Q-deformation and semidualization in 3D quantum gravity

We explore in detail the role in euclidean 3D quantum gravity of quantum Born reciprocity or 'semidualization'. The latter is an algebraic operation defined using quantum group methods that interchanges position and momentum. Using this we are able to clarify the structural relationships between the effective noncommutative geometries that have been discussed in the context of 3D gravity. We show that the spin model based on D(U(su₂)) for quantum gravity without cosmological constant is the semidual of a quantum particle on a 3-sphere, while the bicrossproduct (DSR) C[R ²?R] U(su₂)model based on is the semidual of a quantum particle on hyperbolic space. We show further how the different models are all specific limits of q-deformed models with , q = e^h v??/mp where m_p is the Planck mass and ? is the cosmological constant, and argue that semidualization interchanges m_p ? l_c, where l_c is the cosmological length scale . We investigate the physics of semidualization by studying representation theory. In both the spin model and its semidual we show that irreducible representations have a physical picture as solutions of a respectively noncommutative/curved wave equation. We explain, moreover, that the q-deformed model, at a certain algebraic level, is self-dual under semidualization. © 2009 IOP Publishing Ltd.