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Symplectic vector space

About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.


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Book
01 Dec 2000
TL;DR: In this paper, the moment map of the Poisson bracket is used to measure the moment of a moment in the moment space of differentiable manifolds and vector bundles and Lie groups and Lie algebras.
Abstract: Some aspects of theoretical mechanics Symplectic algebra Symplectic manifolds Hamiltonian vectorfields and the Poisson bracket The moment map Quantization Differentiable manifolds and vector bundles Lie groups and Lie algebras A little cohomology theory Representations of groups Bibliography Index Symbols.

95 citations

Journal ArticleDOI
01 Mar 1976
TL;DR: In this paper, Liberman et al. gave a construction of closed symplectic manifolds with no Kaehler structure, and showed that such manifolds do not have even odd Betti numbers.
Abstract: This is a construction of closed symplectic manifolds with no Kaehler structure. A symplectic manifold is a manifold of dimension 2k with a closed 2-form a such that ak is nonsingular. If M2k is a closed symplectic manifold, then the cohomology class of a is nontrivial, and all its powers through k are nontrivial. M also has an almost complex structure associated with a, up-to homotopy. It has been asked whether every closed symplectic manifold has also a Kaehler structure (the converse is immediate). A Kaehler manifold has the property that its odd dimensional Betti numbers are even. H. Guggenheimer claimed [1], [2] that a symplectic manifold also has even odd Betti numbers. In the review [3] of [1], Liberman noted that the proof was incomplete. We produce elementary examples of symplectic manifolds which are not Kaehler by constructing counterexamples to Guggenheimer's assertion. There is a representation p of Z E Z in the group of diffeomorphisms of T2 defined by (1, 0) -P4 id, (0,1 I 0o 1l where [81 ]" denotes the transformation of T2 covered by the linear transformation of R2. This representation determines a bundle M4 over T with fiber T2: M4 = T2 XZ9Z T2, where Z E Z acts on T2 by covering transformations, and on T2 by p (M4 can also be seen as R4 modulo a group of affine transformations). Let Q1 be the standard volume form for T2. Since p preserves 21, this defines a closed 2-form i2 on M4 which is nonsingular on each fiber. Let p be projection to the base: then it can be checked that S21 + P*' 1 is a symplectic form. (It is, in general, true that "'j + Kp* 21 is a symplectic form, for any closed Q'1 which is a volume form for each fiber, and K sufficiently large.) But H1 (M4) = Z @ Z @ Z, so M4 is not a Kaehler manifold. Many more examples can be constructed. In the same vein, if M2k is a closed symplectic manifold, and if N2k+2 fibers over M2k with the fundamental class of the fiber not homologous to zero in N, then N is also a symplectic manifold. If, for instance, the Euler characteristic of the fiber is not zero, this Received by the editors July 31, 1974. AMS (MOS) subject classifications (1970). Primary 57D15, 58H05. C American Mathematical Society 1976

94 citations

Journal ArticleDOI
TL;DR: In this article, a Hamiltonian theory for 2D soliton equations is developed, in which the spaces of doubly periodic operators on which a hierarchy of commuting flows can be introduced are identified, and these flows are Hamiltonian with respect to a universal symplectic form.
Abstract: We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form $\omega={1\over 2}\r_{\infty} \d k$. We also construct other higher order symplectic forms and compare our formalism with the case of 1D solitons. Restricted to spaces of finite-gap solitons, the universal symplectic form agrees with the symplectic forms which have recently appeared in non-linear WKB theory, topological field theory, and Seiberg-Witten theories. We take the opportunity to survey some developments in these areas where symplectic forms have played a major role.

93 citations

Book ChapterDOI
01 Sep 2007
TL;DR: In this paper, Ekeland and Hofer introduced the concept of the so-called "symplectic capacities", which are invariants of Riemannian manifolds that admit various curvature invariants.
Abstract: A symplectic manifold (M, ω) is a smooth manifold M endowed with a nondegenerate and closed 2-form ω. By Darboux’s Theorem such a manifold looks locally like an open set in some ℝ2n≅ ℂnwith the standard symplectic form and so symplectic manifolds have no local invariants. This is in sharp contrast to Riemannian manifolds, for which the Riemannian metric admits various curvature invariants. Symplectic manifolds do however admit many global numerical invariants, and prominent among them are the so-called symplectic capacities. Symplectic capacities were introduced in 1990 by I. Ekeland and H. Hofer (although the first capacity was in fact constructed by M. Gromov). Since then, lots of new capacities have been defined and they were further studied in. Surveys on symplectic capacities are. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. This is illustrated in Section 2, where we discuss some examples of symplectic capacities and describe a few consequences of their existence. In Section 3 we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities.

92 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that if the regularity assumptions are dropped, the reduced space M is a union of symplectic manifolds, and that the symplectic manifold t together in a nice way.
Abstract: Let (M;!) be a Hamiltonian G-space with a momentum map F : M ! g: It is well-known that if is a regular value of F and G acts freely and properly on the level set F 1 (G); then the reduced space M := F 1 (G )=G is a symplectic manifold. We show that if the regularity assumptions are dropped the space M is a union of symplectic manifolds, and that the symplectic manifolds t together in a nice way. In other words the reduced space is a symplectic stratied space. This extends results known for the Hamiltonian action of compact groups.

92 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202310
202221
202113
20208
201910
201818