  Loop Quantum Gravity is now a well established approach to quantum gravity.
One of the main challenges still faced by the theory is constructing a
consistent dynamics which would lead back to the standard dynamics of the
gravitational field at large scales. Here we will review the recent U(N)
framework for Loop Quantum Gravity and the new spinor representation (that
provides a classical setting for the U(N) framework). Then, we will apply these
techniques to a simple model in order to propose a dynamics for a symmetry
reduced sector of the theory. Furthermore, we will explore certain analogies of
this model with Loop Quantum Cosmology.
