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The essential features of a quantum group deformation of classical symmetries of General Relativity in the case with non-vanishing cosmological constant $\Lambda$ are presented. We fully describe (anti-)de Sitter non-commutative spacetimes and curved momentum spaces in (1+1) and (2+1) dimensions arising from the $\kappa$-deformed quantum group symmetries. These non-commutative spacetimes are introduced semiclassically by means of a canonical Poisson structure, the Sklyanin bracket, depending on the classical $r$-matrix defining the $\kappa$-deformation, while curved momentum spaces are defined as orbits generated by the $\kappa$-dual of the Hopf algebra of quantum symmetries. Throughout this construction we use kinematical coordinates, in terms of which the physical interpretation becomes more transparent, and the cosmological constant $\Lambda$ is included as an explicit parameter whose $\Lambda \rightarrow 0$ limit provides the Minkowskian case. The generalization of these results to the physically relevant (3+1)-dimensional deformation is also commented., Comment: Based on the contribution presented at the "First Hermann Minkowski Meeting on the Foundations of Spacetime Physics" held in Albena, Bulgaria, May 15-18, 2017

The exploration of the universe has recently entered a new era thanks to the multimessenger paradigm, characterized by a continuous increase in the quantity and quality of experimental data that is obtained by the detection of the various cosmic messengers (photons, neutrinos, cosmic rays and gravitational waves) from numerous origins. They give us information about their sources in the universe and the properties of the intergalactic medium. Moreover, multi-messenger astronomy opens up the possibility to search for phenomenological signatures of quantum gravity. On the one hand, the most energetic events allow us to test our physical theories at energy regimes which are not directly accessible in accelerators; on the other hand, tiny effects in the propagation of very high energy particles could be amplified by cosmological distances. After decades of merely theoretical investigations, the possibility of obtaining phenomenological indications of Planck-scale effects is a revolutionary step in the quest for a quantum theory of gravity, but it requires cooperation between different communities of physicists (both theoretical and experimental). This review, prepared within the COST Action CA18108 ‘‘Quantum gravity phenomenology in the multi-messenger approach", is aimed at promoting this cooperation by giving a state-of-the art account of the interdisciplinary expertise that is needed in the effective search of quantum gravity footprints in the production, propagation and detection of cosmic messengers., Talent Scientific Research Program of College of Physics, Sichuan University 1082204112427, Fostering Program in Disciplines Possessing Novel Features for Natural Science of Sichuan University 2020SCUNL209, 1000 Talent program of Sichuan province 2021, Xunta de Galicia, European Commission European Union ERDF, "Maria de Maeztu'' Units of Excellence program MDM-2016-0692, Red Tematica Nacional de Astroparticulas RED2018-102661-T, La Caixa Foundation 100010434, European Commission 847648 LCF/BQ/PI21/11830030 754510, Ministry of Education, Science & Technological Development, Serbia 451-03-9/2021-14/200124, FSR Incoming Postdoctoral Fellowship Ministry of Education, Science and Technological Development, Serbia 451-03-9/2021-14/200124, University of Rijeka grant uniri-prirod-18-48, Croatian Science Foundation (HRZZ) IP-2016-06-9782, Villum Fonden 29405 DGA-FSE 2020-E2117R, European Regional Development Fund through the Center of Excellence (TK133) "The Dark Side of the Universe'' European Regional Development Fund (ESIF/ERDF), Ministry of Education, Youth & Sports - Czech Republic CoGraDS-CZ.02.1.01/0.0/0.0/15 003/0000437, Blavatnik grant, Basque Government IT-97916 Basque Foundation for Science (IKERBASQUE), European Space Agency C4000120711 4000132310, FNRS (Belgian Fund for Research), Programa de Apoyo a Proyectos de Investigacion e Innovacion Tecnologica (PAPIIT), Universidad Nacional Autonoma de Mexico TA100122, National University of La Plata X909 DICYT 042131GR, National Research, Development & Innovation Office (NRDIO) - Hungary 123996, FQXi, Swiss National Science Foundation (SNSF), European Commission 181461 199307, Netherlands Organization for Scientific Research (NWO) 680-91-119 15MV71, Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT) Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research (KAKENHI) 20H01899 20H05853 JP21F21789, Estonian Research Council PRG356, Julian Schwinger Foundation, Generalitat Valenciana Excellence PROMETEO-II/2017/033 PROMETEO/2018/165, Istituto Nazionale di Fisica Nucleare (INFN), European ITN project HIDDeN H2020-MSCA-ITN-2019//860881-HIDDeN, Swedish Research Council, European Commission 2016-05996 European Research Council (ERC) European Commission 668679, Advanced ERC grant TReX, Ministry of Education, Universities and Research (MIUR) 2017X7X85K, Fonds de la Recherche Scientifique - FNRS 4.4501.18, Ministry of Research, Innovation and Digitization - Romania PN19-030102-INCDFM PN-III-P4ID-PCE-2020-2374, United States Department of Energy (DOE) DE-SC0020262, Ministry of Science, ICT & Future Planning, Republic of Korea 075-15-2020-778, German Academic Scholarship Foundation German Research Foundation (DFG) 408049454 420243324 425333893 445990517 Germany's Excellence Strategy (EXC 2121 "Quantum Universe'') 390833306 390837967 Federal Ministry of Education & Research (BMBF) 05 A20GU2 05 A20PX1, Centro de Excelencia "Severo Ochoa'' SEV-2016-0588, CERCA program of the Generalitat de Catalunya, Agencia de Gestio D'Ajuts Universitaris de Recerca Agaur (AGAUR) Generalitat de Catalunya 2017-SGR-1469 2017-SGR-929 ICCUB CEX2019-000918-M, National Science Centre, Poland 2019/33/B/ST2/00050 2017/27/B/ST2/01902, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPQ) 306414/2020-1, Dicyt-USACH 041931MF, National Science Fund of Bulgaria KP-06-N 38/11 RCN ROMFORSK 302640, Comunidad de Madrid 2018-T1/TIC-10431 2019-T1/TIC-13177 S2018/NMT-4291, UK Research & Innovation (UKRI), Science & Technology Facilities Council (STFC) ST/T000759/1 ST/P000258/1 ST/T000732/1 ST/V005596/1, Portuguese Foundation for Science and Technology UIDB/00618/2020 UIDB/00777/2020 UIDP/00777/2020 CERN/FIS-PAR/0004/2019 PTDC/FIS-PAR/29436/2017 PTDC/FISPAR/31938/2017 PTDC/FIS-OUT/29048/2017 SFRH/BD/137127/2018, Centre National de la Recherche Scientifique (CNRS), LabEx UnivEarthS ANR-10-LABX-0023 ANR18-IDEX-0001, Junta de Andalucia European Commission A-FQM-053-UGR18, Natural Sciences and Engineering Research Council of Canada (NSERC) RGPIN-2021-03644, National Science Centre Poland Sonata Bis 2019/33/B/ST2/00050 DEC-2017/26/E/ST2/00763, Natural Sciences and Engineering Research Council of Canada (NSERC) DGIID-DGA 2015-E24/2, Spanish Research State Agency and Ministerio de Ciencia e Innovacion MCIN/AEI PID2019-104114RB-C32 PID2019-105544GB-I00 PID2019-105614GB-C21 PID2019106515GB-I00 PID2019-106802GB-I00 PID2019-107394GB-I00 PID2019-107844GB-C21 PID2019-107847RB-C41 MCIN/AEI PGC2018-095328-B-I00 PGC2018-094856-B-I00 PGC2018-096663-B-C41 PGC2018-096663-B-C44 PGC2018-094626-BC21 PGC2018-101858-B-I00 FPA2017-84543-P FPA2016-76005-C2-1-P, Spanish 'Ministerio de Universidades' BG20/00228 Spanish Government PID2020-115845GBI00 Generalitat de Catalunya Comunidad de Madrid S2018/NMT-4291 Spanish Government PID2019-105544GB-I00, Perimeter Institute for Theoretical Physics, Government of Canada through the Department of Innovation, Science and Economic Development, Province of Ontario through the Ministry of Colleges and Universities, Centre National de la Recherche Scientifique (CNRS), Netherlands Organization for Scientific Research (NWO), Fundamental Questions Institute (FQXi), European Cooperation in Science and Technology (COST) CA18108, Research Council of University of Guilan, Iniziativa Specifica TEONGRAV Iniziativa Specifica QGSKY Iniziativa Specifica QUAGRAP Iniziativa Specifica GeoSymQFT, the Spanish Research State Agency and Ministerio de Ciencia e Innovacion MCIN/AEI PID2020-115845GBI00 PID2019-108485GB-I00 PID2020-113334GB-I00 PID2020-113701GB-I00 PID2020-113775GB-I00 PID2020-118159GB-C41 PID2020-118159GA-C42 PRE2019-089024, Rothchild grant UID/MAT/00212/2020 FPU18/04571

A novel approach to study the properties of models with quantum-deformed relativistic symmetries relies on a noncommutative space of worldlines rather than the usual noncommutative spacetime. In this setting, spacetime can be reconstructed as the set of events, that are identified as the crossing of different worldlines. We lay down the basis for this construction for the κ-Poincaré model, analyzing the fuzzy properties of κ-deformed time-like worldlines and the resulting fuzziness of the reconstructed events.

We derive the non-relativistic $c\to\infty$ and ultra-relativistic $c\to 0$ limits of the $\kappa$-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the $\kappa$-(A)dS quantum algebra, and quantize the resulting contracted Poisson-Hopf algebras, thus giving rise to the $\kappa$-deformation of the Newtonian (Newton-Hooke and Galilei) and Carrollian (Para-Poincar\'e, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corresponding $\kappa$-Newtonian and $\kappa$-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the $\kappa$-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter $\kappa$, the curvature parameter $\eta$ and the speed of light parameter $c$., Comment: 19 pages. v2 matches version accepted for publication by Phys. Lett. B; some references added

Producción Científica, Curved momentum spaces associated to the k-deformation of the (3+1) de Sitter and Anti-de Sitter algebras are constructed as orbits of suitable actions of the dual Poisson-Lie group associated to the k-deformation with non-vanishing cosmological constant. The k-de Sitter and k-Anti-de Sitter curved momentum spaces are separately analysed, and they turn out to be, respectively, half of the (6+1)-dimensional de Sitter space and half of a space with SO(4, 4) invariance. Such spaces are made of the momenta associated to spacetime translations and the ‘hyperbolic’ momenta associated to boost transformations. The known k-Poincaré curved momentum space is smoothly recovered as the vanishing cosmological constant limit from both of the constructions., Ministerio de Economía, Industria y Competitividad (projects MTM2013-43820-P / MTM2016-79639-P), Junta de Castilla y León (projects BU278U14 / VA057U16), European Cooperation in Science and Technology (Action MP1405 QSPACE)

We bring the concept that quantum symmetries describe theories with nontrivial momentum space properties one step further, looking at quantum symmetries of spacetime in presence of a nonvanishing cosmological constant $\Lambda$. In particular, the momentum space associated to the $\kappa$-deformation of the de Sitter algebra in (1+1) and (2+1) dimensions is explicitly constructed as a dual Poisson-Lie group manifold parametrized by $\Lambda$. Such momentum space includes both the momenta associated to spacetime translations and the `hyperbolic' momenta associated to boost transformations, and has the geometry of (half of) a de Sitter manifold. Known results for the momentum space of the $\kappa$-Poincar\'e algebra are smoothly recovered in the limit $\Lambda\to 0$, where hyperbolic momenta decouple from translational momenta. The approach here presented is general and can be applied to other quantum deformations of kinematical symmetries, including (3+1)-dimensional ones., Comment: 13 pages