关于群胚作用和李双代数胚态射

在这篇文章中，我们讨论了李双代数胚之间的态射，得到了一些李双代数胚之间态射的性质．研究了泊松群胚在泊松流形上的泊松作用，以及这个泊松作用与被作用流形的切李双代数胚到作用泊松群胚的切李双代数胚之间的态射的关系，得到了一些有用的结论。     

Dirac submanifolds and Poisson involutions

Dirac submanifolds are a natural generalization in the Poisson category of the symplectic submanifolds of a symplectic manifold. They correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable loci of Poisson involutions. In this paper, we make a general study of these submanifolds including both local and global aspects.In the second part of the paper, we study Poisson involutions and the induced Poisson structures on their stable loci. In particular, we discuss the Poisson involutions on a special class of Poisson groups, and more generally Poisson groupoids, called symmetric Poisson groups, and symmetric Poisson groupoids. Many well-known examples, including the standard Poisson group structures on semi-simple Lie groups, Bruhat Poisson structures on compact semi-simple Lie groups, and Poisson groupoid structures arising from dynamical r-matrices of semi-simple Lie algebras are symmetric, so they admit a Poisson involution. For symmetric Poisson groups, the relation between the stable locus Poisson structure and Poisson symmetric spaces is discussed. As a consequence, we prove that the Dubrovin Poisson structure on the space of Stokes matrices U₊ (appearing in Dubrovin's theory of Frobenius manifolds) is a Poisson symmetric space.     

Geometry of Poisson structures

The purpose of this paper is to consider certain mechanisms of the emergence of Poisson structures on a manifold. We shall also establish some properties of the bivector field that defines a Poisson structure and investigate geometrical structures on the manifold induced by such fields. Further, we shall touch upon the dualism between bivector fields and differential 2-forms.     

Integration of holomorphic Lie algebroids

We prove that a holomorphic Lie algebroid is integrable if and only if its underlying real Lie algebroid is integrable. Thus the integrability criteria of Crainic–Fernandes (Theorem 4.1 in Crainic, Fernandes in Ann Math 2:157, 2003) do also apply in the holomorphic context without any modification. As a consequence we prove that a holomorphic Poisson manifold is integrable if and only if its real part or imaginary part is integrable as a real Poisson manifold.     

Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras

In a previous paper (C. R. Acad. Sci. Paris Sér. I 333 (2001) 763–768), the author introduced a notion of compatibility between a Poisson structure and a pseudo-Riemannian metric. In this paper, we introduce a new class of Lie algebras called pseudo-Riemannian Lie algebras. The two notions are closely related: we prove that the dual of a Lie algebra endowed with its canonical linear Poisson structure carries a compatible pseudo-Riemannian metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra. Moreover, the Lie algebra obtained by linearizing at a point a Poisson manifold with a compatible pseudo-Riemannian metric is a pseudo-Riemannian Lie algebra. We also give some properties of the symplectic leaves of such manifolds, and we prove that every Poisson manifold with a compatible Riemannian metric is unimodular. Finally, we study Poisson Lie groups endowed with a compatible pseudo-Riemannian metric, and we give the classification of all pseudo-Riemannian Lie algebras of dimension 2 and 3.     

Rigidity of infinitesimal momentum maps

In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, this yields rigidity also for lifted actions to the symplectic groupoid.     

A bi-Hamiltonian system on the Grassmannian

Considering the recent result that the Poisson–Nijenhuis geometry corresponds to the quantization of the symplectic groupoid integrating a Poisson manifold, we discuss the Poisson–Nijenhuis structure on the Grassmannian defined by the compatible Kirillov–Kostant–Souriau and Bruhat–Poisson structures. The eigenvalues of the Nijenhuis tensor are Gelfand–Tsetlin variables, which, as was proved, are also in involution with respect to the Bruhat–Poisson structure. Moreover, we show that the Stiefel bundle on the Grassmannian admits a bi-Hamiltonian structure.     

Codimension one symplectic foliations and regular Poisson structures

In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a b-Poisson manifold as we will see in [9].     

Zeroth Poisson Homology,Foliated Cohomology and Perfect Poisson Manifolds

We prove that, for compact regular Poisson manifolds, the zeroth homology group is isomorphic to the top foliated cohomology group, and we give some applications. In particular, we show that, for regular unimodular Poisson manifolds, top Poisson and foliated cohomology groups are isomorphic. Inspired by the symplectic setting, we define what a perfect Poisson manifold is. We use these Poisson homology computations to provide families of perfect Poisson manifolds.     

On the Action of a Groupoid on a Manifold

In this paper,some properties of reduction for symplectic F-spaces are discussed.The properties of stable subgroups are discussed.We find that the symplectic action of a symplectic groupoid on a symplectic manifold can induce a symplectic map between reduced symplectic manifolds.This symplectic action can be characterized by the action of its induced symplectic groupoid on a symplectic manifold.Lastly,we shall discuss Poisson reduction and give a Poisson reduction theorem.     

The modular class of a regular Poisson manifold and the Reeb class of its symplectic foliation

We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. A Riemannian interpretation of the Reeb class will give some geometric criteria which enables one to tell whether the modular class vanishes or not. It also enables one to construct examples of unimodular Poisson manifolds and others which are not unimodular. Finally, we prove that the first leafwise cohomology space is an invariant of Morita equivalence. To cite this article: A. Abouqateb, M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On Poisson–Malcev Structures

Malcev–Poisson structure on a manifold is analogous to a Poisson structure with the Lie identity replaced by a slightly more general Malcev identity. Examples of such structures arise naturally. In the second part of the paper we study Malcev bialgebras. A theorem of characterization is proved.     

Perturbative manifolds and the Noether generators of nth-order Poisson equations

A manifold that contains small perturbations will induce a perturbed partial differential equation. The partial differential equation that we select is the Poisson equation – in order to explore the interplay between the geometry of the manifold and the perturbations. Specifically, we show how the problem of symmetry determination, for higher-order perturbations, can be elegantly expressed via geometric conditions.     

Poisson流形上微缩算子η的应用

本文根据Poisson流形P的1-形式空间∧1(P)上的微缩算子η及其性质,给出了η-代数胚的定义,进一步得到了微缩算子在η-代数胚及Poisson流形中的一些应用.     

Polynomial Hamiltonian form of general relativity

The phase space of general relativity is extended to a Poisson manifold by inclusion of the determinant of the metric and conjugate momentum as additional independent variables. As a result, the action and the constraints take a polynomial form. We propose a new expression for the generating functional for the Green’s functions. We show that the Dirac bracket defines a degenerate Poisson structure on a manifold and the second-class constraints are the Casimir functions with respect to this structure. As an application of the new variables, we consider the Friedmann universe. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 3, pp. 459–494, September, 2006.     

A note on the Poisson structures of the space of probability measures

The space of probability measures on a Riemannian manifold is endowed with the Fisher information metric. In [4] T. Friedrich showed that this space admits also Poisson structures {, }. In this note, we give directly another proof for the structure {, } being Poisson. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

切Poisson作用及约化

本文证明了单连通Poisson紧李群切作用及约化Poisson作用于Poisson流形，若带有等动量映射，则可通过调整Poisson流形的Poisson结构，变成保Poisson结构的Poisson作用，并且该作用限制到Poisson流形的辛叶片上，相对于新Poisson结构是Hamiltion作用。我们把Meyer-Marsden-Weinstein约化从Hamiltion作用推广到切Poisson作用，包括正则值和非正则值两种形式。     

Intégration Symplectique des Variétés de Poisson Régulières

Asymplectic integration of a Poisson manifold (M, Λ) is a symplectic groupoid (Γ,η) whichrealizes the given Poisson manifold, i.e. such that the space of units Γ₀ with the induced Poisson structure Λ₀ is isomorphic to (M, Λ). This notion was introduced by A. Weinstein in [28] in order to quantize Poisson manifolds by quantizing their symplectic integration. Any Poisson manifold can be integrated by alocal symplectic groupoid ([4], [13]) but already for regular Poisson manifolds there are obstructions to global integrability ([2], [6], [11], [17], [28]). The aim of this paper is to summarize all the known obstructions and present a sufficient topological condition for integrability of regular Poisson manifolds; we will indeed describe a concrete procedure for this integration. Further our criterion will provide necessary and sufficient if we require Γ to be Hausdorff, which is a suitable condition to proceed to Weinstein’s program of quantization. These integrability results may be interpreted as a generalization of the Cartan-Smith proof of Lie’s third theorem in the infinite dimensional case.

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