Linear coisotropic reduction - Imported Knowlpedia Content

Core idea

Let $(V,\omega)$ be a symplectic vector space and $W\subseteq V$ a coisotropic subspace. Its linear coisotropic reduction is the quotient vector space

W_{\mathrm{red}}=W/W^\omega

equipped with

\omega_{\mathrm{red}}([u],[v])=\omega(u,v),

where $W^\omega$ is the symplectic orthogonal complement of $W$ . This formula is independent of representatives because every element of $W^\omega$ pairs trivially with $W$ . Its kernel is zero, so $\omega_{\mathrm{red}}$ is nondegenerate. Hence $(W_{\mathrm{red}},\omega_{\mathrm{red}})$ is again a symplectic vector space. No complement to $W^\omega$ is chosen, so the construction is canonical from $W\subseteq V$ and $\omega$ .

Why the quotient is symplectic

The restriction $\omega|_W$ is generally degenerate, and its kernel is exactly $W\cap W^\omega$ . Coisotropy gives $W^\omega\subseteq W$ , so this kernel is $W^\omega$ . Passing to $W/W^\omega$ removes precisely the null directions and no others. Alternation and bilinearity descend immediately, while the kernel calculation proves nondegeneracy.

If $\dim V=2n$ and $\operatorname{codim}W=k$ , then $\dim W^\omega=k$ and

\dim W_{\mathrm{red}}=2n-2k.

The reduced dimension is therefore even, as required.

Examples and boundary cases

For $W=V$ , one has $W^\omega=\{0\}$ , so the reduction is $V$ itself. If $W$ is Lagrangian, then $W=W^\omega$ , and its reduction is the zero symplectic vector space.

In standard $\mathbb R^{2n}$ with coordinates $(q_i,p_i)$ , let $W$ be defined by $p_1=\cdots=p_k=0$ . Then $W^\omega$ is spanned by $\partial/\partial q_1,\ldots,\partial/\partial q_k$ , and the quotient retains the coordinate pairs $(q_{k+1},p_{k+1}),\ldots,(q_n,p_n)$ .

Relation to manifold reduction

For a coisotropic submanifold, the corresponding null spaces form the characteristic distribution. When that distribution integrates to a sufficiently regular foliation with a smooth leaf space, the quotient inherits a symplectic form by the same descent argument. The linear construction is the tangent-space model, but it avoids the global regularity and Hausdorffness issues of a manifold quotient Cannas da Silva, Chapters 23–24.

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: Chapter 1 and Homework 1 for symplectic orthogonals; Chapters 23–24 for manifold reduction.
Victor Guillemin and Shlomo Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984. DOI record. Relevant: Chapter 1, linear symplectic reduction.