  We study a three dimensional non-commutative 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 the flat Euclidean space
as the quotient of its isometry group ISU(2) by SU(2). We show that the algebra
of functions becomes the non-commutative algebra of SU(2) distributions 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 that: (i) plane waves have bounded momenta;
(ii) to a given momentum are associated several SU(2) elements leading to an
effective description of an element in the algebra in terms of several physical
scalar fields; (iii) their product leads to a deformed addition rule of momenta
consistent with the bound on the spectrum. We generalize to the non-commutative
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.
