symplectic complement

Definition [1, 2]Let $(V,\omega )$ be a symplectic vector space and let $W$ be avector subspace of $V$. Then the symplectic complement of $W$is

$$W^{\omega }=\{x\in V|\omega (x,y)=0\text{for all}y\in W\}.$$

It is easy to see that $W^{\omega }$ is also a vector subspace of $V$.Depending on the relation between $W$ and $W^{\omega }$,$W$ is given different names.

1. If $W\subset W^{\omega }$, then $W$ is an isotropic subspace (of $V$).

2. If $W^{\omega }\subset W$, then $W$ is an coisotropic subspace.

3. If $W\cap W^{\omega }=\{0\}$, then $W$ is an symplectic subspace.

4. If $W=W^{\omega }$, then $W$ is an Lagrangian subspace.

For the symplectic complement, we have thefollowing dimension theorem.

Theorem [1, 2] Let $(V,\omega )$ be a symplectic vectorspace, and let $W$ be a vector subspace of $V$. Then

$$dimV=dim{W}^{\omega }+dimW.$$