Related papers: Integrability of central extensions of the Poisson…

We present a geometric construction of central extensions of covering groups of the group of volume preserving diffeomorphisms, integrating central extensions of the Lie algebra of divergence free vector fields defined by Lichnerowicz…

By considering suitable Poisson groupoids, we develop an approach to obtain Lie group structures on (subgroups of) the Poisson diffeomorphism groups of various classes of Poisson manifolds. As applications, we show that the Poisson…

In this paper we find connection between the Hofer's metric of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold, with an integral symplectic form, and the geometry, defined in a paper by Eliashberg and Polterovich,…

We extend known prequantization procedures for Poisson and presymplectic manifolds by defining the prequantization of a Dirac manifold P as a principal U(1)-bundle Q with a compatible Dirac-Jacobi structure. We study the action of Poisson…

As a generalization of the linear Poisson bracket on the dual space of a Lie algebra, we introduce certain non-linear Poisson brackets which are ``cocycle perturbations'' of the linear Poisson bracket. We show that these special Poisson…

An anologue of the Calabi invariant for Poisson manifolds is considered. For any Poisson manifold $P$, the Poisson bracket on $C^{\infty}(P)$ extends to a Lie bracket on the space $\Omega^{1}(P)$ of all differential one-forms, under which…

We generalize the prequantization central extension of a group of diffeomorphisms preserving a closed 2-form \omega (\omega-invariant diffeomorphisms) to an abelian extension of a group of diffeomorphisms preserving a closed vector valued…

We give a proof of the fact that a simply-connected symplectic homogeneous space $(M,\omega)$ of a connected Lie group $G$ is the universal cover of a coadjoint orbit of a one-dimensional central extension of $G$. We emphasise the r\^ole of…

Using tools from Dirac geometry and through an explicit construction, we show that every Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which…

Let $M$ be a manifold with a closed, integral $(k+1)$-form $\omega$, and let $G$ be a Fr\'echet-Lie group acting on $(M,\omega)$. As a generalization of the Kostant-Souriau extension for symplectic manifolds, we consider a canonical class…

We determine the universal central extension of the Lie algebra of hamiltonian vector fields, thereby classifying its central extensions. Furthermore, we classify the central extensions of the Lie algebra of symplectic vector fields, of the…

This paper develops new aspects of the interplay between shifted symplectic geometry and classical Poisson geometry, focusing on lagrangian morphisms into 2-shifted symplectic groups. We establish a Lie-type correspondence between such…

We present the method for finding of the nonlinear Poisson-Lie groups structures on the vector spaces and for their quantization. For arbitrary central extension of Lie algebra explicit formulas of quantization are proposed.

If q : P -> M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra k of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo…

We establish a 1:1 correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson-Lie group $G$ on a…

We present examples of prequantizations over integral symplectic manifolds which admit infinitely many smoothly trivial contact mapping classes. These classes are given by the connected components of the strict contactomorphism group which…

In this review paper, we present several results on central extensions of the Lie algebra of symplectic (Hamiltonian) vector fields, and compare them to similar results for the Lie algebra of (exact) divergence free vector fields. In…

We review the prequantization procedure in the context of super symplectic manifolds with a symplectic form which is not necessarily homogeneous. In developing the theory of non homogeneous symplectic forms, there is one surprising result:…

Let $\mathbb{G}$ be a Lie group with solvable connected component and finitely-generated component group and $\alpha\in H^2(\mathbb{G},\mathbb{S}^1)$ a cohomology class. We prove that if $(\mathbb{G},\alpha)$ is of type I then the same…

We show that the quantisation of a connected simply-connected Poisson-Lie group admits a left-covariant noncommutative differential structure at lowest deformation order if and only if the dual of its Lie algebra admits a pre-Lie algebra…