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

Symmetry Integrability and Geometry-Methods and Applications

article

Ángel Ballesteros¹  Francisco J. Herranz¹  Catherine Meusburger²  Pedro Naranjo¹ 

[1] Universidad de Burgos;Department Mathematik

关键词: (2+1)-gravity;    deformation;    non-commutative spacetime;    anti-de Sitter;    cosmological constant;    quantum groups;    Poisson–Lie groups;    contraction;   

DOI  :  10.3842/SIGMA.2014.052

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown