  We comment on structural properties of the algebras $\mathfrak{A}_{LQG/LQC}$
underlying loop quantum gravity and loop quantum cosmology, especially the
representation theory, relating the appearance of the (dynamically induced)
superselection structure ($\theta$-sectors) in loop quantum cosmology to
recently proposed representations with non-degenerate background geometries in
loop quantum gravity with Abelian structure group. To this end, we review and
employ the concept of extending a given (observable) algebra with possibly
non-trivial centre to a (charged) field algebra with (global) gauge group.We
also interpret the results in terms of the geometry of the structure group G.
Furthermore, we analyze the Koslowski-Sahlmann representations with
non-degenerate background in the case of a non-Abelian structure group. We find
that these representations can be interpreted from two different, though
related, points view: Either, the standard algebras of loop quantum gravity
need to be extended by a (possibly) central term, or the elementary flux vector
fields need to acquire a shift related to the (classical) background to make
these representations well-defined. Both perspectives are linked by the fact
that the background shift is not an automorphism of the algebras, but rather an
affine transformation. Finally, we show how similar algebraic mechanisms, which
are used to explain the breaking of chiral symmetry and the occurrence of
$\theta$-vacua in quantum field theory, extend to loop quantum gravity. Thus,
opening a path for the discussion of these questions in loop quantum gravity.
