A (2+1) non-commutative Drinfel'd double spacetime with cosmological constant

Abstract

We show that the Drinfel'd double associated to the standard quantum deformation $sl_η(2,R)$ is isomorphic to the (2+1)-dimensional AdS algebra with the initial deformation parameter $η$ related to the cosmological constant $Λ=-η^2$. This gives rise to a generalisation of a non-commutative Minkowski spacetime that arises as a consequence of the quantum double symmetry of (2+1) gravity to non-vanishing cosmological constant. The properties of the AdS quantum double that generalises this symmetry to the case $Λ\neq 0$ are sketched, and it is shown that the new non-commutative AdS spacetime is a nonlinear $Λ$-deformation of the Minkowskian one.

14 pages; some comments and references added