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

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 A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

2017 ◽  
Vol 4 (1) ◽  
pp. 137-154 ◽  
Author(s):  
Yat Sun Poon ◽  
John Simanyi
Keyword(s):  
Spectral Sequence ◽  
Complex Structure ◽  
Adjoint Action ◽  
Poisson Structures ◽  
Dolbeault Cohomology ◽  
Poisson Cohomology ◽  
Abelian Complex Structure ◽  
Nijenhuis Bracket ◽  
Holomorphic Poisson Cohomology ◽  
Type Decomposition

Abstract A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bicomplex where one of the two operators is the classical მ̄-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex manifolds are nilmanifolds with an abelian complex structure. For a particular class of holomorphic Poisson structures, this result leads to a Hodge-type decomposition of the holomorphic Poisson cohomology. We provide examples when the nilmanifolds are 2-step.

scholarly journals Applications of Nijenhuis geometry: non-degenerate singular points of Poisson–Nijenhuis structures

2021 ◽  
Author(s):  
Alexey V. Bolsinov ◽  
Andrey Yu. Konyaev ◽  
Vladimir S. Matveev
Keyword(s):  
Singular Points ◽  
Recursion Operator ◽  
Poisson Structures ◽  
Degeneracy Condition ◽  
Compatible Poisson Structures

AbstractWe study and completely describe pairs of compatible Poisson structures near singular points of the recursion operator satisfying natural non-degeneracy condition.

scholarly journals Poisson structures of near-symplectic manifolds and their cohomology

2021 ◽  
Vol 62 (3) ◽  
pp. 033513
Author(s):  
Panagiotis Batakidis ◽  
Ramón Vera
Keyword(s):  
Symplectic Manifolds ◽  
Poisson Structures

Gel'fand-Dorfman theorem and exact cocycle Poisson structures

1995 ◽  
Vol 34 (3) ◽  
pp. 443-452
Author(s):  
K. H. Bhaskara ◽  
John V. Leahy ◽  
K. Rama
Keyword(s):  
Poisson Structures

Nilpotent Orbits in Simple Lie Algebras and their Transverse Poisson Structures

2008 ◽  
Author(s):  
P. A. Damianou ◽  
H. Sabourin ◽  
P. Vanhaecke ◽  
Rui Loja Fernandes ◽  
Roger Picken
Keyword(s):  
Lie Algebras ◽  
Poisson Structures ◽  
Nilpotent Orbits ◽  
Simple Lie Algebras
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