To solve Equation (84) 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 (84
) 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 semi-direct product
: conjugation on the left-hand side of (84
)
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 (84
) 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
. Then only
and its momentum
are gauge invariant.
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 (84
) 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 [44] for the calculation).
Thus, we obtain identification (19
) used in isotropic loop cosmology. (Such a normalization can only be
obtained in curved models.)
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