PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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Moment map components generate the negative infinitesimal action

Statement

Assume ${A C}_{ω}$ . Let $μ : M \to g^{*}$ be an infinitesimal moment map for a symplectic action, so that $d μ^{ξ} = - ι_{ξ_{M}} ω$ for every $ξ \in g$ . Then for every $ξ$ the Hamiltonian vector field of the component $μ^{ξ}$ is the negative of the fundamental field:

$X_{μ^{ξ}} = - ξ_{M} .$

Facts & Assumptions

Given: ${A C}_{ω}$ , a symplectic action, an infinitesimal moment map $μ$ , and $ξ \in g$ .

[A1]

${A C}_{ω}$ is countable choice; it is used only through the fundamental-field dependency of [F1].

[F1]

The component moment equation reads $d μ^{ξ} = - ι_{ξ_{M}} ω$ . Moment map, component Hamiltonians and infinitesimal moment maps.

[F2]

$ι$ is linear in its vector-field slot, so $ι_{- ξ_{M}} ω = - ι_{ξ_{M}} ω$ . Hamiltonian vector field and Hamiltonian function.

[F3]

For every smooth $H$ there is a unique smooth vector field $X_{H}$ satisfying $ι_{X_{H}} ω = d H$ . Hamiltonian vector fields exist uniquely for smooth functions.

Proof

technique · direct

1.1

By [F1] and [F2], $d μ^{ξ} = - ι_{ξ_{M}} ω = ι_{- ξ_{M}} ω$ : the covector $d μ^{ξ}$ is obtained by contracting $ω$ with the field $- ξ_{M}$ .

F1F2given

2.1

The field $- ξ_{M}$ is smooth because $ξ_{M}$ is. So $- ξ_{M}$ is a smooth vector field whose contraction with $ω$ equals $d μ^{ξ}$ , the differential of the smooth function $μ^{ξ}$ .

step 1.1

3.1

By [F3] the field $X_{μ^{ξ}}$ with $ι_{X_{μ^{ξ}}} ω = d μ^{ξ}$ exists and is the only such field, and [step 1.1] exhibits $- ξ_{M}$ as such a field; hence $X_{μ^{ξ}} = - ξ_{M}$ .

A1F3step 2.1∎

Depends on

Used by

Dependency tree · two levels

13 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

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)
Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)