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)...

refer to: Symplectic category Symplectic Clifford algebra, see Weyl algebra Symplectic geometry Symplectic group, and corresponding symplectic Lie algebra...

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...

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...

structure of a symplectic manifold. The term "symplectic", introduced by Hermann Weyl, is a calque of "complex"; previously, the "symplectic group" had been...

\omega } , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally...

_{\sigma (n)}}\right)} The unitary group is the 3-fold intersection of the orthogonal, complex, and symplectic groups: U ⁡ ( n ) = O ⁡ ( 2 n ) ∩ GL ⁡ (...

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...

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...

mathematics and physics. Matrix groups or algebraic groups are (roughly) groups of matrices (for example, orthogonal and symplectic groups), and these give most...

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)...

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...

specifically, for every member of the metaplectic group (which is a double cover of the symplectic group) there is a corresponding quadratic Fourier transform...

chromodynamics and the symplectic group Sp(m) finds application in Hamiltonian mechanics and quantum mechanical versions of it. The classical groups are exactly...

special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude...

metaplectic group Mp2n is a double cover of the symplectic group Sp2n means that there are always two elements in the metaplectic group representing...

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...

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...

orthogonal group and the quaternionic symplectic group, respectively. The J-homomorphism is a homomorphism from the homotopy groups of orthogonal groups to stable...

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...

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...

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...

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...

(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...

properties and concepts in symplectic geometry in mathematics. The terms listed here cover the occurrences of symplectic geometry both in topology as...

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...

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...

restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions...

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...

table of some common Lie groups and their associated Lie algebras. The following are noted: the topological properties of the group (dimension; connectedness;...