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.

GRADED POISSON STRUCTURES AND SCHOUTEN–NIJENHUIS BRACKET ON ALMOST COMMUTATIVE ALGEBRAS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250042 ◽  
Author(s):  
FERDINAND NGAKEU
Keyword(s):  
Poisson Bracket ◽  
Abelian Groups ◽  
Poisson Manifolds ◽  
Poisson Structures ◽  
Symplectic Structures ◽  
Commutative Algebras ◽  
Nijenhuis Bracket

We introduce and study the notion of abelian groups graded Schouten–Nijenhuis bracket on almost commutative algebras and show that any Poisson bracket on such algebras is defined by a graded bivector as in the classical Poisson manifolds. As a particular example, we introduce and study symplectic structures on almost commutative algebras. Our result is a generalization of the ℤ2-graded (super)-Poisson structures.

scholarly journals Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Maxence Mayrand
Keyword(s):  
Symplectic Manifold ◽  
Cotangent Bundle ◽  
Lagrangian Submanifolds ◽  
Poisson Manifold ◽  
Lagrangian Submanifold ◽  
Zero Section ◽  
Poisson Structures ◽  
Symplectic Structures ◽  
Symplectic Groupoids ◽  
Hyperkähler Metrics

Abstract The first part of this paper is a generalization of the Feix–Kaledin theorem on the existence of a hyperkähler metric on a neighborhood of the zero section of the cotangent bundle of a Kähler manifold. We show that the problem of constructing a hyperkähler structure on a neighborhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix–Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler structure on a neighborhood of its identity section. More generally, we reduce the existence of a hyperkähler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin’s unobstructedness theorem.

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 Desingularizing $\boldsymbol{b^m}$-Symplectic Structures

2017 ◽  
Vol 2019 (10) ◽  
pp. 2981-2998 ◽  
Author(s):  
Victor Guillemin ◽  
Eva Miranda ◽  
Jonathan Weitsman
Keyword(s):  
Canonical Form ◽  
Symplectic Form ◽  
Poisson Manifold ◽  
Symplectic Structures ◽  
Symplectic Forms ◽  
The Family

Abstract A $2n$-dimensional Poisson manifold $(M ,\Pi)$ is said to be $b^m$-symplectic if it is symplectic on the complement of a hypersurface $Z$ and has a simple Darboux canonical form at points of $Z$ which we will describe below. In this article, we will discuss a desingularization procedure which, for $m$ even, converts $\Pi$ into a family of symplectic forms $\omega_{\epsilon}$ having the property that $\omega_{\epsilon}$ is equal to the $b^m$-symplectic form dual to $\Pi$ outside an $\epsilon$-neighborhood of $Z$ and, in addition, converges to this form as $\epsilon$ tends to zero in a sense that will be made precise in the theorem below. We will then use this construction to show that a number of somewhat mysterious properties of $b^m$-manifolds can be more clearly understood by viewing them as limits of analogous properties of the $\omega_{\epsilon}$’s. We will also prove versions of these results for $m$ odd; however, in the odd case the family $\omega_{\epsilon}$ has to be replaced by a family of “folded” symplectic forms.

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 W-algebras and Duflo isomorphism

2018 ◽  
Vol 17 (03) ◽  
pp. 1850041
Author(s):  
P. Batakidis ◽  
N. Papalexiou
Keyword(s):  
Lie Algebra ◽  
Deformation Quantization ◽  
Poisson Manifold ◽  
Product Formula ◽  
Poisson Structures ◽  
Semisimple Lie Algebra

We prove that when Kontsevich’s deformation quantization is applied on weight homogeneous Poisson structures, the operators in the ∗-product formula are weight homogeneous. In the linear Poisson case for a semisimple Lie algebra [Formula: see text] the Poisson manifold [Formula: see text] is [Formula: see text]. As an application we provide an isomorphism between the Cattaneo–Felder–Torossian reduction algebra [Formula: see text] and the [Formula: see text]-algebra [Formula: see text]. We also show that in the [Formula: see text]-algebra setting, [Formula: see text] is polynomial. Finally, we compute generators of [Formula: see text] as a deformation of [Formula: see text].

scholarly journals Obstructions for Symplectic Lie Algebroids

2020 ◽  
Author(s):  
Ralph L. Klaasse ◽  
◽  
◽  
Keyword(s):  
Lie Algebroids ◽  
Characteristic Classes ◽  
Poisson Structures ◽  
Symplectic Structures ◽  
Topological Obstructions

Several types of generically-nondegenerate Poisson structures can be effectively studied as symplectic structures on naturally associated Lie algebroids. Relevant examples of this phenomenon include log-, elliptic, b<sup>k</sup>-, scattering and elliptic-log Poisson structures. In this paper we discuss topological obstructions to the existence of such Poisson structures, obtained through the characteristic classes of their associated symplectic Lie algebroids. In particular we obtain the full obstructions for surfaces to carry such Poisson structures.

scholarly journals Poly-symplectic geometry and the AKSZ formalism

2021 ◽  
pp. 2150030
Author(s):  
Ivan Contreras ◽  
Nicolás Martínez Alba
Keyword(s):  
Phase Space ◽  
General Theory ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Sigma Model ◽  
Type Theorem ◽  
Poisson Structures ◽  
Action Functional ◽  
Symplectic Structures ◽  
Poisson Sigma Model

In this paper, we extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular, we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action functional and the poly-symplectic structure of the reduced phase space of the poly-Poisson sigma model, from the AKSZ construction.

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 SYSTEMS OF HESS–APPEL'ROT TYPE AND ZHUKOVSKII PROPERTY

2009 ◽  
Vol 06 (08) ◽  
pp. 1253-1304 ◽  
Author(s):  
VLADIMIR DRAGOVIĆ ◽  
BORISLAV GAJIĆ ◽  
BOŽIDAR JOVANOVIĆ
Keyword(s):  
Symplectic Manifold ◽  
General Characteristic ◽  
Poisson Manifold ◽  
Partial Reduction ◽  
Poisson Structures ◽  
Remarkable Property ◽  
Compatible Poisson Structures ◽  
Casimir Functions ◽  
Crucial Assumption ◽  
Body Case

We start with a review of a class of systems with invariant relations, so called systems of Hess–Appel'rot type that generalizes the classical Hess–Appel'rot rigid body case. The systems of Hess–Appel'rot type have remarkable property: there exists a pair of compatible Poisson structures, such that a system is certain Hamiltonian perturbation of an integrable bi-Hamiltonian system. The invariant relations are Casimir functions of the second structure. The systems of Hess–Appel'rot type carry an interesting combination of both integrable and non-integrable properties. Further, following integrable line, we study partial reductions and systems having what we call the Zhukovskii property: These are Hamiltonian systems on a symplectic manifold M with actions of two groups G and K; the systems are assumed to be K-invariant and to have invariant relation Φ = 0 given by the momentum mapping of the G-action, admitting two types of reductions, a reduction to the Poisson manifold P = M/K and a partial reduction to the symplectic manifold N0 = Φ-1(0)/G; final and crucial assumption is that the partially reduced system to N0 is completely integrable. We prove that the Zhukovskii property is a quite general characteristic of systems of Hess–Appel'rot type. The partial reduction neglects the most interesting and challenging part of the dynamics of the systems of Hess–Appel'rot type — the non-integrable part, some analysis of which may be seen as a reconstruction problem. We show that an integrable system, the magnetic pendulum on the oriented Grassmannian Gr+(n, 2) has a natural interpretation within Zhukovskii property and that it is equivalent to a partial reduction of certain system of Hess–Appel'rot type. We perform a classical and algebro-geometric integration of the system in dimension four, as an example of a known isoholomorphic system — the Lagrange bitop. The paper presents a lot of examples of systems of Hess–Appel'rot type, giving an additional argument in favor of further study of this class of systems.

Sign in / Sign up

Export Citation Format

Share Document