On the semiduals of local isometry groups in three-dimensional 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 κ-Poincaré algebras whose associated r-matrices have spacelike, timelike, and lightlike deformation parameters. We confirm and elaborate the interpretation of semiduality proposed by Majid and Schroers ["q-deformation and semi-dualisation in 3d quantum gravity," J. Phys. A 42, 425402 (2009)]10.1088/1751-8113/42/42/425402 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 κ-Poincaré algebra with spacelike deformation parameter are both associated with 3D gravity with vanishing cosmological constant, while the κ-Poincaré algebra with a timelike deformation parameter can only be associated with 3D gravity if one takes the Planck mass to be imaginary.
- Publication:
-
Journal of Mathematical Physics
- Pub Date:
- July 2012
- DOI:
- 10.1063/1.4731229
- arXiv:
- arXiv:1109.4086
- Bibcode:
- 2012JMP....53g3510O
- Keywords:
-
- 04.60.-m;
- 02.10.-v;
- Quantum gravity;
- Logic set theory and algebra;
- General Relativity and Quantum Cosmology;
- High Energy Physics - Theory;
- Mathematical Physics
- E-Print:
- 33 pages, new version with expanded introduction and discussion, similar to version which will appear in Journal of Mathematical Physics