Symplectic Vector Space

Explicitly, a symplectic form is a bilinear form ω : ''V'' × ''V'' → R which is

''Skew-symmetric'': ω(''u'', ''v'') = −ω(''v'', ''u'') for all ''u'', ''v'' ∈ ''V'',

''Nondegenerate'': if ω(''u'', ''v'') = 0 for all ''v'' ∈ ''V'' then ''u'' = 0.

Working in a fixed Basis , ω can be represented by a Matrix . The two conditions above say that this matrix must be Skew-symmetric and Nonsingular . This is ''not'' the same thing as a Symplectic Matrix , which is a different concept discussed below.

A nondegenerate skew-symmetric bilinear form behaves quite differently from a nondegenerate ''symmetric'' bilinear form, such as the dot product on Euclidean vector spaces. With a Euclidean inner product ''g'', we have ''g''(''v'',''v'') > 0 for all nonzero vectors ''v'', whereas a symplectic form ω satisfies ω(''v'',''v'') = 0.

STANDARD SYMPLECTIC SPACE

The standard symplectic space is R^2''n'' with the symplectic form given by the Symplectic Matrix
:

\omega = \begin{bmatrix} 0 & I_n \ -I_n & 0 \end{bmatrix}

where ''I''_''n'' is the ''n'' × ''n'' Identity Matrix . In terms of basis vectors

:

(x_1, \ldots, x_n, y_1, \ldots, y_n)

:
:

\omega(x_i, y_j) = -\omega(y_j, x_i) = \delta_{ij}\,

\omega(x_i, x_j) = \omega(y_i, y_j) = 0\,

.

A modified version of the Gram-Schmidt Process shows that any finite-dimensional symplectic vector space has such a basis, often called a ''Darboux basis''.

There is another way to interpret this standard symplectic form. Since the model space R^''n'' used above carries much canonical structure which might easily lead to misinterpretation, we will use "anonymous" vector spaces instead.
Let ''V'' be a real vector space of dimension ''n'' and ''V''^∗ its Dual Space . Now consider the Direct Sum ''W'' := ''V'' ⊕ ''V''^∗ of these spaces equipped with the following form:

:

\omega(x \oplus \eta, y \oplus \xi) = \xi(x) - \eta(y)

Now choose any Basis (''v''₁, …, ''v''_''n'') of ''V'' and consider its Dual Basis

_1, \ldots, v^---_n).

We can interpret the basis vectors as lying in ''W'' if we write
''x''_''i'' = (''v''_''i'', 0) and ''y''_''i'' = (0, v_''i''^∗). Taken together, these form a complete basis of ''W'',

:

(x_1, \ldots, x_n, y_1, \ldots, y_n)

.

The form

\omega

defined here can be shown to have the same properties as in the beginning of this section.

VOLUME FORM

Let ω be a Form on a ''n''-dimensional real vector space ''V'', ω ∈ Λ²(''V''). Then ω is non-degenerate if and only if ''n'' is even, and ω^''n''/2 = ω ∧ … ∧ ω is a Volume Form . A volume form on a ''n''-dimensional vector space ''V'' is a multiple of the (unique) ''n''-form ''e''₁^∗ ∧ … ∧ ''e''_''n''^∗ where the ''e''_''i'' are standard basis vectors on ''V''.

For the standard basis defined in the previous section, we have

_1\wedge\ldots \wedge x^---_n

_1\wedge \ldots \wedge y^---_n.

By reordering, one can write

_1\wedge y^---_1\wedge \ldots \wedge x^---_n

Authors variously define ω^''n'' or (−1)^''n''/2ω^''n'' as the standard volume form. An occasional factor of ''n''! may also appear, depending on whether the definition of the Alternating Product contains a factor of ''n''! or not. The volume form defines an Orientation on the symplectic vector space (''V'', ω).

SYMPLECTIC MAP

preserves the symplectic form, that is, if $f^---
ho=\omega$ . The pullback form is defined by

ho(u,v)= ho(f(u),f(v))

and thus ''f'' is a symplectic map Iff

:

ho(f(u),f(v))=\omega(u,v)

for all ''u'' and ''v'' in ''V''. In particular, symplectic maps are volume-preserving, orientation-preserving, and are Isomorphism s.

SYMPLECTIC GROUP

If ''V'' = ''W'', then a symplectic map is called a linear symplectic transformation of ''V''. In particular, in this case one has that

:

\omega(f(u),f(v)) = \omega(u,v)

,

and so the Linear Transformation ''f'' preserves the symplectic form. The set of all symplectic transformations forms a Group and in particular a Lie Group , called the Symplectic Group and denoted by Sp(''V'') or sometimes Sp(''V'',ω). In matrix form symplectic transformations are given by Symplectic Matrices .

SUBSPACES

Let ''W'' be a Linear Subspace of ''V''. Define the symplectic complement of ''W'' to be the subspace
:

W^{\perp} = \{v\in V \mid \omega(v,w) = 0 \mbox{ for all } w\in W\}

The symplectic complement satisfies
:

(W^{\perp})^{\perp} = W

and
:

\dim W + \dim W^\perp = \dim V

However, unlike Orthogonal Complement s, ''W''^⊥ ∩ ''W'' need not be 0. We distinguish four cases:

''W'' is symplectic if ''W''^⊥ ∩ ''W'' = {0}. This is true Iff ω restricts to a nondegenerate form on ''W''. A symplectic subspace with the restricted form is a symplectic vector space in its own right.

''W'' is isotropic if ''W'' ⊆ ''W''^⊥. This is true iff ω restricts to 0 on ''W''. Any one-dimensional subspace is isotropic.

''W'' is coisotropic if ''W''^⊥ ⊆ ''W''. ''W'' is coisotropic if and only if ω descends to a nondegenerate form on the Quotient Space ''W''/''W''^⊥. Equivalently ''W'' is coisotropic iff ''W''^⊥ is isotropic. Any Codimension -one subspace is coisotropic.

''W'' is Lagrangian if ''W'' = ''W''^⊥. A subspace is Lagrangian iff it is both isotropic and coisotropic. In a finite-dimensional vector space, a Lagrangian subspace is an isotropic one whose dimension is half that of ''V''. Every isotropic subspace can be extended to a Lagrangian one.

Referring to the canonical vector space R^2''n'' above,

the subspace spanned by {''x''₁, ''y''₁} is symplectic

the subspace spanned by {''x''₁, ''x''₂} is isotropic

the subspace spanned by {''x''₁, ''x''₂, …, ''x''_''n'', ''y''₁} is coisotropic

the subspace spanned by {''x''₁, ''x''₂, …, ''x''_''n''} is Lagrangian.

	CATEGORIES ABOUT SYMPLECTIC VECTOR SPACE

	linear algebra

	symplectic geometry

	bilinear forms

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Information About

Symplectic Vector Space