Three dimensional quantum geometry and deformed symmetry

We study a three dimensional noncommutative space emerging in the context of three dimensional Euclidean quantum gravity. Our starting point is the assumption that the isometry group is deformed to the Drinfeld double D(SU(2)). We generalize to the deformed case the construction of E3 as the quotient of its isometry group ISU(2) by SU(2). We show that the algebra of functions on E3 becomes the noncommutative algebra of SU(2) distributions, C(SU(2))∗, endowed with the convolution product. This construction gives the action of ISU(2) on the algebra and allows the determination of plane waves and coordinate functions. In particular, we show the following: (i) plane waves have bounded momenta; (ii) to a given momentum are associated several SU(2) elements leading to an effective description of ϕ∊C(SU(2))∗ in terms of several physical scalar fields on E3; (iii) their product leads to a deformed addition rule of momenta consistent with the bound on the spectrum. We generalize to the noncommutative setting the “local” action for a scalar field. Finally, we obtain, using harmonic analysis, another useful description of the algebra as the direct sum of the algebra of matrices. The algebra of matrices inherits the action of ISU(2): rotations leave the order of the matrices invariant, whereas translations change the order in a way we explicitly determine.