scholarly journals On Theory of logarithmic Poisson Cohomology

2017 ◽  
Vol 9 (4) ◽  
pp. 209
Author(s):  
Joseph Dongho ◽  
Alphonse Mbah ◽  
Shuntah Roland Yotcha
Keyword(s):  
Lie Algebra ◽  
Poisson Structure ◽  
Commutative Algebra ◽  
Linear Representation ◽  
Poisson Cohomology ◽  
Associative Commutative Algebra

We define the notion of logarithmic Poisson structure along a non zero ideal $\cali$ of an associative, commutative algebra $\cal A$ and prove that each logarithmic Poisson structure induce a skew symmetric 2-form and a Lie-Rinehart structure on the $\cal A$-module $\Omega_K(\log \cali)$ of logarithmic K\"{a}hler differential. This Lie-Rinehart structure define a representation of the underline Lie algebra. Applying the machinery of Chevaley-Eilenberg and Palais, we define the notion of logarithmic Poisson cohomology which is a measure obstructions of Linear representation of the underline Lie algebra for which the grown ring act by multiplication.

ANTI-COMMUTATIVE ALGEBRAS WITH SKEW-SYMMETRIC IDENTITIES

2009 ◽  
Vol 08 (02) ◽  
pp. 157-180 ◽  
Author(s):  
A. S. DZHUMADIL'DAEV
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Algebra ◽  
Symmetric Polynomial ◽  
Commutative Algebras ◽  
Associative Commutative Algebra ◽  
Symmetric Identities ◽  
Generalized Lie Algebras

Generalizing Lie algebras, we consider anti-commutative algebras with skew-symmetric identities of degree > 3. Given a skew-symmetric polynomial f, we call an anti-commutative algebra f-Lie if it satisfies the identity f = 0. If sn is a standard skew-symmetric polynomial of degree n, then any s4-Lie algebra is f-Lie if deg f ≥ 4. We describe a free anti-commutative super-algebra with one odd generator. We exhibit various constructions of generalized Lie algebras, for example: given any derivations D, F of an associative commutative algebra U, the algebras (U, D ∧ F) and (U, id ∧ D2) are s4-Lie. An algebra (U, id ∧ D3 - 2D ∧ D2) is s'5-Lie, where s'5 is a non-standard skew-symmetric polynomial of degree 5.

The Darboux coordinate system and Holstein–Primakoff representations on Kähler manifolds

2006 ◽  
Vol 84 (10) ◽  
pp. 891-904
Author(s):  
J R Schmidt
Keyword(s):  
Coordinate System ◽  
Lie Groups ◽  
Lie Algebra ◽  
Poisson Structure ◽  
Kähler Manifolds ◽  
Coadjoint Orbits ◽  
Kahler Manifolds ◽  
Canonical Transformations ◽  
Kahler Geometry

The Kahler geometry of minimal coadjoint orbits of classical Lie groups is exploited to construct Darboux coordinates, a symplectic two-form and a Lie–Poisson structure on the dual of the Lie algebra. Canonical transformations cast the generators of the dual into Dyson or Holstein–Primakoff representations.PACS Nos.: 02.20.Sv, 02.30.Ik, 02.40.Tt

scholarly journals A Drift-Free Left Invariant Control System on the Lie Group SO(3)×R3×R3

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Camelia Pop
Keyword(s):  
Control System ◽  
Numerical Integration ◽  
Lie Algebra ◽  
Lie Group ◽  
Poisson Structure ◽  
Dual Space ◽  
Free System ◽  
Geometrical Properties ◽  
Free Left ◽  
Invariant Control

A controllable drift-free system on the Lie group G=SO(3)×R3×R3 is considered. The dynamics and geometrical properties of the corresponding reduced Hamilton’s equations on g∗,·,·- are studied, where ·,·- is the minus Lie-Poisson structure on the dual space g∗ of the Lie algebra g=so(3)×R3×R3 of G. The numerical integration of this system is also discussed.

On Reductive Lie Admissible Algebras

1971 ◽  
Vol 23 (2) ◽  
pp. 325-331 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Commutative Algebra ◽  
Jacobi Identity ◽  
Direct Sum ◽  
Derivation Algebra ◽  
Image Position

A Lie admissible algebra is a non-associative algebra A such that A− is a Lie algebra where A− denotes the anti-commutative algebra with vector space A and with commutation [X, Y] = XY – YX as multiplication; see [1; 2; 5]. Next let L−(X): A− → A−: Y → [X, Y] and H = {L−(X): X ∊ A−}; then, since A− is a Lie algebra, we see that H is contained in the derivation algebra of A− and consequently the direct sum g = A − ⊕ H can be naturally made into a Lie algebra with multiplication [PQ] given by: P = X + L−(U), Q = Y + L−(V) ∊ g, thenand note that for any P, [PP] = 0 so that [PQ] = −[QP] and the Jacobi identity for g follows from the fact that A− is Lie.

MOMENTUM MAPPINGS AND POISSON COHOMOLOGY

1996 ◽  
Vol 07 (03) ◽  
pp. 329-358 ◽  
Author(s):  
VIKTOR L. GINZBURG
Keyword(s):  
Poisson Structure ◽  
Poisson Manifold ◽  
Existence Problem ◽  
Necessary And Sufficient Condition ◽  
Momentum Mapping ◽  
Lie Algebra Cohomology ◽  
Poisson Cohomology ◽  
Poisson Actions ◽  
Poisson Lie Groups ◽  
Necessary And Sufficient

We analyze the question of existence and uniqueness of equivariant momentum mappings for Poisson actions of Poisson Lie groups. A necessary and sufficient condition for the equivariant momentum mapping to be unique is given. The existence problem is solved under some extra hypotheses, for example, when the action preserves the Poisson structure. In this case, the problem is closely related to the triviality of the induced group action on the Poisson cohomology. This action is shown to be trivial whenever the group is compact or semisimple. Conceptually, these results rely upon a version of “Poisson calculus” developed here to make one-forms on a Poisson manifold induce a “flow” preserving the Poisson structure. In the general case, obstructions to the existence of an infinitesimal version of an equivariant momentum mapping are found. Using Lie algebra cohomology with coefficients in Fréchet modules, we show that the obstructions vanish, and the infinitesimal mapping exists, when the group is compact semisimple. We also prove the rigidity of compact group actions preserving the Poisson structure on a compact manifold and calculate the Poisson cohomology of the Poisson homogeneous space [Formula: see text].

scholarly journals On a Non Logsymplectic Logarithmic Poisson Structure with Poisson Cohomology Isomorphic to the Associated Logarithmic Poisson Cohomology

2017 ◽  
Vol 9 (1) ◽  
pp. 109
Author(s):  
Joseph Dongho
Keyword(s):  
Poisson Structure ◽  
Poisson Structures ◽  
Cohomology Groups ◽  
Poisson Cohomology

The main purpose of this article is to show that there are non logsymplectic Poisson structures whose Poisson cohomology groups are isomorphic to corresponding logarithmic Poisson cohomology groups.

scholarly journals Normalisation holomorphe de structures de Poisson

2009 ◽  
Vol 30 (4) ◽  
pp. 1165-1199
Author(s):  
PHILIPP LOHRMANN
Keyword(s):  
Singular Point ◽  
Normal Form ◽  
Lie Algebra ◽  
Poisson Structure ◽  
Linear Part ◽  
The Other ◽  
Diophantine Condition ◽  
Quadratic Part ◽  
Other Hand ◽  
The One

AbstractWe show that a Poisson structure whose linear part vanishes can be holomorphically normalized in a neighbourhood of its singular point $0\in \Bbb C^n$ if, on the one hand, a Diophantine condition on a Lie algebra associated to the quadratic part is satisfied and, on the other hand, the normal form satisfies some formal conditions.

DIFFERENTIAL CALCULUS ON THE LOGARITHMIC EXTENSION OF THE QUANTUM 3D SPACE AND WEYL ALGEBRA

2011 ◽  
Vol 08 (08) ◽  
pp. 1667-1678 ◽  
Author(s):  
MUTTALIP OZAVSAR ◽  
GURSEL YESILOT
Keyword(s):  
Lie Algebra ◽  
Series Expansion ◽  
Commutative Algebra ◽  
Differential Calculus ◽  
Vector Fields ◽  
Deformation Parameter ◽  
Weyl Algebra ◽  
3D Space

Noncommutative derivative operators acting on the quantum 3D space in the sense of Manin are introduced. Furthermore, the quantum 3D space is extended by the series expansion of the logarithm of the grouplike generator in the quantum 3D space. We give its differential calculus and the corresponding Weyl algebra. We also obtain algebra of Cartan–Maurer forms on this extension and the corresponding Lie algebra of vector fields. All noncommutative results are found to reduce to those of the standard commutative algebra when the deformation parameter of the quantum 3D space is set to one.

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