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 $s{l}_{\eta }(2,R)$ is isomorphic to the (2+1)-dimensional AdS algebra with the initial deformation parameter $\eta$ related to the cosmological constant $\Lambda =-{\eta }^2$. 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 $\Lambda \ne 0$ are sketched, and it is shown that the new non-commutative AdS spacetime is a nonlinear $\Lambda$-deformation of the Minkowskian one.

14 pages; some comments and references added

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• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
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• Extending general covariance: Moyal-type noncommutative manifolds
• From Lorentzian to Galilean (2+1) gravity: Drinfel'd doubles, quantisation and noncommutative spacetimes
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• A Short Essay on Quantum Black Holes and Underlying Noncommutative Quantized Space-Time
• Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups
• Fuzzy worldlines with $\kappa$-Poincaré symmetries