Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 052, 26 pages      arXiv:1403.4773      https://doi.org/10.3842/SIGMA.2014.052
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

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

Ángel Ballesteros a, Francisco J. Herranz a, Catherine Meusburger b and Pedro Naranjo a
a) Departamento de Física, Universidad de Burgos, E-09001 Burgos, Spain
b) Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg, Cauerstr. 11, D-91058 Erlangen, Germany

Received March 09, 2014, in final form May 13, 2014; Published online May 18, 2014

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.

Key words: (2+1)-gravity; deformation; non-commutative spacetime; anti-de Sitter; cosmological constant; quantum groups; Poisson-Lie groups; contraction.

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