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We construct the full quantum algebra, the corresponding Poisson-Liestructure and the associated quantum spacetime for a family of quantumdeformations of the isometry algebras of the (2+1)-dimensional anti-de Sitter(AdS), de Sitter (dS) and Minkowski spaces. These deformations correspond to aDrinfel'd double structure on the isometry algebras that are motivated by theirrole in (2+1)-gravity. The construction includes the cosmological constant$\Lambda$ as a deformation parameter, which allows one to treat these cases ina 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 toa twisted quantum Poincar\'e algebra. The resulting non-commutative spacetimeis a nonlinear $\Lambda$-deformation of the $\kappa$-Minkowski one plus anadditional contribution generated by the twist. For the AdS case, we relatethis quantum deformation to two copies of the standard (Drinfel'd-Jimbo)quantum deformation of the Lorentz group in three dimensions, which allows oneto determine the impact of the twist.

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