PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Symplectic reduction of a coisotropic vector subspace

Statement

If $W$ is coisotropic in $(V, ω)$ , then $W / W^{ω}$ has the symplectic form

$\overline{ω} ([u], [v]) = ω (u, v) .$

Facts & Assumptions

Given: A symplectic vector space $(V, ω)$ and a coisotropic subspace $W$ .

[F1]

Coisotropic means $W^{ω} \subseteq W$ . Isotropic, coisotropic, symplectic, and Lagrangian subspaces.

Proof

technique · direct

1.1F1algebra

The quotient is defined by [F1]. Replacing $u$ by $u + a$ with $a \in W^{ω}$ , or $v$ by $v + b$ with $b \in W^{ω}$ , does not change $ω (u, v)$ because $u, v \in W$ . Hence $\overline{ω}$ is well-defined, bilinear, and alternating.

2.1F1step 1.1∎

If $[u]$ lies in its radical, then $ω (u, v) = 0$ for every $v \in W$ , so $u \in W^{ω}$ and $[u] = 0$ . Thus the descended form is nondegenerate. When $W = V$ this recovers $V$ ; when $W$ is Lagrangian the quotient is the zero symplectic space.

Depends on

Isotropic, coisotropic, symplectic, and Lagrangian subspaces

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)