Symplectic orthogonal complement - Imported Knowlpedia Content

Definition

Let $(V,\omega)$ be a symplectic vector space and let $W\subseteq V$ be a linear subspace. The symplectic orthogonal complement of $W$ is

W^\omega=\{v\in V:\omega(v,w)=0\text{ for every }w\in W\}.

It is a linear subspace determined by both $W$ and the symplectic form. Because $\omega$ is nondegenerate and $V$ is finite-dimensional,

\dim W+\dim W^\omega=\dim V \qquad\text{and}\qquad (W^\omega)^\omega=W.

Unlike the orthogonal complement for an inner product, $W$ and $W^\omega$ need not meet trivially and need not form a direct sum.

Algebraic properties

The operation reverses inclusions: if $A\subseteq B$ , then $B^\omega\subseteq A^\omega$ . It also exchanges sums and intersections:

(A+B)^\omega=A^\omega\cap B^\omega, \qquad (A\cap B)^\omega=A^\omega+B^\omega.

The second equality uses finite dimensionality; the first follows directly from the defining vanishing conditions.

Classification of subspaces

The relative position of $W$ and $W^\omega$ determines the standard classes of symplectic linear subspaces. The subspace is isotropic when $W\subseteq W^\omega$ , coisotropic when $W^\omega\subseteq W$ , and Lagrangian when $W=W^\omega$ . The restriction $\omega|_W$ is nondegenerate exactly when $W\cap W^\omega=\{0\}$ .

Examples

In $\mathbb R^{2n}$ with coordinates $(q_1,\ldots,q_n,p_1,\ldots,p_n)$ and standard symplectic form, the span of the $q_i$ is its own symplectic orthogonal and is therefore Lagrangian. The symplectic orthogonal of $V$ is $\{0\}$ , and that of $\{0\}$ is $V$ .

Conventions and scope

The notation $W^\omega$ records the dependence on the form and is preferable when several bilinear forms are present. Some authors write $W^\perp$ , $W^{\perp_\omega}$ , or “skew orthogonal.” In infinite-dimensional spaces, dimension formulas and the double-complement identity can fail without topological closure or additional nondegeneracy hypotheses. See Cannas da Silva, §1.1.

References

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