symplectic vector space

In mathematics, a symplectic vector space is a vector space V equipped with a nondegenerate, skew-symmetric, bilinear form ω called the symplectic form. 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. (Note that this is not the same thing as a symplectic matrix.) If V is finite-dimensional then its dimension must necessarily be even since every skew-symmetric matrix of odd size has determinant zero.

Standard symplectic space

The standard symplectic space is R²ⁿ with the symplectic form given by the block 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\,

There is another way to interpret this standard symplectic form. Since the model space Rⁿ 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 \oplus V^*

of these spaces equipped with the following form:

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

Now choose any basis

(x_1, \ldots, x_n)

of V and consider its dual basis (see dual space for an explanation of how to obtain the dual basis)

(x^*_1, \ldots, x^*_n)

. We can interpret the basis vectors as lying in W if we write

(x_i, 0)

and

(0, x^*_i)

instead of

x_i

and

x^*_i

. Combining these bases of V and V* (in this order!) in this way, we obtain a basis of W. If we relabel that basis again to

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

, the form

\omega

has the same properties as in the beginning of this section.

Symplectic transformations

A linear symplectic transformation of V is a linear transformation A : V → V such that

ω(Au, Av) = ω(u, v).

That is, it is a linear transformation which preserves the symplectic form. The group of all symplectic transformations of V is called the symplectic group, denoted Sp(V). 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 complements,

W^\perp \cap W

need not be 0. We distinguish four cases. The subspace W ⊆ V is said to be:

symplectic if $W^\perp \cap W = \{0\}$
isotropic if $W \sube W^\perp$
coisotropic if $W^\perp\sube W$
Lagrangian if $W = W^\perp$

A subspace W is symplectic iff ω restricts to a nondegenerate form on W. W is isotropic iff ω restricts to 0 on W. A Lagrangian subspace is an isotropic one that is half the dimension of V. Every isotropic subspace can be extended to a Lagrangian one.

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