A coordinate-free definition for Wick-type symbols is given for symplectic manifolds by means of the Fedosov procedure. The main ingredient of this approach is a bilinear symmetric form defined on the complexified tangent bundle of the symplectic manifold and subject to some set of algebraic and differential conditions. It is precisely the structure which describes a deviation of the Wick-type ...

0.1. In this paper we construct a number of geometrically meaningful 1– and 2– cocycles of infinite dimensional Lie groups and Lie algebras. The groups are groups of diffeomorphisms of smooth manifolds and their subgroups, such as the group of symplectomorphisms of a symplectic manifold; the Lie algebras are algebras of vector fields. Coefficient spaces of the cocycles are spaces of various ten...

This paper is about the role of Planck’s constant, h̄, in the geometric quantization of Poisson manifolds using symplectic groupoids. In order to construct a strict deformation quantization of a given Poisson manifold, one can use all possible rescalings of the Poisson structure, which can be combined into a single “Heisenberg-Poisson” manifold. The new coordinate on this manifold is identified ...

We establish that Hitchin’s connection exist for any rigid holomorphic family of Kähler structures on any compact pre-quantizable symplectic manifold which satisfies certain simple topological constraints. Using Toeplitz operators we prove that Hitchin’s connection induces a unique formal connection on smooth functions on the symplectic manifold. Parallel transport of this formal connection pro...

We relate the Bohr-Sommerfeld conditions established in λ-microlocal analysis to formal deformation quantization of symplectic manifolds by classifying star products adapted to some Lagrangian sub-manifold L, i.e. products preserving the classical vanishing ideal IL of L up to IL-preserving equivalences.

Infinite dimensional Poisson structures play a big role in the theory of infinite dimensional Lie algebras 1 , in the theory of integrable system 2 , and in field theory 3 . But for instance, in 2 , the test functional space where the hydrodynamic Poisson structure acts continuously is not conveniently defined. In 4, 5 we have defined such a test functional space in the case of a linear Poisson...

We prove the existence of a deformation quantization for integrable Poisson structures on R 3 and give a generalization for a special class of three dimensional manifolds. The program of deformation quantization of the function algebra on a sym-plectic manifold extends naturally to manifolds with nonregular Poisson structures. In contrast to symplectic manifolds the existence of star products o...

We express the difference between Poisson bracket and deformed bracket for Kontsevich deformation quantization on any Poisson manifold by means of second derivative of the formality quasiisomorphism. The counterpart on star products of the action of formal diffeomorphisms on Poisson formal bivector fields is also investigated. Mathematics Subject Classification (2000) : 16S80, 53D17, 53D55, 58A50.

A generalised notion of Kac-Moody algebra is defined using smooth maps from a compact real manifold M to finite-dimensional Lie group, by means complete orthonormal bases for Hermitian inner product on the and Fourier expansion. The Peter–Weyl theorem case manifolds related groups coset spaces discussed, appropriate Hilbert space L2(M) square-integrable functions are constructed. It shown that ...

The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold M is presented as a second class constrained surface in the fibre bundle T * ρ M which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The...