On the semiduals of local isometry groups in 3d gravity

Abstract

We use factorisations of the local isometry groups arising in 3d gravity for Lorentzian and Euclidean signatures and any value of the cosmological constant to construct associated bicrossproduct quantum groups via semidualisation. In this way we obtain quantum doubles of the Lorentz and rotation groups in 3d, as well as kappa-Poincare algebras whose associated r-matrices have spacelike, timelike and lightlike deformation parameters. We confirm and elaborate the interpretation of semiduality proposed in [13] as the exchange of the cosmological length scale and the Planck mass in the context of 3d quantum gravity. In particular, semiduality gives a simple understanding of why the quantum double of the Lorentz group and the kappa-Poincare algebra with spacelike deformation parameter are both associated to 3d gravity with vanishing cosmological constant, while the kappa-Poincare algebra with a timelike deformation parameter can only be associated to 3d gravity if one takes the Planck mass to be imaginary.

arXiv comment33 pages, new version with expanded introduction and discussion, similar to version which will appear in Journal of Mathematical Physics

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Cited by in corpus (12)

• Quantum Spacetime Phenomenology
• Curved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensions
• The DSR-deformed relativistic symmetries and the relative locality of 3D quantum gravity
• Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti-)de Sitter groups
• Curved momentum spaces from quantum groups with cosmological constant
• Quasitriangular structure and twisting of the 2+1 bicrossproduct model
• Hermitian realizations of kappa-Minkowski spacetime
• Classical r-matrices for the generalised Chern-Simons formulation of 3d gravity
• Causality and momentum conservation from relative locality
• Classical r-matrices via semidualisation
• Deformed phase spaces with group valued momenta
• Quantum symmetries in 2+1 dimensions: Carroll, (a)dS-Carroll, Galilei and (a)dS-Galilei