Topic
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•
26 Jul 2006
TL;DR: Symplectic spinors have been used in many applications, e.g., Lie Derivative and Quantization, Symplectic Connections, Symmlectic Dirac Operators, and Second Order Operators as discussed by the authors.
Abstract: Background on Symplectic Spinors.- Symplectic Connections.- Symplectic Spinor Fields.- Symplectic Dirac Operators.- An Associated Second Order Operator.- The Kahler Case.- Fourier Transform for Symplectic Spinors.- Lie Derivative and Quantization.
67 citations
TL;DR: In this paper, it was shown that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical manifold has non-vanishing symplectic homology.
Abstract: We show that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical symplectic manifold has non-vanishing symplectic homology. As a consequence, we establish the existence of contractible closed characteristics on any thickening of the boundary of the neighborhood. When applied to twisted geodesic flows on compact symplectically aspherical manifolds, this implies the existence of contractible periodic orbits for a dense set of low energy values.
67 citations
TL;DR: In this paper, the authors prove that a finite group acting on a symplectic complex vector space can be generated by "symplectic reflectionsd"', i.e., symplectomorphisms with fixed space of codimension 2 in the vector space.
Abstract: Let G be a finite group acting on a symplectic complex vector space V Assume that the quotient V/G has a holomorphic symplectic resolution We prove that G is generated by "symplectic reflectionsd"', ie symplectomorphisms with fixed space of codimension 2 in V Symplectic resolutions are always semismall A crepant resolution of V/G is always symplectic We give a symplectic version of Nakamura conjectures
67 citations
Posted Content•
TL;DR: In this paper, the existence of Calabi quasi-morphisms and quasi-state on symplectic toric Fano 4-manifolds with semi-simple quantum homology was studied.
Abstract: We review and streamline our previous results and the results of Y.Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4-manifolds. We present also new results due to D.McDuff: she observed that for the existence of quasi-morphisms/quasi-states it suffices to assume that the quantum homology contains a field as a direct summand, and she showed that this weaker condition holds true for one point blow-ups of non-uniruled symplectic manifolds.
67 citations
TL;DR: In this paper, the authors provided an explicit description of the symplectic leaves of a simply connected connected semisimple complex Lie group equipped with the standard Poisson-Lie structure.
Abstract: We provide an explicit description of symplectic leaves of a simply connected connected semisimple complex Lie group equipped with the standard Poisson-Lie structure. This sharpens previously known descriptions of the symplectic leaves as connected components of certain varieties. Our main tool is the machinery of twisted generalized minors. They also allow us to present several quasi-commuting coordinate systems on every symplectic leaf. As a consequence, we construct new completely integrable systems on some special symplectic leaves.
65 citations