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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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TL;DR: In this article, it was shown that there is no finite-dimensional representation by skew-hermitian matrices of a basic algebra of observables B on a noncompact symplectic manifold M for any Lie subalgebra of the Poisson algebra C^\infty(M).
Abstract: We prove that there is no faithful finite-dimensional representation by skew-hermitian matrices of a ``basic algebra of observables'' B on a noncompact symplectic manifold M. Consequently there exists no finite-dimensional quantization of any Lie subalgebra of the Poisson algebra C^\infty(M) containing B.

10 citations

Journal ArticleDOI
TL;DR: In this article, the authors propose a technique to decompose a time series into the sum of a small number of independent and interpretable components based on symplectic geometry theory, which consists of embedding, symplectic QR decomposition, and diagonal averaging steps.
Abstract: We present a technique to decompose a time series into the sum of a small number of independent and interpretable components based on symplectic geometry theory. The proposed symplectic geometry spectrum analysis technique consists of embedding, symplectic QR decomposition of the matrix into an orthogonal matrix and a triangular matrix, grouping, and diagonal averaging steps. As an example application, the noisy Lorenz series demonstrate the effectiveness of this technique in nonlinear prediction.

10 citations

Book ChapterDOI
TL;DR: In this article, it was shown that the eigenvalues of a general positive path can move off the unit circle and that any such path can be extended to have endpoint with all eigen values on the circle.
Abstract: A positive path in the linear symplectic group Sp(2n) is a smooth path which is everywhere tangent to the positive cone. These paths are generated by negative definite (time-dependent) quadratic Hamiltonian functions on Euclidean space. A special case are autonomous positive paths, which are generated by time-independent Hamiltonians, and which all lie in the set u of diagonalizable matrices with eigenvalues on the unit circle. However, as was shown by Krein, the eigenvalues of a general positive path can move off the unit circle. In this paper, we extend Krein’s theory: we investigate the general behavior of positive paths which do not encounter the eigenvalue 1, showing, for example, that any such path can be extended to have endpoint with all eigenvalues on the circle. We also show that in the case 2n = 4 there is a close relation between the index of a positive path and the regions of the symplectic group that such a path can cross. Our motivation for studying these paths came from a geometric squeezing problem [16] in symplectic topology. However, they are also of interest in relation to the stability of periodic Hamiltonian systems [9] and in the theory of geodesies in Riemannian geometry [4].

10 citations

Posted Content
TL;DR: In this article, the authors studied different aspects of the curvature flow on locally homogeneous manifolds, including long-time existence, solitons, regularity and convergence, and developed two classes of Lie groups, which are relatively simple from a structural point of view but geometrically rich and exotic: solvable Lie groups with a codimension one abelian normal subgroup and a construction attached to each left symmetric algebra.
Abstract: Recently, J. Streets and G. Tian introduced a natural way to evolve an almost-Kahler manifold called the symplectic curvature flow, in which the metric, the symplectic structure and the almost-complex structure are all evolving. We study in this paper different aspects of the flow on locally homogeneous manifolds, including long-time existence, solitons, regularity and convergence. We develop in detail two classes of Lie groups, which are relatively simple from a structural point of view but yet geometrically rich and exotic: solvable Lie groups with a codimension one abelian normal subgroup and a construction attached to each left symmetric algebra. As an application, we exhibit a soliton structure on most of symplectic surfaces which are Lie groups. A family of ancient solutions which develop a finite time singularity was found; neither their Chern scalar nor their scalar curvature are monotone along the flow and they converge in the pointed sense to a (non-Kahler) shrinking soliton solution on the same Lie group.

10 citations

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