Abstract 
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[eng] Geometry and physics have a long history of mutual interactions. Among those interactions coming from quantum mechanics, the early “Weyl quantization” of the phase space observables paved the way for the understanding of some deep relations between Poisson geometry and quantum algebras. As two outgrowths, of prime importance in the last decades were the Moyal and Weyl products, respectively in formal and nonformal deformation quantization. The notion of a midpoint of two given points appears naturally in the Weyl quantization and product, suggesting to consider symplectic symmetric spaces with midpoint map.
A basic conjecture of Weinstein in this context relates the associativity of nonformal products of functions with the symplectic area of socalled “double triangles”. The aim of this thesis is to improve the understanding of the structure of solvable symplectic symmetric spaces and to provide the necessary tools for an inductive construction of quantizations of these spaces satisfying the above conjecture.
For a general symmetric space, we relate the existence of a midpoint map to the existence and uniqueness of geodesics joining two points, and characterize the symmetric spaces for which each triangle has a unique double triangle. We introduce symplectic reduction and double extension for symplectic symmetric spaces, and uncover a “primitive structure” canonically attached to any such space, allowing the construction of Tstar extensions à la Bordemann. We then study the behaviour of the main ingredient in our proposed quantization procedure, the Severa projection, under the abovementioned operations. We end the thesis by providing explicit computations for a new class of examples yielding in particular new universal deformation formulæ for the actions of some symplectic Lie groups on Fréchet algebras.
