical -matrices and Drinfel' d-doubles play in the context of 2 + 1 quantum gravity [76][77][78][79][80][81][82][83][84] has been studied in detail in the works [85][86][87][88][89]. Also, the deformed Casimir operators (57) (or (64)) can be used to provide modified dispersion relations, which should be related to those appearing in several phenomenological approaches to quantum gravity (see [90][91][92][93]).…”

Quantum groups and noncommutative spacetimes with cosmological constant

“…Moreover, if we compute the Sklyanin bracket for the spacetime coordinates we will obtain the 'semiclassical' version of the noncommutative spacetime associated to the quantum group G z , and the main algebraic features induced from the spacetime noncommutativity can be straightforwardly appreciated at this Poisson level, such as, for instance, the role played by the cosmological constant. Moreover, if the classical spacetime is a homogeneous space M = G/H, where H is a certain subgroup of G, and the 1-cocycle δ fulfils the so-called coisotropy condition (see [25] and references therein)…”

“…In this paper we report on recent results [19,20,21,22,23,24,25] concerning the construction of noncommutative (A)dS spacetimes, which have been obtained by making use of the theory of quantum groups. We recall that the so-called 'quantum' deformations of Lie groups and algebras (see [26,27,28,29,30,31,32] and references therein) present many features that make them suitable to be considered in a Quantum Gravity scenario:…”

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

“…13 In terms of these generators the quantum algebra U q (sl(2; R), which will be denoted by U (r st ) (o(2, 1)), can be reformulated as follows. The quantum deformation of U (o(2, 1)), corresponding to the classical r -matrix (4.12), is an unital associative algebra U (r st ) (o(2, 1)) with the generators {J 1 , J 2 , q ±ı J 3 } and the defining relations (k = 1, 2):…”

“…It should be noted that the full list of the non-equivalent classical r -matrices for sl(2; R) and o(2, 1) Lie algebras has been obtained early by different methods [9,10] (see also [11][12][13]), however, the complete list of the non-equivalent Hopf quantizations for these Lie algebras has not been presented in the literature. Furthermore, there was put forward the incorrect hypothesis that the isomorphic Lie algebra su(1, 1) and sl(2; R) do not have any isomorphic quasitriangular Lie bialgebras (see [14]).…”