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

Ballesteros, Angel  (Departamento de Física, Universidad de Burgos, Burgos, E-09001, Spain) ; Herranz, Francisco J. (Departamento de Física, Universidad de Burgos, Burgos, E-09001, Spain) ; Meusburger, Catherine (Department Mathematik, FAU Erlangen-Nürnberg, Cauerstr. 11, Erlangen, D-91058, Germany)

22 March 2014

Abstract: We show that the Drinfel'd double associated to the standard quantum deformation ${\mathit{sl}}_{\eta }\left(2,\mathbb{R}\right)$ is isomorphic to the $\left(2+1\right)$ -dimensional AdS algebra with the initial deformation parameter η 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 $\left(2+1\right)$ 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 Λ -deformation of the Minkowskian one.

Published in: Physics letters B 732 (2014) 201-209
DOI: 10.1016/j.physletb.2014.03.036

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