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Noncommutative relativistic spacetimes and worldlines from 2 + 1 quantum (Anti-)de sitter groups Ballesteros Castañeda, Ángel Bruno, N. Rossano Herranz Zorrilla, Francisco José Physics Física The -deformation of the (2 + 1)D anti-de Sitter, Poincaré, and de Sitter groups is presented through a unified approach in which the curvature of the spacetime (or the cosmological constant) is considered as an explicit parameter. The Drinfel’d-double and the Poisson–Lie structure underlying the -deformation are explicitly given, and the three quantum kinematical groups are obtained as quantizations of such Poisson–Lie algebras. As a consequence, the noncommutative (2 + 1)D spacetimes that generalize the -Minkowski space to the (anti-)de Sitter ones are obtained. Moreover, noncommutative 4D spaces of (time-like) geodesics can be defined, and they can be interpreted as a novel possibility to introduce noncommutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related to 2 + 1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related to the Planck length, and the existence of a kind of “duality” between the cosmological constant and the Planck scale is also envisaged. Ministerio de Econom´ıa y Competitividad (MINECO, Spain) under Grants MTM2013-43820-P and MTM2016-79639-P (AEI/FEDER, UE), by Junta de Castilla y Le´on (Spain) under Grants BU278U14 and VA057U16, and by the Action MP1405 QSPACE from the European Cooperation in Science and Technology (COST) 2018-02-06T08:15:38Z 2018-02-06T08:15:38Z 2017 info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion 1687-7357 http://hdl.handle.net/10259/4730 10.1155/2017/7876942 eng Advances in High Energy Physics. 2017, V. 2017, art. ID 7876942 https://doi.org/10.1155/2017/7876942 info:eu-repo/grantAgreement/MINECO/MTM2013-43820-P info:eu-repo/grantAgreement/MINECO/MTM2016-79639-P info:eu-repo/grantAgreement/JCyL/BU278U14 info:eu-repo/grantAgreement/JCyL/VA057U16 info:eu-repo/grantAgreement/UE/COST-MP1405 Attribution 4.0 International http://creativecommons.org/licenses/by/4.0/ info:eu-repo/semantics/openAccess application/pdf Hindawi Publishing Corporation https://riubu.ubu.es/bitstream/10259/4730/7/Ballesteros-AHEP_2017.pdf.jpg Hispana TEXT http://creativecommons.org/licenses/by/4.0/ RIUBU. Repositorio Institucional de la Universidad de Burgos http://hdl.handle.net/10259/4730