  In the context of Loop Quantum Gravity (LQG), we study the fate of Thiemann
complexifier in homogeneous and isotropic FRW cosmology. The complexifier is
the dilatation operator acting on the canonical phase space for gravity and
generates the canonical transformations shifting the Barbero-Immirzi parameter.
We focus on the closed algebra consisting in the complexifier, the 3d volume
and the Hamiltonian constraint, which we call the CVH algebra. In standard
cosmology, for gravity coupled to a scalar field, the CVH algebra is identified
as a su(1,1) Lie algebra, with the Hamiltonian as a null generator, the
complexifier as a boost and the su(1,1) Casimir given by the matter density.
The loop gravity cosmology approach introduces a regularization length scale
$\lambda$ and regularizes the gravitational Hamiltonian in terms of SU(2)
holonomies. We show that this regularization is compatible with the CVH
algebra, if we suitably regularize the complexifier and inverse volume factor.
The regularized complexifier generates a generalized version of the Barbero's
canonical transformation which reduces to the classical one when $\lambda
\rightarrow 0$. This structure allows for the exact integration of the actions
of the Hamiltonian constraints and the complexifier. This straightforwardly
extends to the quantum level: the cosmological evolution is described in terms
of SU(1,1) coherent states and the regularized complexifier generates unitary
transformations. The Barbero-Immirzi parameter is to be distinguished from the
regularization scale $\lambda$, it can be rescaled unitarily and the Immirzi
ambiguity ultimately disappears from the physical predictions of the theory.
Finally, we show that the complexifier becomes the effective Hamiltonian when
deparametrizing the dynamics using the scalar field as a clock, thus
underlining the deep relation between cosmological evolution and scale
