Towards (3+1) gravity through Drinfel'd doubles with cosmological constant

Abstract

We present the generalisation to (3 + 1) dimensions of a quantum deformation of the (2 + 1) (Anti)-de Sitter and Poincaré Lie algebras that is compatible with the conditions imposed by the Chern-Simons formulation of (2 + 1) gravity. Since such compatibility is automatically fulfilled by deformations coming from Drinfel'd double structures, we believe said structures are worth being analysed also in the (3 + 1) scenario as a possible guiding principle towards the description of (3 + 1) gravity. To this aim, a canonical classical r-matrix arising from a Drinfel'd double structure for the three (3 + 1) Lorentzian algebras is obtained. This r-matrix turns out to be a twisted version of the one corresponding to the (3 + 1) κ-deformation, and the main properties of its associated noncommutative spacetime are analysed. In particular, it is shown that this new quantum spacetime is not isomorphic to the κ-Minkowski one, and that the isotropy of the quantum space coordinates can be preserved through a suitable change of basis of the quantum algebra generators. Throughout the paper the cosmological constant appears as an explicit parameter, thus allowing the (flat) Poincaré limit to be straightforwardly obtained.

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

“…This term does not appear either in the (3 + 1)D -Poincaré algebra (with = 0) or in the (2 + 1)D -(A)dS algebras. A twisted version of (66) with a second deformation parameter was considered in [97] by imposing some physical requirements, and the very same classicalmatrix has been derived in [98] from a Drinfel' d-double approach; namely,…”

Basic quantizations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic $$ {\mathfrak{o}}^{\star }(4) $$ symmetries

“…The quantum deformations of o(4 − k, k) (k = 1, 2) have been extensively studied as describing the NC geometry of 2 + 1-dimensional QG with cosmological constant Λ = 0 [73][74][75][76]. The classical action of D = 3 gravity can be introduced geometrically as the gauge theory described by D = 3 Chern-Simons (CS) model.…”

“…If we search for quantum deformations of Fock-Rosly construction, it appears that only classical r-matrices obtained from Drinfeld double (DD) structures [10,11] are allowed [79][80][81]. The DD structures and corresponding classical r-matrices for o(3, 1) and o(2, 2) algebras were recently constructed and classified [74][75][76]. One can check that such quantum deformations, which are well adjusted to the description of quantum-deformed D = 3 gravity, are generated by the subclass of classical r-matrices, listed in [20,22] and quantized in this paper.…”

Quantum groups and noncommutative spacetimes with cosmological constant

“…The answer is affirmative for the twisted κ-AdS ω deformation, whose DD structure has been recently presented in [23,24]. In the basis (2), the corresponding classical r-matrix reads…”

“…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:…”