Some remarks on the integration of the Poisson algebra

Journal of Symplectic Geometry, 2004

Reviews in Mathematical Physics, 2003

We present a general definition of the Poisson bracket between differential forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories and, more generally, on exact multisymplectic manifolds. It is well defined for a certain class of differential forms that we propose to call Poisson forms and turns the space of Poisson forms into a Lie superalgebra.

International Journal of Theoretical Physics, 1995

Using a unitary solution of the classical Yang-Baxter equation on a Lie algebraG we describe a particular way of constructing homogeneous quadratic Poisson structures on the dual of aG-moduleV and study some local features of the symplectic foliation due to the involutive distribution of the Hamiltonian vector fields. We also give some examples where the symplectic leaves are explicitly calculated.

Poisson-Lie groups and complete integrability. I. Drinfeld bigebras, dual extensions and their canonical representations Annales de l'I. H. P., section A, tome 49, n o 4 (1988), p. 433-460 <http://www.numdam.org/item?id=AIHPA_1988__49_4_433_0> © Gauthier-Villars, 1988, tous droits réservés. L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

arXiv (Cornell University), 2019

Using tools from Dirac geometry and through an explicit construction, we show that every Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which relates, for a principal bundle M → M/H, integrations of a Dirac structure on M/H to H-admissible integrations of its pullback Dirac structure on M by presymplectic groupoids. Our construction gives a distinguished class of explicit real or holomorphic pre-symplectic and symplectic groupoids over semi-simple Lie groups and some of their homogeneous spaces, including their symmetric spaces, conjugacy classes, and flag varieties. In a more general framework, we also show integrability of all homogeneous spaces of LA ∨-Lie groups in the sense of E. Meinrenken. Contents 1. Introduction 1 2. Integrations of Lie algebroids via pullbacks 4 3. Lie algebroids associated to equivariant Lie algebra triples 12 4. Integrations of Dirac structures via pullbacks 18 5. Every Poisson homogeneous space is integrable 22 6. Explicit integrations 25 7. Examples: old and new 36 Appendix A. Deriving the formula for the pre-symplectic form on G(L) 40 References 44

arXiv (Cornell University), 2018

In this paper, we discuss the geometric integration of hamiltonian systems on Poisson manifolds, in particular, in the case, when the Poisson structure is induced by a Lie algebra, that is, it is a Lie-Poisson structure. A Hamiltonian system on a Poisson manifold (P, Π) is a smooth manifold P equipped with a bivector field Π satisfying [Π, Π ] = 0 (Jacobi identity), inducing the Poisson bracket on C ∞ (P), {f, g} ≡ Π(df, dg) where f, g ∈ C ∞ (P). For any f ∈ C ∞ (P) the Hamiltonian vector field is defined by X f (g) = {g, f }. The Hamiltonian vector fields X f generate an integrable generalized distribution on P and the leaves of this foliation are symplectic. The flow of any hamiltonian vector field preserves the Poisson structure, it fixes each leaf and the hamiltonian itself is a first integral. It is important to characterize numerical methods preserving some of these fundamental properties of the hamiltonian flow on Poisson manifolds (geometric integrators). We discuss the difficulties of deriving these Poisson methods using standard techniques and we present some modern approaches to the problem.

2002

In this note the long standing problem of the definition of a Poisson bracket in the framework of a multisymplectic formulation of classical field theory is solved. The new bracket operation can be applied to forms of arbitary degree. Relevant examples are discussed and important properties are stated with proofs sketched.

2004

Based on the structure of Casimir elements associated with general Hopf algebras there are constructed Liouville-Arnold integrable flows related with naturally induced Poisson structures on arbitrary co-algebra and their deformations. Some interesting special cases including the oscillatory Heisenberg-Weil algebra related co-algebra structures and adjoint with them integrable Hamiltonian systems are considered.

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.

Russian Mathematical Surveys - RUSS MATH SURVEY-ENGL TR, 1992

CONTENTSIntroduction § 1. Main theoremsChapter I. Algebra § 2. Moyal deformations of the Poisson bracket and *-product on \mathbb R^{2n} § 3. Algebraic construction § 4. Central extensions § 5. ExamplesChapter II. Deformations of the Poisson bracket and *-product on an arbitrary symplectic manifold § 6. Formal deformations: definitions § 7. Graded Lie algebras as a means of describing deformations § 8. Cohomology computations and their consequences § 9. Existence of a *-productChapter III. Extensions of the Lie algebra of contact vector fields on an arbitrary contact manifold §10. Lagrange bracket §11. Extensions and modules of tensor fieldsAppendix 1. Extensions of the Lie algebra of differential operatorsAppendix 2. Examples of equations of Korteweg-de Vries typeReferences Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Abstract Text Return: Query Results Return items starting with number Query Form Dat...