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Topic: Poisson manifold

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	Poisson manifold - Wikipedia, the free encyclopedia
		A complex Poisson manifold is a Poisson manifold with a complex or almost complex structure J such that the complex structure preserves the bivector:
		The symplectic leaves of a complex Poisson manifold are pseudo-Kähler manifolds.
		Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser, 1994.
	en.wikipedia.org /wiki/Poisson_manifold (556 words)

	[No title]
		While a direct quantization of the reduced symplectic manifolds and concordant induced representations of Poisson algebras may be possible in certain examples, a systematic approach intending to mimic the classical reduction/induction procedure in some quantum fashion ought to start from a quantization of the `unconstrained' system.
		Indeed, the term `Morita equivalence of Poisson manifolds' \ci{Xu91} was clearly inspired by the terminology of (strong) `Morita equivalence of operator algebras' \ci{Rie74,Rie82}.
		Now form the symplectic manifold $S'=(S_{P}\til{S})/\F$ as in Definition \ref{def1} (assuming that the leaf space of the foliation is indeed a manifold) and Theorem \ref{cis} (that is, $S_P\til{S}$ consists of those pairs $(x,y)\in S\times \til{S}$ for which $J(x)=\hat{J}(y)$, and the foliation $\F$ is generated by $X_{J^f}-X_{\hat{J}}^f$, $f\in\cin(P)$).
	www.ma.utexas.edu /mp_arc/papers/93-289 (9809 words)

	Program PG05
		By the Ginzburg-Weinstein theorem, $U(n)^$ is isomorphic to $u(n)^$ as a Poisson manifold.
		A Riemann-Poisson manifold is a Riemannian manifold endowed with a Poisson tensor parallel with respect to the contravariant Levi-Civita connection associated with the metric and the Poisson tensor.
		obstruction to the Poisson linearizability at a symplectic leaf of nonzero dimension.
	poisson.zetamu.com /ProgramSMR1665.html (2325 words)

	Groupoids Minicourse March 14/15 2001
		Poisson groupoids were introduced by Weinstein in 1988 as a bridge between Poisson Lie groups and symplectic groupoids.
		The linearization of a Poisson groupoid is a Lie bialgebroid; the abstract concept, defined by Mackenzie and Ping Xu (1994) is a smooth form of strong differential Gerstenhaber algebra.
		The cotangent bundle of a Poisson Lie group G is an object intermediate between the Lie bialgebra which is its infintiesimal invariant, and the matched pair, or symplectic double groupoid, which represents its global form.
	www.maths.qmw.ac.uk /~majid/minicourse.html (652 words)

	Abstracts
		The latter generate a (Poisson or Gerstenhaber) algebra of functionals with values in the $S^1$-equivariant cohomology of the loop space of the manifold on which the theory is defined.
		It is proved that, in the case of $GL_n$ with standard representation, the (Poisson or BV) bracket of two generalized Wilson loops applied to two cycles is the same as the generalized Wilson loop applied to the string bracket of the cycles.
		As we have shown in a previous paper, the perturbative quantization of this model is related to Kontsevich's deformation quantization of Poisson manifolds and to his formality theorem.
	www.math.unizh.ch /~asc/abs.html (1406 words)

	Manchester Geometry Seminar (Site not responding. Last check: 2007-11-06)
		Formally, a Poisson structure on a manifold M is a Lie algebra structure on the real vector space of smooth functions on M satisfying, additionally, the Leibniz rule with respect to the ordinary multiplication.
		Conversely, a Poisson manifold is foliated by symplectic leaves.
		A "Poisson reduction" is a procedure which allows to decrease the order of a Hamiltonian system with symmetries by constructing a new Poisson bracket on a manifold of smaller dimension.
	www.maths.man.ac.uk /~tv/Seminar/1999-2000/kirill.html (364 words)

	Geometric methods in the theory of nonlinear waves and their applications. : Genova Research Unit
		Poisson geometry appears to be crucial in the study of integrable systems, both finite and infinite dimensional, and is the starting point in order to generalize the concept of integrabilty to quantum systems.
		Indeed, the choice of a Poisson bivector on a surface X (in other terms, the choice of a global section of the anticanonical bundle) determines natural Poisson structures both on the moduli space of stable sheaves and on the Hilbert scheme of points of X [Bo, Mar].
		According to the terminology fixed in [Fr], a special Kaehler manifold is a Kaehler (or more generally, pseudo-Kaehler) manifold equipped with a flat torsionless connection such that the covariant derivative of the Kaehler form and the covariant differential of the complex structure are both equal to zero.
	www.sissa.it /fm/course/genova2004.html (1558 words)

	Poisson Structures (Site not responding. Last check: 2007-11-06)
		Poisson structures arose in the works of Lagrange, Poisson and Lie in XIX century; their modern formulation is due to Lichnerowicz, Kirillov and Weinstein.
		This is a brief introduction to the algebra of Poisson brackets and constitutes my contribution to 1999/2000 YAS Seminar at Tor Vergata University.
		Bashkara-Viswanath: Poisson algebras and Poisson manifolds Pitman, 1988
	www.math.unifi.it /~caressa/math/poisson.html (324 words)

	On Embedding the 1:1:2 Resonance Space in a Poisson Manifold (Site not responding. Last check: 2007-11-06)
		The Hamiltonian actions of $\S^{1}$ on the symplectic manifold $\R^{6}$ in the $1:1:-2$ and $1:1:2$ resonances are studied.
		The orbit space and the reduced orbit space are singular Poisson spaces with smooth structures determined by the invariant functions.
		It is shown that the Poisson structure on the orbit space, for both the $1:1:2$ and the $1:1:-2$ resonance, cannot be extended to $V$, and that the Poisson structure on the reduced orbit space $J^{- 1}(0)/\S^{1}$ for the $1:1:-2$ resonance cannot be extended to the hyperplane $V_{J}$.
	www.univie.ac.at /EMIS/journals/ERA-AMS/1995-02-001/1995-02-001.html (190 words)

	Document sans titre
		We will explain Vorobjev's model for the linearization of a Poisson manifold on a neighborhood of a symplectic leaf of arbitrary dimension.
		integral manifolds are Prymians of dimension $3g-3$, defined on the holomorphic-symplectic manifold ${\cal T}^*{\cal SU}_X (2,\xi)$, the cotangent bundle to the moduli space of vector bundles of rank 2 and fixed odd determinant $\xi$, over a Riemann surface $X$ of genus $g>1$.
		I will show how the state sum invariants of three manifolds (with boundaries) are recovered in canonical loop quantum gravity in three dimensions in the definition of the physical scalar product of the theory.
	www.math.psu.edu /cgmp/seminars/seminarsSpring2004.htm (1345 words)

	Xu: Poisson cohomology of regular Poisson manifolds
		The main purpose of this paper is to suggest a method of computing Poisson cohomology of a Poisson manifold by means of symplectic groupoids.
		The key idea is to convert the problem of computing Poisson cohomology to that of computing de Rham cohomology of certain manifolds.
		In particular, we shall derive an explicit formula for the Poisson cohomology of a regular Poisson manifold where the symplectic foliation is a trivial fibration.
	math-doc.ujf-grenoble.fr /numdam-bin/item?id=AIF_1992__42_4_967_0 (257 words)

	Problemi Attuali di Fisica Teorica 06 \| The String Coffee Table
		Abstract: The Poisson sigma model is a bidimensional field theory having as target manifold a Poisson manifold.
		Kontsevich formula for the deformation quantization of the target manifold is interpretable as the perturbative expansion of a particular correlator of the model.
		In this seminar, we clarify the meaning of the inegrality condition of the Poisson tensor which appears both in the integration of the gauge transformations of the model and in the geometric quantization of the target manifold.
	golem.ph.utexas.edu /string/archives/000749.html (807 words)

	Poisson manifold - Education - Information - Educational Resources - Encyclopedia - Music (via CobWeb/3.1 ... (Site not responding. Last check: 2007-11-06)
		A Poisson manifold is a differential manifold M such that the algebra of smooth functions over it,
		A manifold M with a smooth bivector field η can be turned into a Poisson manifold via =η(df,dg) provided η(η(df,dg),dh)+η(η(dg,dh),df)+η(η(dh,df),dg) for all f, g, h.
		For a symplectic manifold, η is nothing other than the inverse of the symplectic form ω, which exists because it is invertible.
	education.music.us.cob-web.org:8888 /P/Poisson-manifold.htm (312 words)

	geoms05 (Site not responding. Last check: 2007-11-06)
		A symplectic groupoid is a symplectic manifold with a submersive Poisson morphism onto a Poisson manifold.
		But not every Poisson manifold can be associated with such a symplectic groupoid.
		Based on the detailed analysis of these new metrics, we obtain good understanding of all of the known classical complete Kahler metrics, in particular the Kahler–Einstein metric from which we prove that the logarithmic cotangent bundle of the moduli space is stable in the sense of Mumford.
	math.arizona.edu /~foth/geoms05.html (1665 words)

	Re: Poisson groups/actions (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-11-06)
		A symplectic manifold is a special sort of Poisson manifold, one where the Poisson brackets are "nondegenerate".
		A Poisson manifold as defined in my previous post is the same as a manifold equipped with a 2-vector field P satisfying [P,P] = 0.
		The relation between this P and the Poisson bracket is {f,g} = , where the angle brackets here stand for the obvious pairing between 2-vector fields and 2-forms.
	www.lns.cornell.edu.cob-web.org:8888 /spr/1999-03/msg0015189.html (519 words)

	On Embedding the 1 : 1 : 2 Resonance Space in a Poisson Manifold (ResearchIndex) (via CobWeb/3.1 ... (Site not responding. Last check: 2007-11-06)
		The Hamiltonian actions of S 1 on the symplectic manifold R 6 in the 1 : 1 : \Gamma2 and 1 : 1 : 2 resonances are studied.
		Associated to each action is a Hilbert basis of polynomials defining an embedding of the orbit space into a Euclidean space V and of the reduced orbit space J \Gamma1 (0)=S 1 into a hyperplane V J of V, where J is the quadratic momentum map for the action.
		The orbit space and the reduced orbit space are singular Poisson spaces with smooth structures determined by...
	citeseer.ist.psu.edu.cob-web.org:8888 /268756.html (385 words)

	Re: Poisson groups/actions
		Well, it's a smooth manifold G together with a smooth function m: G x G -> G called "multiplication", a smooth function i: G -> G called the "inverse" and a smooth function 1: {} -> G from the one-point set {} to G, sending * to the "unit" element of G...
		In this case, a Poisson action of G on M turns out to be simply an ordinary action of the Lie group G on M that preserves the Poisson brackets of functions on M. This is what people always used to mean when they talked about a "group of symmetries" of some classical phase space.
		As before, a Poisson action of G on M is a Poisson map A: G x M -> M, but now this is more interesting since the Poisson brackets on G enter into the game.
	www.lns.cornell.edu /spr/1999-02/msg0015032.html (1105 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		are exact differentials, one obtains the concept of the Poisson bracket of two functions on
		Mostly, for a symplectic structure on a manifold the defining
		denote the vector field on a symplectic manifold
	eom.springer.de /s/s091860.htm (314 words)

	Réunion d'hiver 2002 de la SMC
		I will discuss the relationship between two notions of equivalence in Poisson geometry: one is gauge equivalence, that appears as the Poisson counterpart of Morita equivalence of star-product algebras via Kontsevich's formality correspondence; the other is Xu's Morita equivalence for integrable Poisson manifolds, that is a refinement of Weinstein's notion of dual pairs.
		This is done when the original manifold is a product of two-dimentional spheres or more generally when it is a product of manifolds such that the cohomology ring of each of them is generated by a degree two class.
		The ring of smooth functions on S is a Poisson algebra isomorphic to the algebra of smooth G-invariant functions on P.
	camel.math.ca /Reunions/hiver02/abs/sg.html (1848 words)

	Xu Abstract (Site not responding. Last check: 2007-11-06)
		Poisson manifolds appear as general phase spaces in classical mechanics.
		The idea of realizing a Poisson bracket by non-degenerate or symplectic structure can be traced back to S. Lie in the 19th century.
		The existence of symplectic realizations for arbitrary Poisson manifolds was proved independently by Karasev and Weinstein in late 80's.
	www.math.uiuc.edu /Colloquia/01FA/xu_dec06-01.html (149 words)

	Involutive orbits of non-Noether symmetry groups
		Razmadze Institute of Mathematics, 1 Aleksidze Street, Tbilisi 0193, Georgia We consider set of functions on Poisson manifold related by continues one-parameter group of transformations.
		The present paper is an attempt to describe class of one-parameter group of transformations of Poisson manifold that possess involutive orbits and may be related to Hamiltonian integrable systems.
		Groups of transformations of Poisson manifold that possess involutive orbits play important role in some integrable models where conservation laws form orbit of non-Noether symmetry group.
	www.geocities.com /chavchan/21/xml/orbits.xml (990 words)

	Mathematics 428, Section M1, Quantization (Site not responding. Last check: 2007-11-06)
		It will be centered around a few key results: Fedosov's geometric construction of a star-product on a symplectic manifold, Kontsevich's construction of a star-product on a Poisson manifold, the formality theorem, its ramifications, and perhaps a path integral formula for the star-product.
		De Wilde M. and Lecomte P.B.A. "Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds," Lett.
		The idea of non-commutative space-time and correspondigly of non-zero Poisson brackets between the space-time coordinates in the quasi-classical limit is very old indeed.
	www.math.uiuc.edu /~stoyanov/math428-M1 (825 words)

	[No title]
		In a standard manner Poisson bivector field defines a Lie bracket on the algebra of observables (smooth real-valued functions on phase space) called Poisson bracket:
		Generally speaking the conservation laws associated with symmetry might appear to be neither independent nor involutive.
		is ordinary exterior differential and the expression (57) is its well known representation in terms of Poisson bivector field).
	geocities.com /csssite/samp1.xml (1959 words)

	No_2 (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-11-06)
		One starts with a (free and proper) action of a Lie group G on a configuration manifold Q, considers its natural cotangent lift to T
		Q and then one seeks realizations of the corresponding symplectic or Poisson reduced space.
		We further develop this theory by explicitly identifying the symplectic leaves of the Poisson manifold
	www.comptes.carleton.ca.cob-web.org:8888 /Vol_22/No_2.html (834 words)

	Poisson Manifold -- from Wolfram MathWorld
		A smooth manifold with a Poisson bracket defined on its
		Weisstein, Eric W. "Poisson Manifold." From MathWorld--A Wolfram Web Resource.
		Show your math savvy with a MathWorld T-shirt.
	mathworld.wolfram.com /PoissonManifold.html (48 words)

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