  We give a general definition of spin foam models, and then of models of 4d
quantum gravity based on constraining BF theory. We highlight the construction
and quantization ambiguities entering model building, among which the choice of
quantization map applied to the B variables carrying metric information after
imposing simplicity constraints, and the different strategies for imposing the
latter constraints. We then construct a new spin foam model for 4d quantum
gravity, using the flux representation of states and amplitudes, based on the
Duflo quantization map and the associated non-commutative Fourier transform for
Lie groups. The advantages of the new model are the geometrically transparent
way in which constraints are imposed, and the underlying mathematical
properties of the Duflo map itself. Last the presence of a closed analytical
formula for the model's amplitudes is another valuable asset for future
applications.
