Twisted (2+1) κ-AdS Algebra, Drinfel'd Doubles and Non-Commutative Spacetimes

Abstract

Abstract. We construct the full quantum algebra, the corresponding Poisson-Lie structure and the associated quantum spacetime for a family of quantum deformations of the isometry algebras of the (2+1)-dimensional anti-de Sitter (AdS), de Sitter (dS) and Minkowski spaces. These deformations correspond to a Drinfel'd double structure on the isometry algebras that are motivated by their role in (2+1)-gravity. The construction includes the cosmological constant Λ as a deformation parameter, which allows one to treat these cases in a common framework and to obtain a twisted version of both space-and time-like κ-AdS and dS quantum algebras; their flat limit Λ → 0 leads to a twisted quantum Poincaré algebra. The resulting non-commutative spacetime is a nonlinear Λ-deformation of the κ-Minkowski one plus an additional contribution generated by the twist. For the AdS case, we relate this quantum deformation to two copies of the standard (Drinfel'd-Jimbo) quantum deformation of the Lorentz group in three dimensions, which allows one to determine the impact of the twist.

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

“…This is indeed ω independent but, as it was shown in [44,48], when the all-orders quantum group is computed, the quantum spacetime with nonvanishing ω is a nonlinear algebra whose higher-order contributions explicitly depend on the cosmological constant.…”

“…In fact, the (2 þ 1) κ-AdS ω deformations are generated by the classical r-matrix given by (see [43,44])…”

“…Notice that this coproduct is written in a "bicrossproducttype" basis that generalizes the one corresponding to the (2 þ 1) κ-AdS ω algebra [44,45]. As it can be easily checked, the κ-Poincaré coproduct (2) is obtained from the above expressions in the limit ω → 0.…”

“…(iii) The Minkowski spacetime M 3þ1 ≡ ISOð3; 1Þ= SOð3; 1Þ arises when ω ¼ Λ ¼ 0. Explicit ambient space coordinates for these three maximally symmetric Lorentzian spacetimes can be obtained by making use of a suitable realization of the Lie groups obtained by exponentiation of a faithful representation of the AdS ω Lie algebra [see [43][44][45] for details in the (2 þ 1)-dimensional case].…”

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]).…”