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In the context of Loop Quantum Gravity (LQG), we study the fate of Thiemanncomplexifier in homogeneous and isotropic FRW cosmology. The complexifier isthe dilatation operator acting on the canonical phase space for gravity andgenerates the canonical transformations shifting the Barbero-Immirzi parameter.We focus on the closed algebra consisting in the complexifier, the 3d volumeand the Hamiltonian constraint, which we call the CVH algebra. In standardcosmology, for gravity coupled to a scalar field, the CVH algebra is identifiedas a su(1,1) Lie algebra, with the Hamiltonian as a null generator, thecomplexifier 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 CVHalgebra, if we suitably regularize the complexifier and inverse volume factor.The regularized complexifier generates a generalized version of the Barbero'scanonical transformation which reduces to the classical one when $\lambda\rightarrow 0$. This structure allows for the exact integration of the actionsof the Hamiltonian constraints and the complexifier. This straightforwardlyextends to the quantum level: the cosmological evolution is described in termsof SU(1,1) coherent states and the regularized complexifier generates unitarytransformations. The Barbero-Immirzi parameter is to be distinguished from theregularization scale $\lambda$, it can be rescaled unitarily and the Immirziambiguity ultimately disappears from the physical predictions of the theory.Finally, we show that the complexifier becomes the effective Hamiltonian whendeparametrizing the dynamics using the scalar field as a clock, thusunderlining the deep relation between cosmological evolution and scale

The Thiemann Complexifier and the CVH algebra for Classical and Quantum FLRW Cosmology