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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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11 citations
TL;DR: The generators of the Lie algebra of the symplectic groupsp(2n, R) are, recurrently, realized by means of polynomials in the quantum canonical variablesp ≥ 0 andq ≥ 0.
Abstract: The generators of the Lie algebra of the symplectic groupsp(2n, R) are, recurrently, realized by means of polynomials in the quantum canonical variablesp
i andq
i. These realizations are skew-Hermitian, the Casimir operators are realized by constant multiples of identity elements, and, depending on the number of the canonical pairs used, they depend ond, d=1, 2, ...,n free real parameters.
11 citations
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TL;DR: In this article, the Beauville-Bogomolov lattice is computed for a simple singular symplectic manifold of dimension 4, obtained as a partial desingularization of the quotient of a K3 surface with an anti-symplectic involution.
Abstract: The Beauville-Bogomolov lattice is computed for a simplest singular symplectic manifold of dimension 4, obtained as a partial desingularization of the quotient $S^{[2]}/\iota$, where $S^{[2]}$ is the Hilbert square of a K3 surface $S$ and $\iota$ is a symplectic involution on it. This result applies, in particular, to the singular symplectic manifolds of dimension 4, constructed by Markushevich-Tikhomirov as compactifications of families of Prym varieties of a linear system of curves on a K3 surface with an anti-symplectic involution.
11 citations
01 Jan 2005
TL;DR: A survey of momentum maps and dual pairs can be found in this article, with a focus on the current interest in symplectic and Poisson groupoids, as well as the definitions and properties which will be used in what follows.
Abstract: In this survey, we will try to indicate some important ideas, due in large part to Alan Weinstein, which led from the study of momentum maps and dual pairs to the current interest in symplectic and Poisson groupoids. We hope that it will be useful for readers new to the subject; therefore, we begin by recalling the definitions and properties which will be used in what follows. More details can be found in [4], [26], [47].
11 citations
TL;DR: In this article, it was shown that any compact nilmanifold with a symplectic structure has a nilpotent Lie group as its universal covering such that as a solvable Lie group it has at most two steps.
Abstract: In this paper, we obtain an inductive process to classify all the compact nilmanifolds with symplectic structures. As a consequence, we give a counter-example to a many years question that any compact nilmanifold with a symplectic structure has a nilpotent Lie group as its universal covering such that as a solvable Lie group it has at most two steps.
11 citations