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 Poisson structures on almost complex Lie algebroids

2014 ◽  
Vol 11 (08) ◽  
pp. 1450069 ◽  
Author(s):  
Paul Popescu
Keyword(s):  
Lie Algebroids ◽  
Complex Manifolds ◽  
Poisson Structures ◽  
Almost Complex Manifolds ◽  
Almost Complex

In this paper, we extend the almost complex Poisson structures from almost complex manifolds to almost complex Lie algebroids. Examples of such structures are also given and the almost complex Poisson morphisms of almost complex Lie algebroids are studied.

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.

Dual Structures on the Prolongations of a Lie Algebroid

2013 ◽  
Vol 59 (2) ◽  
pp. 373-390
Author(s):  
Liviu Popescu
Keyword(s):  
Covariant Derivative ◽  
Lie Algebroids ◽  
Legendre Transformation ◽  
Lie Algebroid ◽  
Symplectic Structures ◽  
Nonlinear Connection ◽  
Mechanical Structure

Abstract In the present paper we study the properties of dual structures on the prolongations of a Lie algebroid. We introduce the dynamical covariant derivative on Lie algebroids and prove that the nonlinear connection induced by a regular Lagrangian is compatible with the metric and symplectic structures. The notions of mechanical structure and semi-Hamiltonian section are introduced on the prolongation of the Lie algebroid to its dual bundle and their properties are investigated. Finally, we prove the equivalence between the metric nonlinear connection and semi-Hamiltonian section, using the Legendre transformation induced by a regular Hamiltonian.

scholarly journals Lie Algebroids, Holonomy and Characteristic Classes

2002 ◽  
Vol 170 (1) ◽  
pp. 119-179 ◽  
Author(s):  
Rui Loja Fernandes
Keyword(s):  
Lie Algebroids ◽  
Characteristic Classes

scholarly journals Modular Classes of Regular Twisted Poisson Structures on Lie Algebroids

2007 ◽  
Vol 80 (2) ◽  
pp. 183-197 ◽  
Author(s):  
Yvette Kosmann-Schwarzbach ◽  
Milen Yakimov
Keyword(s):  
Lie Algebroids ◽  
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 Modular classes revisited

2014 ◽  
Vol 11 (09) ◽  
pp. 1460042 ◽  
Author(s):  
Janusz Grabowski
Keyword(s):  
Geometric Approach ◽  
Geometric Interpretation ◽  
Lie Algebroids ◽  
Poisson Geometry ◽  
Dirac Structure ◽  
Poisson Structures ◽  
Courant Algebroids ◽  
Modular Class ◽  
Novel Approach

We present a graded-geometric approach to modular classes of Lie algebroids and their generalizations, introducing in this setting an idea of relative modular class of a Dirac structure for certain type of Courant algebroids, called projectable. This novel approach puts several concepts related to Poisson geometry and its generalizations in a new light and simplifies proofs. It gives, in particular, a nice geometric interpretation of modular classes of twisted Poisson structures on Lie algebroids.

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