Poisson Vector Fields Research Articles

Deformation quantization and the action of Poisson vector fields

As one knows, for every Poisson manifold M there exists a formal noncommutative deformation of the algebra of functions on it; it is determined in a unique way (up to an equivalence relation) by the given Poisson bivector. Let a Lie algebra g act by derivations on the functions on M. The main question, which we shall address in this paper is whether it is possible to lift this action to the derivations on the deformed algebra. It is easy to see, that when dimension of g is 1, the only necessary and sufficient condition for this is that the given action is by Poisson vector fields. However, when dimension of g is greater than 1, the previous methods do not work. In this paper we show how one can obtain a series of homological obstructions for this problem, which vanish if there exists the necessary extension.

Hamiltonian dynamical systems and geometry of surfaces in 3-D

Hamiltonian vector field, Poisson vector field and the gradient of Hamiltonian function defines Darboux frame along an integral curve of a Hamiltonian dynamical system on a surface whose normal vector field corresponds to the Poisson structure for a given Hamiltonian system. We show that the existence of compatible Poisson structures determined by the normal legs of the Darboux frame is resolved to the characteristic equation for the Weingarten map. We also show that a Hamiltonian dynamical system in three dimensions has bi-Hamiltonian representation determined by the normal legs of Frenet-Serret triad if and only if an integral curve of Hamiltonian vector field is both a geodesic and a line of curvature.

Bicrossed products induced by Poisson vector fields and their integrability

First we show that, associated to any Poisson vector field [Formula: see text] on a Poisson manifold [Formula: see text], there is a canonical Lie algebroid structure on the first jet bundle [Formula: see text] which, depends only on the cohomology class of [Formula: see text]. We then introduce the notion of a cosymplectic groupoid and we discuss the integrability of the first jet bundle into a cosymplectic groupoid. Finally, we give applications to Atiyah classes and [Formula: see text]-algebras.

Poisson Vector Fields on Weil Bundles

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

A Method to Generate First Integrals from Infinitesimal Symmetries

We propose a method to construct first integrals of a dynamical system, starting with a given set of linearly independent infinitesimal symmetries. In the case of two infinitesimal symmetries, a rank two Poisson structure on the ambient space it is found, such that the vector field that generates the dynamical system, becomes a Poisson vector field. Moreover, the symplectic leaves and the Casimir functions of the associated Poisson manifold are characterized. Explicit conditions that guarantee Hamilton–Poisson realizations of the dynamical system are also given.

DRESSING-MODULAR VECTOR FIELD OF POISSON–LIE GROUP

We show that on any Poisson–Lie group, the effect of the dressing vector fields on volume forms generates a class on Poisson cohomology. The comparison with the modular class give rise to a complete Poisson vector field related to the modular group.

The Cohomology of the Lie Algebras of Formal Poisson Vector Fields and Laplace Operators

We review a Laplace operator on the cochain complex of the Lie algebra of the formal Poisson vector fields to obtain an analogy of the Hodge theorem for its cohomology. We present intermediate results of the computation on the plane.

The Maxwell operator on q-Minkowski space and q-hyperboloid algebras

We introduce an analog of the Maxwell operator on a q-Minkowski space algebra (treated as a particular case of the so-called reflection equation algebra) and on certain of its quotients. We treat the space of ‘quantum differential forms’ as a projective module in the spirit of the Serre–Swan approach. Also, we use ‘braided tangent vector fields’ which are q-analogs of Poisson vector fields associated with the Lie bracket sl(2).

The relative cohomology of formal contact vector fields with respect to formal Poisson vector fields

We review the method due to Gel’fand and Fuks to show the finite dimensionality of the cohomology ring of the Lie algebra of formal contact vector fields. We apply this method to prove the relative cohomology of it with respect to formal Poisson vector fields is trivial.

Poisson Action and Formality

Using a formality on a Poisson manifold, we construct a star product and for each Poisson vector field a derivation of this star product. Starting with a Poisson action of a Lie group, we are able under a natural cohomological assumption to define a representation of its Lie algebra in the space of derivations of the star product. Finally, we use these results to define some generically tangential star products on duals of Lie algebra as in [1] but in a more realistic context.

Computation of cohomology of Lie (super)algebra: algorithm, implementation and new results

Abstract A new algorithmic approach to computation of cohomologies of Lie (super)algebras is described briefly. This approach, based on splitting the full cochain complex into “minimal” subcomplexes, is implemented in the C language. The results of computing the cohomology in the trivial module for the Lie superalgebra SLe(2) are presented. This superalgebra, being an odd counterpart to the Lie algebras of Poisson and Hamilton vector fields, is concerned with the Batalin–Vilkovisky formalism.

A Method of Splitting Cochain Complexes for Computing Cohomology: Lie Algebra of Hamiltonian Vector Fields H(2|0)

Computation of homology or cohomology is inherently a problem of high combinatorial complexity. Recently, we have proposed a new algorithm for computing cohomology of Lie (super)algebras. This algorithm is based on splitting a complete cochain complex into minimal subcomplexes. The algorithm is implemented in C as a program LieCohomology. This paper presents results of computation of cohomology in a trivial module for a Lie algebra of Hamiltonian vector fields H(2v0). We demonstrate that the new approach is much more efficient than the traditional one. In particular, we have revealed some new cohomology classes for the H(2v0) algebra and the related Lie algebra of the Poisson vector fields Po(2v0).

Conformal Hamiltonian systems

Vector fields whose flow preserves a symplectic form up to a constant, such as simple mechanical systems with friction, are called “conformal”. We develop a reduction theory for symmetric conformal Hamiltonian systems, analogous to symplectic reduction theory. This entire theory extends naturally to Poisson systems: given a symmetric conformal Poisson vector field, we show that it induces two reduced conformal Poisson vector fields, again analogous to the dual pair construction for symplectic manifolds. Conformal Poisson systems form an interesting infinite-dimensional Lie algebra of foliate vector fields. Manifolds supporting such conformal vector fields include cotangent bundles, Lie–Poisson manifolds, and their natural quotients.

Computation of Cohomologies of Lie Superalgebras: Algorithm and Implementation

In this work, an algorithm and its implementation in C for computing cohomologies of Lie algebras and superalgebras are presented. Using the program developed, we computed certain cohomologies of the Poisson and Hamiltonian vector fields and of their analogs based on the odd Poisson bracket (antibracket).

The outer derivation of a complex Poisson manifold

Abstract We introduce the canonical outer derivation (or vector field) on a Poisson manifold. This is a Poisson vector field well-defined modulo hamiltonian vector fields.We study this outer derivation by geometric and sheaf-theoretic methods, mostly for holomorphic Poisson manifolds.