On the space of generalized fluxes for loop quantum gravity

We show that the space of generalized fluxes-momentum space-for loop quantum gravity cannot be constructed by Fourier transforming the projective limit construction of the space of generalized connections-position space-due to the non-Abelianess of the gauge group SU(2). From the Abelianization of SU(2), U(1)(3), we learn that the space of generalized fluxes turns out to be an inductive limit, and we determine the consistency conditions the fluxes should satisfy under coarse graining of the underlying graphs. We comment on the applications to loop quantum cosmology (LQC), in particular, how the characterization of the Bohr compactification of the real line as a projective limit opens the way for a similar analysis for LQC.