Strict Deformation Quantization for Actions of a Class of Symplectic Lie Groups

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.Comment: 17 pages, LaTeX, reference added, minor corrections in the Introductio

Homotopes of Symmetric Spaces I. Construction by Algebras with Two Involutions

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".Comment: V2: minor correction

Affine connections and symmetry jets

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.Comment: 94 pages, 15 figure

Homotopes of Symmetric Spaces II. Structure Variety and Classification

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.Comment: V2: minor corrections and modification

Bargmann-Fock realization of the noncommutative torus

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

Universal Deformation Formulae for Three-Dimensional Solvable Lie groups

We apply methods from strict quantization of solvable symmetric spaces to obtain universal deformation formulae for actions of every three-dimensional solvable Lie group. We also study compatible co-products by generalizing the notion of smash product in the context of Hopf algebras. We investigate in particular the dressing action of the `book' group on SU(2)

Rankin-Cohen brackets and formal quantization

In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results \cite{cm:deformation}. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation \cite{z:deformation} of modular forms, and relate this deformation to the Weyl-Moyal product. We also show that the projective structure introduced by Connes and Moscovici is equivalent to the existence of certain geometric data in the case of foliation groupoids. Using the methods developed by the second author \cite{t1:def-gpd}, we reconstruct a universal deformation formula of the Hopf algebra \calh_1 associated to codimension one foliations. In the end, we prove that the first Rankin-Cohen bracket $RC_1$ defines a noncommutative Poisson structure for an arbitrary \calh_1 action.Comment: 21 pages, minor changes and typos correcte