# symplectic complement

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

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

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. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

For the symplectic complement, we have the following dimension theorem.

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

 $\dim V=\dim W^{\omega}+\dim W.$

## References

• 1 D. McDuff, D. Salamon, Introduction to Symplectic Topology, Clarendon Press, 1997.
• 2 R. Abraham, J.E. Marsden, Foundations of Mechanics, 2nd ed., Perseus Books, 1978.
Title symplectic complement SymplecticComplement 2013-03-22 13:32:25 2013-03-22 13:32:25 matte (1858) matte (1858) 8 matte (1858) Definition msc 15A04 symplectic complement isotropic subspace coisotropic subspace symplectic subspace Lagrangian subspace