In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, F) and Sp(n)...
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refer to: Symplectic category Symplectic Clifford algebra, see Weyl algebra Symplectic geometry Symplectic group, and corresponding symplectic Lie algebra...
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In mathematics, a symplectic vector space is a vector space V {\displaystyle V} over a field F {\displaystyle F} (for example the real numbers R {\displaystyle...
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In mathematics, a symplectic matrix is a 2 n × 2 n {\displaystyle 2n\times 2n} matrix M {\displaystyle M} with real entries that satisfies the condition...
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structure of a symplectic manifold. The term "symplectic", introduced by Hermann Weyl, is a calque of "complex"; previously, the "symplectic group" had been...
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\omega } , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally...
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_{\sigma (n)}}\right)} The unitary group is the 3-fold intersection of the orthogonal, complex, and symplectic groups: U ( n ) = O ( 2 n ) ∩ GL (...
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In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers. The construction...
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special linear group SL(2, C) and to the symplectic group Sp(2, C). These isomorphisms allow the Lorentz group to act on a large number of mathematical...
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mathematics and physics. Matrix groups or algebraic groups are (roughly) groups of matrices (for example, orthogonal and symplectic groups), and these give most...
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Siegel parabolic subgroup (category Algebraic groups)
the symplectic group with abelian radical, given by the matrices of the symplectic group whose lower left quadrant is 0 (for the standard symplectic form)...
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rotation group, SO(3, R) SO(8) indefinite orthogonal group unitary group symplectic group list of finite simple groups list of simple Lie groups Representations...
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specifically, for every member of the metaplectic group (which is a double cover of the symplectic group) there is a corresponding quadratic Fourier transform...
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chromodynamics and the symplectic group Sp(m) finds application in Hamiltonian mechanics and quantum mechanical versions of it. The classical groups are exactly...
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special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude...
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metaplectic group Mp2n is a double cover of the symplectic group Sp2n means that there are always two elements in the metaplectic group representing...
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representation of O(16). Symplectic gauge groups could also be considered. For example, Sp(8) (which is called Sp(4) in the article symplectic group) has a representation...
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a paramodular group is a special sort of arithmetic subgroup of the symplectic group. It is a generalization of the Siegel modular group, and has the same...
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Bott periodicity theorem (category Topology of Lie groups)
orthogonal group and the quaternionic symplectic group, respectively. The J-homomorphism is a homomorphism from the homotopy groups of orthogonal groups to stable...
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unitary symplectic matrices, Sp(r) and as its associated centerless group the Lie group PSp(r) = Sp(r)/{I, −I} of projective unitary symplectic matrices...
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the orthogonal group, O(V), which preserves a non-degenerate quadratic form on V, symplectic group, Sp(V), which preserves a symplectic form on V (a non-degenerate...
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orthogonal group O(n, K), special orthogonal group SO(n, K), and symplectic group Sp(n, K)) are Lie groups that act on the vector space Kn. The group operations...
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Lagrangian Grassmannian (category Symplectic geometry)
Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is 1/2n(n + 1) (where the dimension of...
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Random matrix (redirect from Gaussian Symplectic Ensemble)
(Hij)n i,j=1. Its distribution is invariant under conjugation by the symplectic group, and it models Hamiltonians with time-reversal symmetry but no rotational...
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properties and concepts in symplectic geometry in mathematics. The terms listed here cover the occurrences of symplectic geometry both in topology as...
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Momentum map (redirect from Symplectic quotient)
specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic...
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Siegel upper half-space (redirect from Siegel modular group)
special case g = 1. The group of automorphisms preserving the complex structure of the manifold is isomorphic to the symplectic group Sp(2g, R). Just as the...
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restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions...
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calculate the integrals over the unitary group as a power series in 1/d. For orthogonal and symplectic groups the Weingarten functions were evaluated by...
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table of some common Lie groups and their associated Lie algebras. The following are noted: the topological properties of the group (dimension; connectedness;...
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