Related papers: Formal symplectic groupoid of a deformation quanti…
We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of…
A method of constructing a class of bihamiltonian structures is presented. Elements of this class are generalizations of the so-called bihamiltonian structures of general position on odd-dimensional manifolds. The method consists in a…
We develop an approach to the deformation quantization on the real plane with an arbitrary Poisson structure which based on Weyl symmetrically ordered operator products. By using a polydifferential representation for deformed coordinates…
We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold…
This set of notes corresponds to a mini-course given in September 2018 in Bedlewo; it does not contain any new result; it complements -- with intersection -- the introduction to formal deformation quantization and group actions,…
We introduce new invariants associated to collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these…
We study symplectic forms on hypersurface algebroids. These are a broad generalization of the $b^{k}$-Poisson structures studied extensively by Miranda, Scott, and collaborators, and their geometry is intimately related to the group of…
Our paper develops a theory of Poisson slices and a uniform approach to their partial compactifications. The theory in question is loosely comparable to that of symplectic cross-sections in real symplectic geometry.
We prove that any coadjoint orbit with real eigenvalues of a complex semisimple Lie group, equipped with the real part of the canonical holomorphic symplectic form, is symplectomorphic to the cotangent bundle of a (partial) flag manifold.…
Fedosov's simple geometrical construction for deformation quantization of symplectic manifolds is generalized in three ways without introducing new variables: (1) The base manifold is allowed to be a supermanifold. (2) The star product does…
This is an expository note on Fedosov's construction of deformation quantization. Given a symplectic manifold and a connection on it, we show how to calculate the star-product step by step. We draw simple diagrams to solve the recursive…
We introduce poly-symplectic groupoids, which are natural extensions of symplectic groupoids to the context of poly-symplectic geometry, and define poly-Poisson structures as their infinitesimal counterparts. We present equivalent…
We give several characterisations of groupoids determined by involutive automorphisms on semilattices of groups.
We introduce non-smooth symplectic forms on manifolds and describe corresponding Poisson structures on the algebra of Colombeau generalized functions. This is achieved by establishing an extension of the classical map of smooth functions to…
It is shown that the de Rham complex of a symplectic manifold $M$ satisfying the hard Lefschetz condition is formal. Moreover, it is shown that the differential Gerstenhaber-Batalin-Vilkoviski algebra associated to such a symplectic…
The main idea of this note is to describe the integration procedure for poly-Poisson structures, that is, to find a poly-symplectic groupoid integrating a poly-Poisson structure, in terms of topological field theories, namely via the…
The description of all deformation quantizations with separation of variables on a Kaehler manifold obtained in our earlier paper is used to identify the Fedosov star-product of Wick type constructed by M. Bordemann and S. Waldmann. This…
We introduce a method to resolve a symplectic orbifold into a smooth symplectic manifold. Then we study how the formality and the Lefschetz property of the symplectic resolution are compared with that of the symplectic orbifold. We also…
We expose the basics of the Fedosov quantization procedure, placed in the general framework of symplectic ringed spaces. This framework also includes some Poisson manifolds with non regular Poisson structures, presymplectic manifolds,…
A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of non-stationary Poisson brackets, i.e. corresponding…