Coisotropic subspace

Let $(V,\omega)$ be a symplectic vector space and $W\subseteq V$ a linear subspace. The subspace $W$ is coisotropic if

W^\omega\subseteq W,

where $W^\omega$ is its symplectic orthogonal complement. The subspace $W^\omega$ is then the characteristic subspace of $W$ . The restriction $\omega|_W$ has kernel exactly $W^\omega$ , so it descends to a nondegenerate alternating form on the quotient $W/W^\omega$ . Hence $W/W^\omega$ is naturally a symplectic vector space; this quotient is the linear model for coisotropic reduction.

Equivalent conditions

The following conditions are equivalent: $W$ is coisotropic; $W^\omega\subseteq W$ ; the symplectic orthogonal $W^\omega$ is isotropic; and $\ker(\omega|_W)$ has dimension equal to $\operatorname{codim}W$ . In particular, a coisotropic subspace of a $2n$ -dimensional symplectic vector space has dimension at least $n$ .

Examples and boundary cases

The whole space $V$ is coisotropic because $V^\omega=\{0\}$ . Every hyperplane in a finite-dimensional symplectic vector space is coisotropic. A Lagrangian subspace is both isotropic and coisotropic, since $W=W^\omega$ . By contrast, a proper symplectic subspace is not coisotropic: nondegeneracy gives $W\cap W^\omega=\{0\}$ , so $W^\omega\subseteq W$ would force $W=V$ .

Reduction

For $u,v\in W$ , define $\bar\omega([u],[v])=\omega(u,v)$ on $W/W^\omega$ . This is well defined because changing either representative by an element of $W^\omega$ leaves the value unchanged. Its kernel is zero by construction. The same mechanism globalizes, under regularity hypotheses, to the reduction of a coisotropic submanifold by its characteristic foliation.

Conventions and scope

Some sources write “co-isotropic.” The definition here is purely linear and finite-dimensional. For a coisotropic submanifold $C$ of a symplectic manifold, the condition is imposed pointwise as $(T_pC)^\omega\subseteq T_pC$ , and forming a smooth quotient requires additional global hypotheses.

References

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