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[parent] symplectic complement (Definition)

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

Bibliography

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.




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Also defines:  symplectic complement, isotropic subspace, coisotropic subspace, symplectic subspace, Lagrangian subspace

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dimension theorem for symplectic complement (proof) (Proof) by matte
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Cross-references: theorem, dimension, relation, easy to see, vector subspace, symplectic vector space
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This is version 5 of symplectic complement, born on 2003-04-02, modified 2004-02-28.
Object id is 4139, canonical name is SymplecticComplement.
Accessed 7454 times total.

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AMS MSC15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)

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