Symplectic subspace - Imported Knowlpedia Content

Definition

Let $(V,\omega)$ be a finite-dimensional symplectic vector space. A linear subspace $W\subseteq V$ is a symplectic subspace if the restricted alternating form $\omega|_{W\times W}$ is nondegenerate. Equivalently,

W\cap W^\omega=\{0\},

where $W^\omega$ is the symplectic orthogonal complement. With the restricted form, $W$ is itself a symplectic vector space, so it has even dimension. The term describes nondegeneracy of the restriction; it does not mean merely that $W$ lies inside a symplectic vector space.

Orthogonal splitting

For a symplectic subspace $W$ , dimension counting and $W\cap W^\omega=0$ give

V=W\oplus W^\omega.

The restriction of $\omega$ to $W^\omega$ is also nondegenerate. Thus a symplectic subspace always has a canonical complementary symplectic subspace determined by the ambient form.

Equivalent tests

The following conditions are equivalent: the restricted form on $W$ is nondegenerate; $W\cap W^\omega=0$ ; and the map $W\to W^*$ , $w\mapsto\omega(w,\mathord{-})|_W$ , is an isomorphism. In a symplectic basis for $V$ , the span of any collection of complete pairs $e_i,f_i$ is symplectic.

Contrasts

A nonzero isotropic subspace cannot be symplectic, since its restricted form is zero. A coisotropic subspace instead satisfies $W^\omega\subseteq W$ and may have a nontrivial kernel for its restricted form. The zero subspace is symplectic under the usual vacuous nondegeneracy convention.

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §1.1, symplectic orthogonals and subspaces.
Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: Chapter 2, linear symplectic geometry.