The Symplectic Foliation of a Poisson Manifold

1994 ◽  
pp. 19-30
Author(s):  
Izu Vaisman
Keyword(s):  
Poisson Manifold

scholarly journals Poisson Principal Bundles

2021 ◽  
Author(s):  
Shahn Majid ◽  
◽  
Liam Williams ◽  
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
Poisson Structure ◽  
Differential Calculus ◽  
Principal Bundle ◽  
Poisson Manifold ◽  
Spin Connection ◽  
Poisson Geometry ◽  
Hopf Fibration ◽  
Principal Bundles

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space X is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the q-Hopf fibration on the standard q-sphere. We also construct the Poisson level of the spin connection on a principal bundle.

EQUIVARIANT POISSON COHOMOLOGY AND A SPECTRAL SEQUENCE ASSOCIATED WITH A MOMENT MAP

1999 ◽  
Vol 10 (08) ◽  
pp. 977-1010 ◽  
Author(s):  
VIKTOR L. GINZBURG
Keyword(s):  
Tensor Product ◽  
Spectral Sequence ◽  
Vector Fields ◽  
Equivariant Cohomology ◽  
Poisson Manifold ◽  
Momentum Mapping ◽  
Poisson Cohomology ◽  
Leray Spectral Sequence ◽  
Differential Complex ◽  
Extensive Introduction

We introduce and study a new spectral sequence associated with a Poisson group action on a Poisson manifold and an equivariant momentum mapping. This spectral sequence is a Poisson analog of the Leray spectral sequence of a fibration. The spectral sequence converges to the Poisson cohomology of the manifold and has the E2-term equal to the tensor product of the cohomology of the Lie algebra and the equivariant Poisson cohomology of the manifold. The latter is defined as the equivariant cohomology of the multi-vector fields made into a G-differential complex by means of the momentum mapping. An extensive introduction to equivariant cohomology of G-differential complexes is given including some new results and a number of examples and applications are considered.

scholarly journals TRANSVERSAL HARMONIC THEORY FOR TRANSVERSALLY SYMPLECTIC FLOWS

2008 ◽  
Vol 84 (2) ◽  
pp. 233-245 ◽  
Author(s):  
HONG KYUNG PAK
Keyword(s):  
Compact Manifold ◽  
Hodge Structure ◽  
Contact Manifold ◽  
Symplectic Manifolds ◽  
Poisson Manifold ◽  
Harmonic Forms ◽  
Cosymplectic Manifolds ◽  
Almost Cosymplectic Manifold ◽  
Differential Complex ◽  
Hard Lefschetz Theorem

AbstractWe develop the transversal harmonic theory for a transversally symplectic flow on a manifold and establish the transversal hard Lefschetz theorem. Our main results extend the cases for a contact manifold (H. Kitahara and H. K. Pak, ‘A note on harmonic forms on a compact manifold’, Kyungpook Math. J.43 (2003), 1–10) and for an almost cosymplectic manifold (R. Ibanez, ‘Harmonic cohomology classes of almost cosymplectic manifolds’, Michigan Math. J.44 (1997), 183–199). For the point foliation these are the results obtained by Brylinski (‘A differential complex for Poisson manifold’, J. Differential Geom.28 (1988), 93–114), Haller (‘Harmonic cohomology of symplectic manifolds’, Adv. Math.180 (2003), 87–103), Mathieu (‘Harmonic cohomology classes of symplectic manifolds’, Comment. Math. Helv.70 (1995), 1–9) and Yan (‘Hodge structure on symplectic manifolds’, Adv. Math.120 (1996), 143–154).

scholarly journals On the Lie-formality of Poisson manifolds

2008 ◽  
Vol 2 (2) ◽  
pp. 361-384 ◽  
Author(s):  
G. Sharygin ◽  
D. Talalaev
Keyword(s):  
Lie Algebra ◽  
Present Note ◽  
Poisson Manifold ◽  
Poisson Manifolds ◽  
De Rham ◽  
Graded Lie Algebra ◽  
Differential Graded Lie Algebra

AbstractIn the present note we prove formality of the differential graded Lie algebra of de Rham forms on a smooth Poisson manifold.

Moduli Space of Flat Connections as a Poisson Manifold

1997 ◽  
Vol 11 (26n27) ◽  
pp. 3195-3206 ◽  
Author(s):  
V. V. Fock ◽  
A. A. Rosly
Keyword(s):  
Riemann Surface ◽  
Lie Groups ◽  
Poisson Structure ◽  
Moduli Space ◽  
Poisson Manifold ◽  
Gauge Fields ◽  
Two Dimensional ◽  
Flat Connections ◽  
Lattice Gauge ◽  
Poisson Lie Groups

In this talk we describe the Poisson structure of the moduli space of flat connections on a two dimensional Riemann surface in terms of lattice gauge fields and Poisson–Lie groups.

Bicrossed products induced by Poisson vector fields and their integrability

2016 ◽  
Vol 13 (03) ◽  
pp. 1650022
Author(s):  
Samson Apourewagne Djiba ◽  
Aïssa Wade
Keyword(s):  
Vector Field ◽  
Cohomology Class ◽  
Vector Fields ◽  
Poisson Manifold ◽  
Lie Algebroid ◽  
Jet Bundle

First we show that, associated to any Poisson vector field [Formula: see text] on a Poisson manifold [Formula: see text], there is a canonical Lie algebroid structure on the first jet bundle [Formula: see text] which, depends only on the cohomology class of [Formula: see text]. We then introduce the notion of a cosymplectic groupoid and we discuss the integrability of the first jet bundle into a cosymplectic groupoid. Finally, we give applications to Atiyah classes and [Formula: see text]-algebras.

scholarly journals The outer derivation of a complex Poisson manifold

1999 ◽  
Vol 1999 (506) ◽  
pp. 181-189
Author(s):  
G. Zuckerman ◽  
J. L Brylinski
Keyword(s):  
Poisson Manifold ◽  
Outer Derivation

scholarly journals Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry

2021 ◽  
Author(s):  
Vladislav G Kupriyanov ◽  
Richard J Szabo
Keyword(s):  
Gauge Theories ◽  
Differential Forms ◽  
Poisson Manifold ◽  
New Approach ◽  
Gauge Sector ◽  
Gauge Transformations ◽  
Gauge Algebra ◽  
Gauge Symmetries ◽  
Noncommutative Gauge Theories ◽  
Differential Graded Poisson Algebras

Abstract We formulate general definitions of semi-classical gauge transformations for noncommutative gauge theories in general backgrounds of string theory, and give novel explicit constructions using techniques based on symplectic embeddings of almost Poisson structures. In the absence of fluxes the gauge symmetries close a Poisson gauge algebra and their action is governed by a $P_\infty$-algebra which we construct explicitly from the symplectic embedding. In curved backgrounds they close a field dependent gauge algebra governed by an $L_\infty$-algebra which is not a $P_\infty$-algebra. Our technique produces new all orders constructions which are significantly simpler compared to previous approaches, and we illustrate its applicability in several examples of interest in noncommutative field theory and gravity. We further show that our symplectic embeddings naturally define a $P_\infty$-structure on the exterior algebra of differential forms on a generic almost Poisson manifold, which generalizes earlier constructions of differential graded Poisson algebras, and suggests a new approach to defining noncommutative gauge theories beyond the gauge sector and the semi-classical limit based on $A_\infty$-algebras.

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 Strict deformation quantization of the state space of Mk(ℂ) with applications to the Curie–Weiss model

2020 ◽  
Vol 32 (10) ◽  
pp. 2050031 ◽  
Author(s):  
Klaas Landsman ◽  
Valter Moretti ◽  
Christiaan J. F. van de Ven
Keyword(s):  
Phase Space ◽  
State Space ◽  
Deformation Quantization ◽  
Classical Limit ◽  
Poisson Manifold ◽  
Quantum Spin ◽  
The State ◽  
Symmetric Convex ◽  
Algebraic Vector ◽  
Classical Phase Space

Increasing tensor powers of the [Formula: see text] matrices [Formula: see text] are known to give rise to a continuous bundle of [Formula: see text]-algebras over [Formula: see text] with fibers [Formula: see text] and [Formula: see text], where [Formula: see text], the state space of [Formula: see text], which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of [Formula: see text] à la Rieffel, defined by perfectly natural quantization maps [Formula: see text] (where [Formula: see text] is an equally natural dense Poisson subalgebra of [Formula: see text]). We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its [Formula: see text] symmetry is spontaneously broken in the thermodynamic limit [Formula: see text]. If this limit is taken with respect to the macroscopic observables of the model (as opposed to the quasi-local observables), it yields a classical theory with phase space [Formula: see text] (i.e. the unit three-ball in [Formula: see text]). Our quantization map then enables us to take the classical limit of the sequence of (unique) algebraic vector states induced by the ground state eigenvectors [Formula: see text] of this model as [Formula: see text], in which the sequence converges to a probability measure [Formula: see text] on the associated classical phase space [Formula: see text]. This measure is a symmetric convex sum of two Dirac measures related by the underlying [Formula: see text]-symmetry of the model, and as such the classical limit exhibits spontaneous symmetry breaking, too. Our proof of convergence is heavily based on Perelomov-style coherent spin states and at some stage it relies on (quite strong) numerical evidence. Hence the proof is not completely analytic, but somewhat hybrid.

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