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.

Poisson manifolds and the schouten bracket

1988 ◽  
Vol 22 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Yu. M. Vorob'ev ◽  
M. V. Karasev
Keyword(s):  
Poisson Manifolds ◽  
Schouten Bracket

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.

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 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.

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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