Abstract:
This work is a contribution to the area of Strict Quantization (in the sense of Rieffel) in the presence of curvature and non-Abelian group actions. More precisely, we use geometry to obtain explicit oscillatory integral formulae for strongly invariant strict deformation quantizations of a class of solvable symplectic symmetric spaces. Each of these quantizations gives rise to a field of (pre)-C*-algebras whose fibers are function algebras which are closed under the deformed product. The symmetry group of the symmetric space acts on each fiber by C*-algebra automorphisms.

Abstract:
We define a universal deformation formula (UDF) for the actions of the affine group on Frechet algebras. More precisely, starting with any associative Frechet algebra which the affine group acts on in a strongly continuous and isometrical manner, the UDF produces a family of topological associative algebra structures on the space of smooth vectors of the action deforming the initial product. The deformation field obtained is based over an infinite dimensional parameter space naturally associated with the space of pseudo-differential operators on the real line. This note also presents some geometrical aspects of the UDF and in particular its relation with hyperbolic geometry.

Abstract:
Ricci-type symplectic manifolds have been introduced and extensively studied by M. Cahen et al.. In this note, we describe their deformation quantizations in the split solvable symmetric case. In particular, we introduce the notion of non-formal tempered deformation quantization on such a space. We show that the set of tempered deformation quantizations is in one-to-one correspondence with the space of Schwartz operator multipliers on the real line. Moreover we prove that every invariant formal star product on a split Ricci-type solvable symmetric space is an asymptotic expansion of a tempered non-formal quantization. This note illustrates and partially reviews through an example a problematic studied by the author regarding non-formal quantization in presence of large groups of symmetries.

Abstract:
This is the pdf -version of the author's Ph.D. thesis (1995, ULB, Belgium). The notion of symeplectic symmertic space is introduced and studied via Lie theoretical and symplectic geoemetrical methods. The first chapter concerns basic poperties, however, an explicit formula for the Loos connection in the symplectic framework is given. In the second chapter, one proves a decomposition result analogous to de Rham - Wu's decomposition theorem for pseudo-Riemaniann symmertic spaces; this result is established for the general category of Lie triple systems as well. The chapter three deals with the reductive case; a structure theorem analoguous to Borel- de Sibenthal's theorem for Riemanniann spaces is established at the root system level. The three last chapters deal with various classes of symplectic symmetric spaces; results such as the uniqueness of a compact factor as well as a complete classification in dimension four are presented.

Abstract:
We present explicit universal strict deformation quantization formulae for actions of Iwasawa subgroups AN of SU(1,n). This answers a question raised by Rieffel.

Abstract:
We investigate a special kind of contraction of symmetric spaces (respectively, of Lie triple systems), called homotopy. In this first part of a series of two papers we construct such contractions for classical symmetric spaces in an elementary way by using associative algebras with several involutions. This construction shows a remarkable duality between the underlying "space" and the "deformation parameter".

Abstract:
We classify homotopes of classical symmetric spaces (studied in Part I of this work). Our classification uses the fibered structure of homotopes: they are fibered as symmetric spaces, with flat fibers, over a non-degenerate base; the base spaces correspond to inner ideals in Jordan pairs. Using that inner ideals in classical Jordan pairs are always complemented (in the sense defined by O. Loos and E. Neher), the classification of homotopes is obtained by combining the classification of inner ideals with the one of isotopes of a given inner ideal.

Abstract:
Let $\mathbb B$ be a Lie group admitting a left-invariant negatively curved K\"ahlerian structure. Consider a strongly continuous action $\alpha$ of $\mathbb B$ on a Fr\'echet algebra $\mathcal A$. Denote by $\mathcal A^\infty$ the associated Fr\'echet algebra of smooth vectors for the action $\alpha$. In the Abelian case $\mathbb B=\mathbb R^{2n}$ and $\alpha$ isometric, Marc Rieffel proved that Weyl's operator symbol composition formula yields a deformation through Fr\'echet algebra structures ${\star_{\theta}^\alpha}_{\theta\in\mathbb R}$ on $\mathcal A^\infty$. When $\mathcal A$ is a $C^\star$-algebra, every deformed algebra $(\mathcal A^\infty,\star^\alpha_\theta)$ admits a compatible pre-$C^\star$-structure. In this paper, we prove both analogous statements in the general negatively curved K\"ahlerian group and (non-isometric) "tempered" action case. The construction relies on the one hand on combining a non-Abelian version of oscillatory integral on tempered Lie groups with geometrical objects coming from invariant WKB-quantization of solvable symplectic symmetric spaces, and, on the second hand, in establishing a non-Abelian version of the Calder\`on-Vaillancourt Theorem. In particular, we give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.

Abstract:
We establish a bijective correspondence between affine connections and a class of semi-holonomic jets of local diffeomorphisms of the underlying manifold called symmetry jets in the text. The symmetry jet corresponding to a torsion free connection consists in the family of $2$-jets of the geodesic symmetries. Conversely, any connection is described in terms of the geodesic symmetries by a simple formula involving only the Lie bracket of vector fields. We then formulate, in terms of the symmetry jet, several aspects of the theory of affine connections and obtain geometric and intrinsic descriptions of various related objects involving the gauge groupoid of the frame bundle. In particular, the property of uniqueness of affine extension admits an equivalent formulation as the property of existence and uniqueness of a certain groupoid morphism. Moreover, affine extension may be carried out at all orders and this allows for a description of the tensors associated to an affine connections, namely the torsion, the curvature and their covariant derivatives of all orders, as obstructions for the affine extension to be holonomic. In addition this framework provides a nice interpretation for the absence of other tensors.

Abstract:
We give an interpretation of the Bargman transform as a correspondence between state spaces that is analogous to commonly considered intertwiners in representation theory of finite groups. We observe that the non-commutative torus is nothing else that the range of the star-exponential for the Heisenberg group within the Kirillov's orbit method context. We deduce from this a realization of the non-commutative torus as acting on a Fock space of entire functions.