Singular reduction of Nambu–Poisson manifolds

2017 ◽  
Vol 14 (09) ◽  
pp. 1750128 ◽  
Author(s):  
Apurba Das
Keyword(s):  
Poisson Manifold ◽  
Reduction Theorem ◽  
Poisson Manifolds ◽  
Singular Case ◽  
Hamiltonian Flow ◽  
Reduction Procedure ◽  
Singular Reduction ◽  
Poisson Reduction ◽  
Reduced Space ◽  
Suitable Notion

The version of Marsden–Ratiu Poisson reduction theorem for Nambu–Poisson manifolds by a regular foliation have been studied by Ibáñez et al. In this paper, we show that this reduction procedure can be extended to the singular case. Under a suitable notion of Hamiltonian flow on the reduced space, we show that a set of Hamiltonians on a Nambu–Poisson manifold can also be reduced.

scholarly journals On the Lie-formality of Poisson manifolds

2008 ◽  
Vol 2 (2) ◽  
pp. 361-384 ◽  
Author(s):  
G. Sharygin ◽  
D. Talalaev
Keyword(s):  
Lie Algebra ◽  
Present Note ◽  
Poisson Manifold ◽  
Poisson Manifolds ◽  
De Rham ◽  
Graded Lie Algebra ◽  
Differential Graded Lie Algebra

AbstractIn the present note we prove formality of the differential graded Lie algebra of de Rham forms on a smooth Poisson manifold.

scholarly journals The Homology Class of a Poisson Transversal

2018 ◽  
Vol 2020 (10) ◽  
pp. 2952-2976
Author(s):  
Pedro Frejlich ◽  
Ioan Mărcuț
Keyword(s):  
Homology Class ◽  
Poisson Manifold ◽  
Poisson Manifolds ◽  
Poisson Structures ◽  
Symplectic Structures ◽  
Symplectic Case

Abstract This note is devoted to the study of the homology class of a compact Poisson transversal in a Poisson manifold. For specific classes of Poisson structures, such as unimodular Poisson structures and Poisson manifolds with closed leaves, we prove that all their compact Poisson transversals represent nontrivial homology classes, generalizing the symplectic case. We discuss several examples in which this property does not hold, as well as a weaker version of this property, which holds for log-symplectic structures. Finally, we extend our results to Dirac geometry.

scholarly journals Quasi-Poisson Manifolds

2002 ◽  
Vol 54 (1) ◽  
pp. 3-29 ◽  
Author(s):  
A. Alekseev ◽  
Y. Kosmann-Schwarzbach ◽  
E. Meinrenken
Keyword(s):  
Poisson Manifold ◽  
Inner Product ◽  
Poisson Manifolds ◽  
Moment Maps ◽  
Schouten Bracket ◽  
Invariant Element

AbstractA quasi-Poisson manifold is a G-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in associated to an invariant inner product. We introduce the concept of the fusion of such manifolds, and we relate the quasi-Poisson manifolds to the previously introduced quasi-Hamiltonian manifolds with group-valued moment maps.

scholarly journals Covariant reduction of classical Hamiltonian Field Theories: From D’Alembert to Klein–Gordon and Schrödinger

2020 ◽  
Vol 35 (23) ◽  
pp. 2050214
Author(s):  
F. M. Ciaglia ◽  
F. Di Cosmo ◽  
A. Ibort ◽  
G. Marmo ◽  
L. Schiavone
Keyword(s):  
Equations Of Motion ◽  
Symplectic Reduction ◽  
Field Theories ◽  
Reduction Procedure ◽  
Departure Point ◽  
Four Dimensions ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space ◽  
Klein Gordon ◽  
Reduced Space

A novel reduction procedure for covariant classical field theories, reflecting the generalized symplectic reduction theory of Hamiltonian systems, is presented. The departure point of this reduction procedure consists in the choice of a submanifold of the manifold of solutions of the equations describing a field theory. Then, the covariance of the geometrical objects involved, will allow to define equations of motion on a reduced space. The computation of the canonical geometrical structure is performed neatly by using the geometrical framework provided by the multisymplectic description of covariant field theories. The procedure is illustrated by reducing the D’Alembert theory on a five-dimensional Minkowski space-time to a massive Klein–Gordon theory in four dimensions and, more interestingly, to the Schrödinger equation in 3 + 1 dimensions.

scholarly journals REDUCTION AND CONSTRUCTION OF POISSON QUASI-NIJENHUIS MANIFOLDS WITH BACKGROUND

2010 ◽  
Vol 07 (04) ◽  
pp. 539-564
Author(s):  
FLÁVIO CORDEIRO ◽  
JOANA M. NUNES DA COSTA
Keyword(s):  
Reduction Procedure ◽  
Gauge Transformations ◽  
Poisson Reduction

We extend the Falceto–Zambon version of Marsden–Ratiu Poisson reduction to Poisson quasi-Nijenhuis structures with background on manifolds. We define gauge transformations of Poisson quasi-Nijenhuis structures with background, study some of their properties and show that they are compatible with reduction procedure. We use gauge transformations to construct Poisson quasi-Nijenhuis structures with background.

scholarly journals NON-ABELIAN POISSON MANIFOLDS FROM D-BRANES

2004 ◽  
Vol 19 (34) ◽  
pp. 2541-2548
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Lie Algebra ◽  
Gauge Group ◽  
Matrix Multiplication ◽  
Poisson Manifold ◽  
Poisson Manifolds ◽  
Classical Dynamics ◽  
Planck’S Constant ◽  
C Algebra ◽  
Simple Lie Algebra ◽  
Finite Dimensional

Superimposed D-branes have matrix-valued functions as their transverse coordinates, since the latter take values in the Lie algebra of the gauge group inside the stack of coincident branes. This leads to considering a classical dynamics where the multiplication law for coordinates and/or momenta, being given by matrix multiplication, is non-Abelian. Quantization further introduces noncommutativity as a deformation in powers of Planck's constant ℏ. Given an arbitrary simple Lie algebra [Formula: see text] and an arbitrary Poisson manifold ℳ, both finite-dimensional, we define a corresponding C⋆-algebra that can be regarded as a non-Abelian Poisson manifold. The latter provides a natural framework for a matrix-valued classical dynamics.

The outer derivation of a complex Poisson manifold

1999 ◽  
Vol 1999 (506) ◽  
pp. 181-189 ◽  
Author(s):  
J. L Brylinski ◽  
G Zuckerman
Keyword(s):  
Vector Field ◽  
Poisson Manifold ◽  
Hamiltonian Vector ◽  
Poisson Manifolds ◽  
Outer Derivation

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.

Constrained Hamiltonian Dynamics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Lie Algebra ◽  
Symplectic Geometry ◽  
Quantum Physics ◽  
Vector Fields ◽  
Hamiltonian Dynamics ◽  
Poisson Algebra ◽  
Geodesic Equation ◽  
Poisson Manifolds ◽  
C Algebras ◽  
Poisson Reduction

This chapter focuses on autonomous geometrical mechanics, using the language of symplectic geometry. It discusses manifolds (including Kähler manifolds, Riemannian manifolds and Poisson manifolds), tangent bundles, cotangent bundles, vector fields, the Poincaré–Cartan 1-form and Darboux’s theorem. It covers symplectic transforms, the Marsden–Weinstein symplectic quotient, presymplectic and symplectic 2-forms, almost symplectic structures, symplectic leaves and foliation. It also discusses contact structures, musical isomorphisms and Arnold’s theorem, as well as integral invariants, Nambu structures, the Nambu bracket and the Lagrange bracket. It describes Poisson bi-vector fields, Poisson structures, the Lie–Poisson bracket and the Lie–Poisson reduction, as well as Lie algebra, the Lie bracket and Lie algebra homomorphisms. Other topics include Casimir functions, momentum maps, the Euler–Poincaré equation, fibre derivatives and the geodesic equation. The chapter concludes by looking at deformation quantisation of the Poisson algebra, using the Moyal bracket and C*-algebras to develop a quantum physics.

scholarly journals On the existence of a Hofer type metric for Poisson manifolds

2016 ◽  
Vol 27 (09) ◽  
pp. 1650075
Author(s):  
Tomasz Rybicki
Keyword(s):  
Poisson Manifold ◽  
Poisson Manifolds ◽  
Important Class ◽  
Hamiltonian Group ◽  
Symplectic Groupoid ◽  
Hofer Metric

An analogue of the Hofer metric [Formula: see text] on the Hamiltonian group [Formula: see text] of a Poisson manifold [Formula: see text] can be defined, but there is the problem of its nondegeneracy. First, we observe that [Formula: see text] is a genuine metric on [Formula: see text], when the union of all proper leaves of the corresponding symplectic foliation is dense. Next, we deal with the important class of integrable Poisson manifolds. Recall that a Poisson manifold is called integrable, if it can be realized as the space of units of a symplectic groupoid. Our main result states that [Formula: see text] is a Hofer type metric for every Poisson manifold, which admits a Hausdorff integration.

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