The Experts below are selected from a list of 288 Experts worldwide ranked by ideXlab platform
Florian Schätz - One of the best experts on this subject based on the ideXlab platform.
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BFV-Complex and Higher Homotopy Structures
Communications in Mathematical Physics, 2008Co-Authors: Florian SchätzAbstract:We present a connection between the BFV-complex (abbreviation for Batalin-Fradkin-Vilkovisky complex) and the strong homotopy Lie algebroid associated to a coisotropic subManifold of a Poisson Manifold. We prove that the latter structure can be derived from the BFV-complex by means of homotopy transfer along contractions. Consequently the BFV-complex and the strong homotopy Lie algebroid structure are L _∞ quasi-isomorphic and control the same formal deformation problem. However there is a gap between the non-formal information encoded in the BFV-complex and in the strong homotopy Lie algebroid respectively. We prove that there is a one-to-one correspondence between coisotropic subManifolds given by graphs of sections and equivalence classes of normalized Maurer-Cartan elemens of the BFV-complex. This does not hold if one uses the strong homotopy Lie algebroid instead.
M Boucetta - One of the best experts on this subject based on the ideXlab platform.
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the modular class of a regular Poisson Manifold and the reeb class of its symplectic foliation
Comptes Rendus Mathematique, 2003Co-Authors: A Abouqateb, M BoucettaAbstract: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).
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the modular class of a regular Poisson Manifold and the reeb invariant of its symplectic foliation
arXiv: Differential Geometry, 2002Co-Authors: A Abouqateb, M BoucettaAbstract: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. The Riemannian interpretation of those classes will permit us to show that a regular Poisson Manifold whose symplectic foliation is of codimension one is unimodular if and only if its symplectic foliation is Riemannian foliation. It permit us also to construct examples of unimodular Poisson Manifolds and other which are not unimodular. Finally, we prove that the first leafwise cohomology is an invariant of Morita equivalence.
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Poisson Manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras
arXiv: Differential Geometry, 2002Co-Authors: M BoucettaAbstract:The notion of Poisson Manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a compatible pseudo-metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra, and that the Lie algebra obtained by linearizing at a point a Poisson Manifold with compatible pseudo-metric is a pseudo-Riemannian Lie algebra. Furthermore, we give some properties of the symplectic leaves of such Manifolds, and we prove that every Poisson Manifold with compatible metric (every Riemann-Lie algebra) is unimodular. As a final, we classify all pseudo-Riemannian Lie algebras of dimension 2 and 3.
David Martinez Torres - One of the best experts on this subject based on the ideXlab platform.
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Zeroth Poisson homology, foliated cohomology and perfect Poisson Manifolds
arXiv: Symplectic Geometry, 2018Co-Authors: David Martinez Torres, Eva MirandaAbstract:We prove that for 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 is a perfect Poisson Manifold. We use these Poisson homology computations to provide families of perfect Poisson Manifolds.
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a Poisson Manifold of strong compact type
Indagationes Mathematicae, 2014Co-Authors: David Martinez TorresAbstract:Abstract We construct a corank one Poisson Manifold which is of strong compact type, i.e., the associated Lie algebroid structure on its cotangent bundle is integrable, and the source 1-connected (symplectic) integration is compact. The construction relies on the geometry of the moduli space of marked K3 surfaces.
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a Poisson Manifold of strong compact type
arXiv: Differential Geometry, 2013Co-Authors: David Martinez TorresAbstract:We sketch the construction of an integrable Poisson Manifold whose canonical integration is compact
Yvette Kosmannschwarzbach - One of the best experts on this subject based on the ideXlab platform.
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the lie bialgebroid of a Poisson nijenhuis Manifold
Letters in Mathematical Physics, 1996Co-Authors: Yvette KosmannschwarzbachAbstract:We describe a new class of Lie bialgebroids associated with Poisson-Nijenhuis structures. Resume. Nous etudions une nouvelle classe de bigebroides de Lie, associes aux structures de Poisson-Nijenhuis. Introduction. Nijenhuis operators have been introduced in the theory of integrable systems in the work of Magri, Gelfand and Dorfman (see the book [4]), and, under the name of hereditary operators, in that of Fuchssteiner and Fokas. Poisson-Nijenhuis structures were defined by Magri and Morosi in 1984 [15] in their study of completely integrable systems. There is a compatibility condition between the Poisson structure and the Nijenhuis structure that is expressed by the vanishing of a rather complicated tensorial expression. In this letter, we shall prove that this condition can be expressed in a very simple way, using the notion of a Lie bialgebroid [14] [7] [12]. A Lie bialgebroid is a pair of vector bundles in duality, each of which is a Lie algebroid, such that the differential defined by one of them on the exterior algebra of its dual is a derivation of the Schouten bracket. Here we show that a Poisson structure and a Nijenhuis structure constitute a Poisson-Nijenhuis structure if and only if the following condition is satisfied: the cotangent and tangent bundles are a Lie bialgebroid when equipped respectively with the bracket of 1-forms defined by the Poisson structure, and with the deformed bracket of vector fields defined by the Nijenhuis structure. Let me add three ”historical” remarks. This result was first conjectured by Magri during a conversation that we held at the time of the Semestre Symplectique at the Centre Emile Borel. Secondly, the Lie bracket of differential 1-forms on a Poisson Manifold, defining the Lie-algebroid structure of its cotangent bundle, was defined by Fuchssteiner in an article of 1982 [6] which is not often cited, though it is certainly one of the first papers to mention this important definition. Thirdly, as A. Weinstein has shown [18] [19], Sophus Lie’s book [11] contains a comprehensive theory of Poisson Manifolds under the name of function groups, including, among many results, a proof of the contravariant form of the Jacobi identity, a proof of the duality between Lie algebra structures and linear Poisson structures on vector spaces, the notions of distinguished functions (Casimir functions) and polar groups (dual pairs), and the existence of canonical coordinates. Moreover, Caratheodory, in his book [2], proves explicitly the tensorial character of the Poisson bivector and gives a rather complete account of this theory, based on a short article by Lie [10] that appeared even earlier than the famous “Theorie der Transformationsgruppen”, Part II, of 1890.
Serge Preston - One of the best experts on this subject based on the ideXlab platform.
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On the Integrability of Orthogonal Distributions in Poisson Manifolds
Balkan Journal of Geometry and Its Applications, 2020Co-Authors: Daniel Fish, Serge PrestonAbstract:In this article we study conditions for the integrability of the distribution defined on a regular Poisson Manifold as the orthogonal complement (with respect to a pseudo-Riemannian metric) to the tangent spaces of the leaves of a symplectic foliation. Integrability criteria in terms of Casimir covectors and in terms of the Nijenhuis Tensor defined by the orthogonal distribution are obtained. Examples of integrability and non-integrability of this distribution are provided. M.S.C. 2000: 53D17, 58D27, 53B21.
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On the Integrability of Orthogonal Distributions in Poisson Manifolds
arXiv: Symplectic Geometry, 2004Co-Authors: Daniel Fish, Serge PrestonAbstract:We study conditions for the integrability of the distribution defined on a regular Poisson Manifold as the orthogonal complement (with respect to some (pseudo)-Riemannian metric) to the tangent spaces of the leaves of a symplectic foliation. Examples of integrability and non-integrability of this distribution are provided.
