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.

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 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
Canonical name	SymplecticComplement
Date of creation	2013-03-22 13:32:25
Last modified on	2013-03-22 13:32:25
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	8
Author	matte (1858)
Entry type	Definition
Classification	msc 15A04
Defines	symplectic complement
Defines	isotropic subspace
Defines	coisotropic subspace
Defines	symplectic subspace
Defines	Lagrangian subspace