PropositionStatement: Literature-sourcedProof: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Finite constrained mode quotient

Statement

Let $V$ be a finite-dimensional real vector space with a nondegenerate alternating bilinear form $ω : V \times V \to R$ . For a linear subspace $W \subseteq V$ , define $W^{ω} : = {v \in V : ω (v, w) = 0 for every w \in W}$ , and assume $W^{ω} \subseteq W$ (so $W$ is coisotropic). Then the quotient vector space $W / W^{ω}$ , whose elements are cosets $[u] : = u + W^{ω}$ , carries the well-defined symplectic form $\overline{ω} ([u], [v]) = ω (u, v)$ . This proposition concerns only finite-dimensional constrained mode models; it does not assert that an infinite-dimensional Einstein–Maxwell solution space has a smooth symplectic quotient.

Statement references

Isotropic, coisotropic, symplectic, and Lagrangian subspaces

Symplectic reduction of a coisotropic vector subspace

Facts & Assumptions

Given: A finite-dimensional real symplectic vector space $(V, ω)$ and a linear subspace $W \subseteq V$ with $W^{ω} \subseteq W$ .

[F1]

Isotropic, coisotropic, symplectic, and Lagrangian subspaces: coisotropic means $W^{ω} \subseteq W$ , where $W^{ω} = {v \in V : ω (v, w) = 0 for every w \in W}$ .

[F2]

Symplectic reduction of a coisotropic vector subspace: for a symplectic vector space and coisotropic subspace $W$ , $W / W^{ω}$ has the well-defined nondegenerate alternating form $\overline{ω} ([u], [v]) = ω (u, v)$ .

Proof

1.1givenF1F2

The hypothesis $W^{ω} \subseteq W$ is exactly the coisotropic condition in [F1]. Thus the quotient and descended form in [F2] apply to this $(V, ω, W)$ .

2.1givenF2step 1.1∎

By [F2], changing either representative by an element of $W^{ω}$ leaves $ω (u, v)$ unchanged, and a class in the radical has representative in $W^{ω}$ , so the descended form is well-defined and nondegenerate. It is alternating by the alternation of $ω$ . Since $W$ is a subspace of finite-dimensional $V$ , $W / W^{ω}$ is finite-dimensional, so it is a finite-dimensional symplectic vector space. No infinite-dimensional quotient regularity is used.

Remarks

For $W = V$ the quotient is $V$ itself; for a Lagrangian constraint subspace $W = W^{ω}$ the quotient is the zero symplectic vector space. These are finite-dimensional boundary cases of the stated result, not evidence for an infinite-dimensional solution-space reduction.

Depends on

Dependency tree · 2 levels

Sources

Eckhard Meinrenken, Symplectic Geometry, Lecture Notes, University of Toronto