scholarly journals Deformation Quantization of Poisson Structures Associated to Lie Algebroids

2009 ◽  
Author(s):  
Nikolai Neumaier
Keyword(s):  
Deformation Quantization ◽  
Lie Algebroids ◽  
Poisson Structures

scholarly journals KONTSEVICH'S UNIVERSAL FORMULA FOR DEFORMATION QUANTIZATION AND THE CAMPBELL–BAKER–HAUSDORFF FORMULA

2000 ◽  
Vol 11 (04) ◽  
pp. 523-551 ◽  
Author(s):  
VINAY KATHOTIA
Keyword(s):  
Lie Algebras ◽  
Deformation Quantization ◽  
Differential Operators ◽  
Main Tool ◽  
Graphical Analysis ◽  
Poisson Structures ◽  
Nilpotent Lie Algebras ◽  
Enveloping Algebra ◽  
Universal Formula ◽  
The Given

We relate a universal formula for the deformation quantization of Poisson structures (⋆-products) on ℝd proposed by Maxim Kontsevich to the Campbell–Baker–Hausdorff (CBH) formula. We show that Kontsevich's formula can be viewed as exp (P) where P is a bi-differential operator that is a deformation of the given Poisson structure. For linear Poisson structures (duals of Lie algebras) his product takes the form exp (C+L) where exp (C) is the ⋆-product given by the universal enveloping algebra via symmetrization, essentially the CBH formula. This is established by showing that the two products are identical on duals of nilpotent Lie algebras where the operator L vanishes. This completely determines part of Kontsevich's formula and leads to a new scheme for computing the CBH formula. The main tool is a graphical analysis for bi-differential operators and the computation of certain iterated integrals that yield the Bernoulli numbers.

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.

Contractions in Deformed Lie-Poisson Structures

1997 ◽  
Vol 12 (01) ◽  
pp. 225-230 ◽  
Author(s):  
V. D. Lyakhovsky ◽  
A. M. Mirolyubov
Keyword(s):  
Significant Role ◽  
Deformation Quantization ◽  
Duality Principle ◽  
Poisson Structures ◽  
Lie Bialgebras ◽  
First Order ◽  
Natural Analogues ◽  
Explicit Realization ◽  
Quantum Duality ◽  
Initial Object

The geometrical description of deformation quantization based on quantum duality principle makes it possible to introduce deformed Lie-Poisson structures — natural analogues of classical Lie bialgebras for the case when the initial object is a quantized group. Special types of contractions — the so called first order contractions — play the significant role in the explicit realization of deformed Lie-Poisson structures. Such contractions can be naturally applied to the Drinfeld double. In the case of (u(1) ⊕ sl(2))q we investigate explicitly the deformed structures induced by the first order contractions.

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 W-algebras and Duflo isomorphism

2018 ◽  
Vol 17 (03) ◽  
pp. 1850041
Author(s):  
P. Batakidis ◽  
N. Papalexiou
Keyword(s):  
Lie Algebra ◽  
Deformation Quantization ◽  
Poisson Manifold ◽  
Product Formula ◽  
Poisson Structures ◽  
Semisimple Lie Algebra

We prove that when Kontsevich’s deformation quantization is applied on weight homogeneous Poisson structures, the operators in the ∗-product formula are weight homogeneous. In the linear Poisson case for a semisimple Lie algebra [Formula: see text] the Poisson manifold [Formula: see text] is [Formula: see text]. As an application we provide an isomorphism between the Cattaneo–Felder–Torossian reduction algebra [Formula: see text] and the [Formula: see text]-algebra [Formula: see text]. We also show that in the [Formula: see text]-algebra setting, [Formula: see text] is polynomial. Finally, we compute generators of [Formula: see text] as a deformation of [Formula: see text].

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 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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