### B.2 Isotropic models

On Bianchi models, additional symmetries can be imposed, which corresponds to a further symmetry
reduction and introduces non-trivial isotropy subgroups. These models with enhanced symmetry can be
treated on an equal footing by writing the symmetry group as a semidirect product , with
the isotropy subgroup and the translational subgroup , which is one of the Bianchi
groups. Composition in this group is defined as , which
depends on the group homomorphism into the automorphism group of
(which will be denoted by the same letter as the representation on used below).
Inverse elements are given by . To determine the form of invariant
connections we have to compute the Maurer-Cartan form on (using the usual notation):
Here the Maurer-Cartan forms on and on appear. We then choose an embedding
, which can most easily be done as . Thus, ,
and a reconstructed connection takes the form , which is the same as for
anisotropic models before (where now are left invariant one-forms on the translation group
). However, here is constrained by equation (69) and we get only a subset as isotropic
connections.
To solve Equation (69) we have to treat LRS (locally rotationally symmetric) models with a single
rotational symmetry and isotropic models separately. In the first case we choose , whereas in
the second case we have ( denotes the linear span). Equation (69) can be written
infinitesimally as

( for LRS, for isotropy). The are generators of , on which
the isotropy subgroup acts by rotation, . This is the derivative of the
representation defining the semidirect product : Conjugation on the left hand side of (69)
is , which follows from the composition in
.

Next, we have to determine the possible conjugacy classes of homomorphisms . For LRS
models their representatives are given by

for (as will be shown in detail below for spherically symmetric connections). For the
components of defined by , Equation (69) takes the form . This
has a non-trivial solution only for , in which case can be written as

with arbitrary numbers , , (the factors of are introduced for the sake of normalization).
Their conjugate momenta take the form

and the symplectic structure is given by

and vanishes in all other cases. There is remaining gauge freedom from the reduced structure group
which rotates the pairs and . Gauge invariant are then only and
its momentum .

In the case of isotropic models we have only two homomorphisms and
up to conjugation (to simplify notation we use the same letters for the homomorphisms as in
the LRS case, which is justified by the fact that the LRS homomorphisms are restrictions of
those appearing here). Equation (69) takes the form for without non-trivial
solutions, and for . Each of the last equations has the same form as for
LRS models with , and their solution is with an arbitrary . In this case
the conjugate momenta can be written as , and we have the symplectic structure
.

Thus, in both cases there is a unique non-trivial sector, and no topological charge appears. The
symplectic structure can again be made independent of by redefining , ,
and , , , . If one computes the isotropic
reduction of a Bianchi IX metric following from the left-invariant 1-forms of SU(2), one obtains a closed
Friedmann-Robertson-Walker metric with scale factor (see, e.g., [36] for the calculation).
Thus, we obtain the identification (18) used in isotropic loop cosmology. (Such a normalization can only be
obtained in curved models.)