The Poisson Bivector and the Schouten-Nijenhuis Bracket

1994 ◽  
pp. 5-17
Author(s):  
Izu Vaisman
Keyword(s):  
Nijenhuis Bracket ◽  
Poisson Bivector

Frölicher–Nijenhuis cohomology on G2- and Spin(7)-manifolds

2018 ◽  
Vol 29 (12) ◽  
pp. 1850075
Author(s):  
Kotaro Kawai ◽  
Hông Vân Lê ◽  
Lorenz Schwachhöfer
Keyword(s):  
Riemannian Manifold ◽  
Differential Form ◽  
Kähler Manifold ◽  
Cohomology Groups ◽  
Kahler Manifold ◽  
Partial Description ◽  
Nijenhuis Bracket

In this paper, we show that a parallel differential form [Formula: see text] of even degree on a Riemannian manifold allows to define a natural differential both on [Formula: see text] and [Formula: see text], defined via the Frölicher–Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential with respect to the canonical parallel [Formula: see text]-form on a [Formula: see text]- and [Formula: see text]-manifold, respectively. We calculate the cohomology groups of [Formula: see text] and give a partial description of the cohomology of [Formula: see text].

scholarly journals The Frölicher-Nijenhuis bracket on some functional spaces

1998 ◽  
Vol 68 (2) ◽  
pp. 97-106 ◽  
Author(s):  
Ivan Kolář ◽  
Marco Modungo
Keyword(s):  
Nijenhuis Bracket

Some applications of the Mangiarotti–Modugno formula for tangent-valued forms

2014 ◽  
Vol 11 (07) ◽  
pp. 1460022
Author(s):  
Ivan Kolář
Keyword(s):  
Lie Algebras ◽  
Classical Approach ◽  
Graded Lie Algebras ◽  
Nijenhuis Bracket ◽  
Fibered Manifolds

First, we present a classical approach to the general connections on arbitrary fibered manifolds. Then we compare this approach with the use of the Frölicher–Nijenhuis bracket by Mangiarotti and Modugno [Graded Lie algebras and connections on a fibered space, J. Math. Pures Appl. 63 (1984) 111–120]. Finally, we demonstrate that the latter viewpoint is very efficient in the theory of torsions of connections on Weil bundles.

The Frölicher–Nijenhuis bracket and the geometry of $$G_2$$ G 2 -and Spin(7)-manifolds

2017 ◽  
Vol 197 (2) ◽  
pp. 411-432 ◽  
Author(s):  
Kotaro Kawai ◽  
Hông Vân Lê ◽  
Lorenz Schwachhöfer
Keyword(s):  
Nijenhuis Bracket

scholarly journals ON RELATION BETWEEN WEYL AND KONTSEVICH QUANTUM PRODUCTS: DIRECT EVALUATION UP TO THE ℏ3-ORDER

2001 ◽  
Vol 16 (10) ◽  
pp. 615-625 ◽  
Author(s):  
A. ZOTOV
Keyword(s):  
Deformation Quantization ◽  
Jacobi Identity ◽  
Star Product ◽  
Star Products ◽  
Moyal Product ◽  
Third Order ◽  
Direct Evaluation ◽  
The Third ◽  
Arbitrary Change ◽  
Poisson Bivector

In his celebrated paper Kontsevich has proved a theorem which manifestly gives a quantum product (deformation quantization formula) and states that changing coordinates leads to gauge equivalent star products. To illuminate his procedure, we make an arbitrary change of coordinates in the Weyl (Moyal) product and obtain the deformation quantization formula up to the third order. In this way, the Poisson bivector is shown to depend on ℏ and not to satisfy the Jacobi identity. It is also shown that the values of coefficients in the formula obtained follow from associativity of the star product.

scholarly journals Lie groupoids and the Frölicher-Nijenhuis bracket

2013 ◽  
Vol 44 (4) ◽  
pp. 709-730 ◽  
Author(s):  
Henrique Bursztyn ◽  
Thiago Drummond
Keyword(s):  
Lie Groupoids ◽  
Nijenhuis Bracket

scholarly journals On computational Poisson geometry I: Symbolic foundations

2021 ◽  
Vol 0 (0) ◽  
pp. 0 ◽  
Author(s):  
Miguel Ángel Evangelista-Alvarado ◽  
José Crispín Ruíz-Pantaleón ◽  
Pablo Suárez-Serrato
Keyword(s):  
Vector Fields ◽  
Hamiltonian Vector ◽  
Poisson Geometry ◽  
Gauge Transformations ◽  
Hamiltonian Vector Fields ◽  
Parametric Families ◽  
Poisson Bivector

<p style='text-indent:20px;'>We present a computational toolkit for (local) Poisson-Nijenhuis calculus on manifolds. Our Python module $\textsf{PoissonGeometry}$ implements our algorithms and accompanies this paper. Examples of how our methods can be used are explained, including gauge transformations of Poisson bivector in dimension 3, parametric Poisson bivector fields in dimension 4, and Hamiltonian vector fields of parametric families of Poisson bivectors in dimension 6.</p>

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.

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