scholarly journals On a Non Logsymplectic Logarithmic Poisson Structure with Poisson Cohomology Isomorphic to the Associated Logarithmic Poisson Cohomology

2017 ◽  
Vol 9 (1) ◽  
pp. 109
Author(s):  
Joseph Dongho
Keyword(s):  
Poisson Structure ◽  
Poisson Structures ◽  
Cohomology Groups ◽  
Poisson Cohomology

The main purpose of this article is to show that there are non logsymplectic Poisson structures whose Poisson cohomology groups are isomorphic to corresponding logarithmic Poisson cohomology groups.

Geometry of Poisson Structures

1995 ◽  
Vol 2 (4) ◽  
pp. 347-359
Author(s):  
Z. Giunashvili
Keyword(s):  
Poisson Structure ◽  
Poisson Structures ◽  
Geometrical Structures

Abstract The purpose of this paper is to consider certain mechanisms of the emergence of Poisson structures on a manifold. We shall also establish some properties of the bivector field that defines a Poisson structure and investigate geometrical structures on the manifold induced by such fields. Further, we shall touch upon the dualism between bivector fields and differential 2-forms.

Generating Function Methods for Coefficient-Varying Generalized Hamiltonian Systems

2014 ◽  
Vol 6 (01) ◽  
pp. 87-106
Author(s):  
Xueyang Li ◽  
Aiguo Xiao ◽  
Dongling Wang
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian Systems ◽  
Generating Function ◽  
Poisson Structure ◽  
Poisson Structures ◽  
Volterra Systems

AbstractThe generating function methods have been applied successfully to generalized Hamiltonian systems with constant or invertible Poisson-structure matrices. In this paper, we extend these results and present the generating function methods preserving the Poisson structures for generalized Hamiltonian systems with general variable Poisson-structure matrices. In particular, some obtained Poisson schemes are applied efficiently to some dynamical systems which can be written into generalized Hamiltonian systems (such as generalized Lotka-Volterra systems, Robbins equations and so on).

scholarly journals On Theory of logarithmic Poisson Cohomology

2017 ◽  
Vol 9 (4) ◽  
pp. 209
Author(s):  
Joseph Dongho ◽  
Alphonse Mbah ◽  
Shuntah Roland Yotcha
Keyword(s):  
Lie Algebra ◽  
Poisson Structure ◽  
Commutative Algebra ◽  
Linear Representation ◽  
Poisson Cohomology ◽  
Associative Commutative Algebra

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.

scholarly journals COMPATIBLE POISSON STRUCTURES OF TODA TYPE DISCRETE HIERARCHY

2005 ◽  
Vol 20 (07) ◽  
pp. 1367-1388 ◽  
Author(s):  
HENRIK ARATYN ◽  
KLAUS BERING
Keyword(s):  
Poisson Structure ◽  
Integrable Hierarchy ◽  
Difference Operators ◽  
Baxter Equation ◽  
Poisson Structures ◽  
Algebra Isomorphism ◽  
R Matrix ◽  
Compatible Poisson Structures ◽  
Special Value

An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and nonlocal families of R-matrix solutions to the modified Yang–Baxter equation. The three R-theoretic Poisson structures and the Suris quadratic bracket are derived. The resulting family of bi-Poisson structures include a seminal discrete bi-Poisson structure of Kupershmidt at a special value.

scholarly journals ON PLANAR AND NON-PLANAR ISOCHRONOUS SYSTEMS AND POISSON STRUCTURES

2010 ◽  
Vol 07 (07) ◽  
pp. 1115-1131 ◽  
Author(s):  
PARTHA GUHA ◽  
A. GHOSE CHOUDHURY
Keyword(s):  
Dynamical Systems ◽  
Poisson Structure ◽  
Poisson Structures ◽  
Isochronous Systems

We construct certain new classes of isochronous dynamical systems based on the recent constructions of Calogero and Leyvraz. We show how a Poisson structure can be ascribed to such equations in ℝ3 and indicate their connection with the Nambu structures.

MOMENTUM MAPPINGS AND POISSON COHOMOLOGY

1996 ◽  
Vol 07 (03) ◽  
pp. 329-358 ◽  
Author(s):  
VIKTOR L. GINZBURG
Keyword(s):  
Poisson Structure ◽  
Poisson Manifold ◽  
Existence Problem ◽  
Necessary And Sufficient Condition ◽  
Momentum Mapping ◽  
Lie Algebra Cohomology ◽  
Poisson Cohomology ◽  
Poisson Actions ◽  
Poisson Lie Groups ◽  
Necessary And Sufficient

We analyze the question of existence and uniqueness of equivariant momentum mappings for Poisson actions of Poisson Lie groups. A necessary and sufficient condition for the equivariant momentum mapping to be unique is given. The existence problem is solved under some extra hypotheses, for example, when the action preserves the Poisson structure. In this case, the problem is closely related to the triviality of the induced group action on the Poisson cohomology. This action is shown to be trivial whenever the group is compact or semisimple. Conceptually, these results rely upon a version of “Poisson calculus” developed here to make one-forms on a Poisson manifold induce a “flow” preserving the Poisson structure. In the general case, obstructions to the existence of an infinitesimal version of an equivariant momentum mapping are found. Using Lie algebra cohomology with coefficients in Fréchet modules, we show that the obstructions vanish, and the infinitesimal mapping exists, when the group is compact semisimple. We also prove the rigidity of compact group actions preserving the Poisson structure on a compact manifold and calculate the Poisson cohomology of the Poisson homogeneous space [Formula: see text].

scholarly journals CLASSIFICATION OF POISSON SURFACES

2005 ◽  
Vol 07 (01) ◽  
pp. 89-95 ◽  
Author(s):  
CLAUDIO BARTOCCI ◽  
EMANUELE MACRÌ
Keyword(s):  
Poisson Structure ◽  
Classification Theorem ◽  
Poisson Structures ◽  
Complex Projective ◽  
Projective Surfaces

We study complex projective surfaces admitting a Poisson structure; we prove a classification theorem and count how many independent Poisson structures there are on a given Poisson surface.

scholarly journals Incidence stratifications on Hilbert schemes of smooth surfaces, and an application to Poisson structures

2016 ◽  
Vol 27 (01) ◽  
pp. 1650006 ◽  
Author(s):  
Ziv Ran
Keyword(s):  
Poisson Structure ◽  
Hilbert Scheme ◽  
Smooth Surface ◽  
Smooth Curve ◽  
Smooth Surfaces ◽  
Poisson Structures ◽  
Hilbert Schemes ◽  
Hilbert Scheme Of Points

Given a smooth curve on a smooth surface, the Hilbert scheme of points on the surface is stratified according to the length of the intersection with the curve. The strata are highly singular. We show that this stratification admits a natural log-resolution, namely the stratified blowup. As a consequence, the induced Poisson structure on the Hilbert scheme of a Poisson surface has unobstructed deformations.

scholarly journals FOLIATIONS ASSOCIATED TO REGULAR POISSON STRUCTURES

2001 ◽  
Vol 03 (03) ◽  
pp. 441-456 ◽  
Author(s):  
MÉLANIE BERTELSON
Keyword(s):  
Poisson Structure ◽  
Symplectic Form ◽  
Poisson Manifold ◽  
Poisson Structures

A regular Poisson manifold can be described as a foliated space carrying a tangentially symplectic form. Examples of foliations are produced here that are not induced by any Poisson structure although all the basic obstructions vanish.

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