Poisson structures on cotangent bundles.
Poisson structures on Weil bundles.
Poisson-Lie groupoids and the contraction procedure
On the level of Lie algebras, the contraction procedure is a method to create a new Lie algebra from a given Lie algebra by rescaling generators and letting the scaling parameter tend to zero. One of the most well-known examples is the contraction from 𝔰𝔲(2) to 𝔢(2), the Lie algebra of upper-triangular matrices with zero trace and purely imaginary diagonal. In this paper, we will consider an extension of this contraction by taking also into consideration the natural bialgebra structures on these...
Poisson–Lie sigma models on Drinfel’d double
Poisson sigma models represent an interesting use of Poisson manifolds for the construction of a classical field theory. Their definition in the language of fibre bundles is shown and the corresponding field equations are derived using a coordinate independent variational principle. The elegant form of equations of motion for so called Poisson-Lie groups is derived. Construction of the Poisson-Lie group corresponding to a given Lie bialgebra is widely known only for coboundary Lie bialgebras. Using...
Prolongation of Poisson -form on Weil bundles
In this paper, denotes a smooth manifold of dimension , a Weil algebra and the associated Weil bundle. When is a Poisson manifold with -form , we construct the -Poisson form , prolongation on of the -Poisson form . We give a necessary and sufficient condition for that be an -Poisson manifold.
Quantification formelle des variétés de Poisson
Quantification pour les paires symétriques et diagrammes de Kontsevich
In this article we use the expansion for biquantization described in [7] for the case of symmetric spaces. We introduce a function of two variables for any symmetric pairs. This function has an expansion in terms of Kontsevich’s diagrams. We recover most of the known results though in a more systematic way by using some elementary properties of this function. We prove that Cattaneo and Felder’s star product coincides with Rouvière’s for any symmetric pairs. We generalize some of Lichnerowicz’s...
Quantization of pencils with a gl-type Poisson center and braided geometry
We consider Poisson pencils, each generated by a linear Poisson-Lie bracket and a quadratic Poisson bracket corresponding to a so-called Reflection Equation Algebra. We show that any bracket from such a Poisson pencil (and consequently, the whole pencil) can be restricted to any generic leaf of the Poisson-Lie bracket. We realize a quantization of these Poisson pencils (restricted or not) in the framework of braided affine geometry. Also, we introduce super-analogs of all these Poisson pencils and...
Quantization of Poisson Hamiltonian systems
In this paper we recall the concept of Hamiltonian system in the canonical and Poisson settings. We will discuss the quantization of the Hamiltonian systems in the Poisson context, using formal deformation quantization and quantum group theories.
Quantum 4-sphere: the infinitesimal approach
We describe how the constructions of quantum homogeneous spaces using infinitesimal invariance and quantum coisotropic subgroups are related. As an example we recover the quantum 4-sphere of [2] through infinitesimal invariance with respect to .
Self-Similarity of Poisson structures on tori
We study the group of diffeomorphisms of a 3-dimensional Poisson torus which preserve the Poisson structure up to a constant multiplier, and the group of similarity ratios.
Several cohomology algebras connected with Poisson structure.
Shoikhet's conjecture and Duflo isomorphism on (co)invariants.
Singular Poisson reduction of cotangent bundles.
We consider the Poisson reduced space (T* Q)/K, where the action of the compact Lie group K on the configuration manifold Q is of single orbit type and is cotangent lifted to T* Q. Realizing (T* Q)/K as a Weinstein space we determine the induced Poisson structure and its symplectic leaves. We thus extend the Weinstein construction for principal fiber bundles to the case of surjective Riemannian submersions Q → Q/K which are of single orbit type.
Singularities of Poisson and Nambu structures
Some liftings of Poisson structures to Weil bundles
We establish a formula for the Schouten-Nijenhuis bracket of linear liftings of skew-symmetric tensor fields to any Weil bundle. As a result we obtain a construction of some liftings of Poisson structures to Weil bundles.
Stability of higher order singular points of Poisson manifolds and Lie algebroids
We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first-order approximation (not necessarily linear) of a given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical Lichnerowicz-Poisson cohomology and the deformation cohomology for Lie algebroids recently introduced by Crainic and Moerdijk. We also provide several examples of stable singular...
Structures de Poisson sur les variétés algébriques de dimension trois
Tangent lifts of higher order of multiplicative Dirac structures
The tangent lifts of higher order of Dirac structures and some properties have been defined in [9] and studied in [11]. By the same way, the tangent lifts of higher order of Poisson structures have been studied in [10] and some applications are given. In particular, the authors have studied the nature of the Lie algebroids and singular foliations induced by these lifting. In this paper, we study the tangent lifts of higher order of multiplicative Poisson structures, multiplicative Dirac structures...
The BV-algebra of a Jacobi manifold
We show that the Gerstenhaber algebra of the 1-jet Lie algebroid of a Jacobi manifold has a canonical exact generator, and discuss duality between its homology and the Lie algebroid cohomology. We also give new examples of Lie bialgebroids over Poisson manifolds.