Cohomology associated to a Poisson structure

Journal of Mathematics, 2023

Let M be a smooth manifold and A a Weil algebra. We discuss the diferential forms in the Weil bundles (M A , π, M), and we established a link between diferential forms in M A and M as well as their cohomology. We also discuss the cohomology in.

Advances in Pure Mathematics, 2015

In this paper, M is a smooth manifold of finite dimension n, A is a local algebra and M A is the associated Weil bundle. We study Poisson vector fields on M A and we prove that all globally hamiltonian vector fields on M A are Poisson vector fields.

arXiv (Cornell University), 2012

2014

Abstract. In this paper we study some algebraic properties of harmonic forms on Poisson manifolds. It is well known that in the classical case (on Riemannian manifolds) the product of harmonic forms is not harmonic. Here we describe the algebraic and analytical mechanisms explaining this fact. We also obtain a condition under which the product of de Rham cohomology classes, which includes harmonic representatives, can be represented by a harmonic form.

Japanese journal of mathematics, 2021

We prove that, for a Poisson vertex algebra V, the canonical injective homomorphism of the variational cohomology of V to its classical cohomology is an isomorphism, provided that V, viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables. Contents 1. Introduction 1 2. Variational PVA cohomology 4 3. Preliminaries on the symmetric group and on graphs 6 4. Classical PVA cohomology 9 5. The Main Theorem 14 6. Sesquilinear Harrison cohomology 15 7. Relation between symmetric sesquilinear Harrison and classical PVA cohomology complexes 21 8. Vanishing of the sesquilinear Harrison cohomology 24 9. Proof of the Main Theorem 5.2 32 References 33 Key words and phrases. Poisson vertex algebra (PVA), classical operad, classical PVA cohomology, variational PVA cohomology, sesquilinear Hochschild and Harrison cohomology.

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

Journal of Mathematics Research, 2017

We define the notion of logarithmic Poisson structure along a non zero ideal $\cali$ of an associative, commutative algebra $\cal A$ and prove that each logarithmic Poisson structure induce a skew symmetric 2-form and a Lie-Rinehart structure on the $\cal A$-module $\Omega_K(\log \cali)$ of logarithmic K\"{a}hler differential. This Lie-Rinehart structure define a representation of the underline Lie algebra. Applying the machinery of Chevaley-Eilenberg and Palais, we define the notion of logarithmic Poisson cohomology which is a measure obstructions of Linear representation of the underline Lie algebra for which the grown ring act by multiplication.

Rendiconti Del Circolo Matematico Di Palermo, February 2004, Volume 53, Issue 1, pp 23-60. DOI: 10.1007/BF02921426, 2004

We give a geometric description of different classes of Poisson modules as introduced in [1]: we start with tensors tangent to leaves on a Poisson manifold, consider Poisson structures on bundles and also an example of Poisson module on a manifold which does not come from any vector bundle; finally we use this language to sketch some integral calculus on Poisson manifolds: we suggest how to introduce integration, homology and cohomology in our setting.

Japanese Journal of Mathematics, 2013

It is well known that the validity of the so called Lenard-Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalgebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix.

2001

We study variuos homological structures associated with Poisson algebra, the canonical differential complex for singular Poisson structure and the analogue of the star operator for such manifolds. Give the interpretation of the classical Koszul differential of exterior forms, as the supercommutator with some second order element. Describe the space of invariant distributions on manifold with singular Poisson structure. 1 Lie superalgebra structure on the space of multiderivations of a commutative algebra. Poisson cohomologies Let A be a real or complex vector space. For each integer k let L k (A) be the space of multilinear antisymmetric maps from A k into A. Let L 0 (A) be A and L(A) be ⊕ ∞ k=0 Lk (A).