Symplectic Linear Maps

DefinitionSymplectomorphism

A linear map $\phi: (V, \omega) \to (V', \omega')$ between symplectic vector spaces is a symplectomorphism (or symplectic map) if:

$\phi^* \omega' = \omega$

That is, $\omega(v, w) = \omega'(\phi(v), \phi(w))$ for all $v, w \in V$ .

The group of symplectic automorphisms of $(V, \omega)$ is denoted $\text{Sp}(V, \omega)$ or $\text{Sp}(2n, \mathbb{R})$ for $V = \mathbb{R}^{2n}$ .

Symplectomorphisms are the structure-preserving maps in symplectic geometry, playing a role analogous to isometries in Riemannian geometry or unitary operators in quantum mechanics. However, the symplectic group has distinctive properties not shared by these other groups.

ExampleSymplectic Group $\text{Sp}(2n, \mathbb{R})$

A matrix $A \in \text{GL}(2n, \mathbb{R})$ belongs to $\text{Sp}(2n, \mathbb{R})$ if and only if:

$A^T J A = J$

where $J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$ is the standard symplectic matrix.

For $n = 1$ , we have $\text{Sp}(2, \mathbb{R}) = \text{SL}(2, \mathbb{R})$ , the $2 \times 2$ matrices with determinant 1.

Remark

Every symplectomorphism has determinant $\pm 1$ . In fact, for connected symplectic groups, $\det A = 1$ for all $A \in \text{Sp}(2n, \mathbb{R})$ . This follows from the fact that $\det(A^T J A) = (\det A)^2 \det J = \det J$ .

The symplectic group is a non-compact Lie group of dimension $n(2n+1)$ . Unlike the orthogonal group $\text{O}(n)$ which is compact, $\text{Sp}(2n, \mathbb{R})$ contains unbounded transformations that preserve the symplectic structure.

DefinitionInfinitesimal Symplectomorphisms

A linear map $A: V \to V$ is infinitesimally symplectic if:

$\omega(Av, w) + \omega(v, Aw) = 0$

for all $v, w \in V$ . The space of such maps forms the symplectic Lie algebra $\mathfrak{sp}(V, \omega)$ .

ExampleHamiltonian Matrices

For $(\mathbb{R}^{2n}, \omega_0)$ , a matrix $H \in \mathfrak{sp}(2n, \mathbb{R})$ satisfies:

$J^T H + H^T J = 0 \quad \Leftrightarrow \quad JH = (JH)^T$

Such matrices have the block form:

$H = \begin{pmatrix} A & B \\ C & -A^T \end{pmatrix}$

where $B = B^T$ and $C = C^T$ are symmetric. The Lie algebra $\mathfrak{sp}(2n, \mathbb{R})$ has dimension $n(2n+1)$ .

The exponential map $\exp: \mathfrak{sp}(2n, \mathbb{R}) \to \text{Sp}(2n, \mathbb{R})$ generates one-parameter subgroups of symplectomorphisms. These correspond to linear Hamiltonian flows, providing the foundation for Hamiltonian mechanics on linear phase spaces.

Remark

The symplectic group sits between the general linear group and the orthogonal group:

$\text{O}(2n) \cap \text{Sp}(2n, \mathbb{R}) = \text{U}(n)$

This intersection gives the unitary group, which preserves both a Riemannian metric and a symplectic form compatible with a complex structure.