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

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 $\Lambda$ 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 $\kappa$-AdS and dS quantum algebras; their flat limit $\Lambda\to 0$ leads to a twisted quantum Poincar\'e algebra. The resulting non-commutative spacetime is a nonlinear $\Lambda$-deformation of the $\kappa$-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.