Poisson Vector Fields on Weil Bundles

The purpose of this note on Poisson cohomology is to show that, if M is a Poisson manifold and if A is a Weil algebra, then the Weil bundle M A is a Poisson manifold. Therefore, we establish a relationship between Poisson cohomology H • {.,.}A (M A , A) with values in A and Poisson cohomology H • {.,.},R (M A ) with real values.

International Mathematical Forum, 2014

Let M be a paracompact smooth manifold of dimension n, A a Weil algebra and M A the Weil bundle associated. We define and describe the notion of d-Poisson cohomology and of d A-Poisson cohomology on M A .

Journal of Symplectic Geometry, 2004

2016

Let M be a paracompact smooth manifold of dimension n, A a Weil algebra and M A the Weil bundle associated. We define and describe the notion of d-Poisson cohomology and of d A -Poisson cohomology on M A .

Annals of Global Analysis and Geometry, 2020

Using the notion of a contravariant derivative, we give some algebraic and geometric characterizations of Poisson algebras associated to the infinitesimal data of Poisson submanifolds. We show that such a class of Poisson algebras provides a suitable framework for the study of the Hamiltonization problem for the linearized dynamics along Poisson submanifolds.

Archivum Mathematicum, 2016

In this paper, M denotes a smooth manifold of dimension n, A a Weil algebra and M A the associated Weil bundle. When (M, ω M) is a Poisson manifold with 2-form ω M , we construct the 2-Poisson form ω A M A , prolongation on M A of the 2-Poisson form ω M. We give a necessary and sufficent condition for that M A be an A-Poisson manifold.

arXiv (Cornell University), 2019

We investigate some infinite dimensional Lie algebras and their associated Poisson structures which arise from a Lie group action on a manifold. If G is a Lie group, g its Lie algebra and M is a manifold on which G acts, then the set of smooth maps from M to g has at least two Lie algebra structures, both satisfying the required property to be a Lie algebroid. We may then apply a construction by Marle to obtain a Poisson bracket on the set of smooth real or complex valued functions on M × g *. In this paper, we investigate these Poisson brackets. We show that the set of examples include the standard Darboux symplectic structure and the classical Lie Poisson brackets, but is a strictly larger class of Poisson brackets than these. Our study includes the associated Hamiltonian flows and their invariants, canonical maps induced by the Lie group action, and compatible Poisson structures. Our approach is mainly computational and we detail numerous examples. The Lie brackets from which our results derive, arose from the consideration of connections on bundles with zero curvature and constant torsion. We give an alternate derivation of the Lie bracket which will be suited to applications to Lie group actions for applications not involving a Riemannian metric. We also begin a study of the infinite dimensional Poisson brackets which may be obtained by considering a central extension of the Lie algebras.

J Reine Angew Math, 1999

We introduce a canonical outer vector field on a Poisson manifold, also due independently to A. Weinstein. We view it as a global section of the sheaf of Poisson vector fields modulo the subsheaf of hamiltonian vector fields. We study this outer derivation mostly in the case of holomorphic Poisson manifolds.

Duke Mathematical Journal, 2002

We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifolds, based on Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection. Let M be a paracompact smooth d-dimensional manifold. The Lie bracket of vector fields extends to a bracket, the Schouten-Nijenhuis bracket, on the graded commutative algebra Γ(M, ∧ • T M ) of multivector fields so that: This bracket defines a graded super Lie algebra structure on Γ(M, ∧ • T M ), with the shifted grading deg A Poisson structure on M is a bivector field α ∈ Γ(M, ∧ 2 T M ) obeying [α, α] = 0. This identity for α, which we can regard as a bilinear form on the cotangent bundle, implies that {f, g} = α(df, dg) is a Poisson bracket on the algebra C ∞ (M ) of smooth real-valued function. If such a bivector field is given, we say that M is a Poisson manifold. Following [2], we introduce the notion of (deformation) quantization of the algebra of functions on a Poisson manifold. A quantization of the algebra of smooth functions C ∞ (M ) on the Poisson manifold M is a topological algebra A over the ring of formal power series R [[ǫ]] in a formal variable ǫ with product ⋆, together with an R-algebra ǫ j P j (f, g) for some bidifferential operators P j : C ∞ (M ) 2 → C ∞ (M ) with P j (f, 1) = P j (1, g) = 0 and P 1 (f, g) -P 1 (g, f ) = 2α(df, dg). If we fix a section as in (ii), we obtain a star product on C ∞ (M ), i.e. a formal series P ǫ = ǫP 1 + ǫ 2 P 2 + • • • whose coefficients P j are bidifferential operators C ∞ (M ) 2 → C ∞ (M ) so that f ⋆ M g := f g + P ǫ (f, g) extends to an associative R[[ǫ]]-bilinear product on C ∞ (M ) [[ǫ]] with unit 1 ∈ C ∞ (M ) and such that f ⋆ M g -g ⋆ M f = 2ǫα(df, dg) mod ǫ 2 .

2010

Let M be a paracompact differentiable manifold, A a local algebra and M^{A} a manifold of infinitely near points on M of kind A. We define the notion of A-Poisson manifold on M^{A}. We show that when M is a Poisson manifold, then M^{A} is an A-Poisson manifold. We also show that if (M,) is a symplectic manifold, the structure