Abstract
Let G be a locally compact group, \({\Gamma\subset G}\) an abelian subgroup and let Ψ be a continuous 2-cocycle on the dual group \({\Hat\Gamma}\). Let B be a C*-algebra and \({\Delta_B\in{\rm Mor}\,(B,B\otimes{\rm C}_0(G))}\) a continuous right coaction. Using Rieffel deformation, we can construct a quantum group \({({\rm C}_0(G)^{\tilde\Psi\otimes\Psi},\Delta^\Psi)}\) and the deformed C*-algebra B Ψ. The aim of this paper is to present a construction of the continuous coaction \({\Delta_B^\Psi}\) of the quantum group \({({\rm C}_0(G)^{\tilde\Psi\otimes\Psi},\Delta^\Psi)}\) on B Ψ. The transition from the coaction Δ B to its deformed counterpart \({\Delta_B^\Psi}\) is nontrivial in the sense that \({\Delta_B^\Psi}\) contains complete information about Δ B . In order to illustrate our construction we apply it to the action of the Lorentz group on the Minkowski space obtaining a C*-algebraic quantum Minkowski space.
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Communicated by Y. Kawahigashi
Supported by the Marie Curie Research Training Network Non-Commutative Geometry MRTN-CT-2006-031962 and by Geometry and Symmetry of Quantum Spaces, PIRSES-GA-2008-230836.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Kasprzak, P. Rieffel Deformation of Group Coactions. Commun. Math. Phys. 300, 741–763 (2010). https://doi.org/10.1007/s00220-010-1093-9
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DOI: https://doi.org/10.1007/s00220-010-1093-9