On the semiduals of local isometry groups in three-dimensional gravity

Abstract

We use factorisations of the local isometry groups arising in 3d gravity for Lorentzian and Euclidean signatures and any value of the cosmological constant to construct associated bicrossproduct quantum groups via semidualisation. In this way we obtain quantum doubles of the Lorentz and rotation groups in 3d, as well as κ-Poincaré algebras whose associated r-matrices have spacelike, timelike and lightlike deformation parameters. We confirm and elaborate the interpretation of semiduality proposed in [13] as the exchange of the cosmological length scale and the Planck mass in the context of 3d quantum gravity. In particular, semiduality gives a simple understanding of why the quantum double of the Lorentz group and the κ-Poincaré algebra with spacelike deformation parameter are both associated to 3d gravity with vanishing cosmological constant, while the κ-Poincaré algebra with a timelike deformation parameter can only be associated to 3d gravity if one takes the Planck mass to be imaginary.

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

“…At the quantum algebra level, we have introduced a new basis (60) for the quantum (A)dS algebras, which is the analogous of the -Poincaré bicrossproduct basis. As far as the (dual) quantum groups are concerned, the results cover the noncommutative (A)dS spacetimes (49) up to secondorder in the deformation parameter, thus generalizing the -Minkowskian spacetime (33) which turns out to be their common first-order seed.…”

“…In particular, starting from the expressions given in [19], a nonlinear transformation involving the generators of the stabilizer subgroup of a time-like line allows us to obtain these quantum algebras in a new basis that generalizes for any the bicrossproduct basis of -Poincaré [16]. These results are analysed in connection with 2 + 1 quantum gravity [38] and a "duality" between curvature/cosmological constant and deformation parameter/Planck length is suggested along the same lines of the so-called "semidualization" approach for Hopf algebras in 2 + 1 quantum gravity [59] associated with the exchange of the cosmological length scale and the Planck mass (see also [60,61]). Finally, some remarks and comments concerning recent findings in this framework close the paper.…”

“…Moreover, the "semidualization" approach in 2 + 1 quantum gravity introduced in [59] (see also [60,61]) provides a more complete Hopf algebraic understanding of this duality between the Planck scale and the cosmological constant = 2 = −Λ and shows its direct connection with the quantum version of the so-called Born reciprocity principle [74] between (now noncommutative) coordinates and (curved) momenta. This framework is also helpful in order to understand the existing constraints on contraction limits involving these two parameters that we will discuss in the sequel.…”

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

“…In fact, the κ-AdS ω algebra can be thought of as a two-parametric deformation, which is ruled by a quantum deformation parameter z ¼ 1=κ (the Planck scale) and a "classical" deformation parameter ω ¼ −Λ (the cosmological constant), which has a well-defined geometrical meaning. As we show in the following, the roles of the two deformation parameters are interchanged when the dual Poisson-Lie group is considered, in the spirit of the semidualization approach to (2 þ 1) quantum gravity [46,47].…”

Deformed phase spaces with group valued momenta

“…A description of a moving defect can be obtained by boosting the conical metric, in this case the three momentum of the particle will be a general element of SL(2, R) [3,51]. Various treatments exist for the description of the phase space of point particles coupled to gravity in three dimensions [3,52,53] and its symmetries [54,55].…”