Symplectic Vector Space - Subspaces

Subspaces

Let W be a linear subspace of V. Define the symplectic complement of W to be the subspace

The symplectic complement satisfies:

However, unlike orthogonal complements, W⊥ ∩ W need not be 0. We distinguish four cases:

• W is symplectic if W⊥ ∩ W = {0}. This is true if and only if 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 if and only if 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 if and only if W⊥ is isotropic. Any codimension-one subspace is coisotropic.
• W is Lagrangian if W = W⊥. A subspace is Lagrangian if and only if 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 R2n above,

• the subspace spanned by {x₁, y₁} is symplectic
• the subspace spanned by {x₁, x₂} is isotropic
• the subspace spanned by {x₁, x₂, ..., x_n, y₁} is coisotropic
• the subspace spanned by {x₁, x₂, ..., x_n} is Lagrangian.