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 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 DEFORMATION QUANTIZATION OF POISSON MANIFOLDS IN THE DERIVATIVE EXPANSION

2009 ◽  
Vol 06 (02) ◽  
pp. 219-224 ◽  
Author(s):  
A. V. BRATCHIKOV
Keyword(s):  
Poisson Bracket ◽  
Lie Group ◽  
Deformation Quantization ◽  
Second Order ◽  
Poisson Manifolds ◽  
Poisson Structures ◽  
Derivative Expansion ◽  
Order Deformation ◽  
Bracket Algebra ◽  
Derivatives Of

Deformation quantization of Poisson manifolds is studied within the framework of an expansion in powers of derivatives of Poisson structures. We construct the Lie group associated with a Poisson bracket algebra which defines a second order deformation in the derivative expansion.

scholarly journals DEFORMATION QUANTIZATION OF CERTAIN NONLINEAR POISSON STRUCTURES

1998 ◽  
Vol 09 (05) ◽  
pp. 599-621 ◽  
Author(s):  
BYUNG-JAY KAHNG
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Deformation Quantization ◽  
Dual Space ◽  
Poisson Brackets ◽  
Poisson Structures ◽  
Central Extensions ◽  
C Algebras ◽  
Lie Bracket ◽  
Compact Quantum Groups

As a generalization of the linear Poisson bracket on the dual space of a Lie algebra, we introduce certain nonlinear Poisson brackets which are "cocycle perturbations" of the linear Poisson bracket. We show that these special Poisson brackets are equivalent to Poisson brackets of central extension type, which resemble the central extensions of an ordinary Lie bracket via Lie algebra cocycles. We are able to formulate (strict) deformation quantizations of these Poisson brackets by means of twisted group C*-algebras. We also indicate that these deformation quantizations can be used to construct some specific non-compact quantum groups.

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 The Z2-graded Schouten–Nijenhuis bracket and generalized super-Poisson structures

1997 ◽  
Vol 38 (7) ◽  
pp. 3735-3749 ◽  
Author(s):  
J. A. de Azcárraga ◽  
J. M. Izquierdo ◽  
A. M. Perelomov ◽  
J. C. Pérez-Bueno
Keyword(s):  
Poisson Structures ◽  
Nijenhuis Bracket

Z-Graded Extensions of Poisson Brackets

1997 ◽  
Vol 09 (01) ◽  
pp. 1-27 ◽  
Author(s):  
Janusz Grabowski
Keyword(s):  
Poisson Bracket ◽  
Jacobi Identity ◽  
Differential Forms ◽  
Poisson Manifold ◽  
Exterior Derivative ◽  
Tensor Fields ◽  
Lie Bracket ◽  
Nijenhuis Bracket ◽  
Hamiltonian Map ◽  
Vector Valued

A Z-graded Lie bracket { , }P on the exterior algebra Ω(M) of differential forms, which is an extension of the Poisson bracket of functions on a Poisson manifold (M,P), is found. This bracket is simultaneously graded skew-symmetric and satisfies the graded Jacobi identity. It is a kind of an 'integral' of the Koszul–Schouten bracket [ , ]P of differential forms in the sense that the exterior derivative is a bracket homomorphism: [dμ, dν]P=d{μ, ν}P. A naturally defined generalized Hamiltonian map is proved to be a homomorphism between { , }P and the Frölicher–Nijenhuis bracket of vector valued forms. Also relations of this graded Poisson bracket to the Schouten–Nijenhuis bracket and an extension of { , }P to a graded bracket on certain multivector fields, being an 'integral' of the Schouten–Nijenhuis bracket, are studied. All these constructions are generalized to tensor fields associated with an arbitrary Lie algebroid.

scholarly journals Holomorphic Jacobi manifolds

2020 ◽  
Vol 31 (03) ◽  
pp. 2050024 ◽  
Author(s):  
Luca Vitagliano ◽  
Aïssa Wade
Keyword(s):  
Poisson Manifolds ◽  
Poisson Structures ◽  
Special Cases ◽  
Jacobi Manifolds ◽  
One To One ◽  
The Relationship

In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as special cases of holomorphic Jacobi manifolds. We show that holomorphic Jacobi structures yield a much richer framework than that of holomorphic Poisson structures. We also discuss the relationship between holomorphic Jacobi structures, generalized contact bundles and Jacobi–Nijenhuis structures.

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