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.,  for the calculation). Thus, we obtain the identification (18) used in isotropic loop cosmology. (Such a normalization can only be obtained in curved models.)
© Max Planck Society and the author(s)