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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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Journal ArticleDOI
TL;DR: The complex symplectic geometry of the space QF(S) of quasi-Fuchsian structures of a compact orientable surface S of genus g > 1 was studied in this paper.
Abstract: We study the complex symplectic geometry of the space QF(S) of quasi-Fuchsian structures of a compact orientable surface S of genus g > 1. We prove that QF(S) is a complex symplectic manifold. The complex symplectic structure is the complexification of the Weil–Petersson symplectic structure of Teichmuller space and is described in terms which look natural from the point of view of hyperbolic geometry.

18 citations

Journal ArticleDOI
TL;DR: In this article, the Marsden-Weinstein reduction procedure for almost symplectic manifolds is implemented and conditions for transformation of non-equivariant momentum maps into equivariant ones by modifying the group action are studied.

18 citations

Journal ArticleDOI
TL;DR: In this article, the authors characterized the symplectic normal space at any point of a smooth Riemannian manifold and showed that this space splits as the direct sum of the cotangent bundle of a linear space and a symplectic linear space coming from reduction of a coadjoint orbit.
Abstract: For the cotangent bundle T*Q of a smooth Riemannian manifold acted upon by the lift of a smooth and proper action by isometries of a Lie group, we characterize the symplectic normal space at any point. We show that this space splits as the direct sum of the cotangent bundle of a linear space and a symplectic linear space coming from reduction of a coadjoint orbit. This characterization of the symplectic normal space can be expressed solely in terms of the group action on the base manifold and the coadjoint representation. Some relevant particular cases are explored. (C) 2007 Elsevier B.V. All rights reserved.

18 citations

Posted Content
TL;DR: In this paper, the trace map and the gluing map are considered as rational maps on Lagrangian Grassmannians, and the singularities of the symplectic reduction are studied.
Abstract: The first part of this paper deals with electrical networks and symplectic reductions. We consider two operations on electrical networks (the "trace map" and the "gluing map") and show that they correspond to symplectic reductions. We also give several general properties about symplectic reductions, in particular we study the singularities of symplectic reductions when considered as rational maps on Lagrangian Grassmannians. This is motivated by [23] where a renormalization map was introduced in order to describe the spectral properties of self-similar lattices. In this text, we show that this renormalization map can be expressed in terms of symplectic reductions and that some of its key properties are direct consequences of general properties of symplectic reductions (and the singularities of the symplectic reduction play an important role in relation with the spectral properties of our operator). We also present new examples where we can compute the renormalization map.

18 citations

Journal ArticleDOI
TL;DR: In this paper, a normal form theory for symplectic maps with non-diagonalizable linear part is obtained by working directly with the group of symplectic diffeomorphisms, and the role of normal form symmetry and reduction of the normalized map is also considered.

18 citations

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