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

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Semisimple Hamiltonian actions have a unique equivariant moment map when one exists

Statement

Assume ${A C}_{ω}$ and let $M$ be connected. Let $g$ be a finite-dimensional real semisimple Lie algebra, and let a Hamiltonian action of a Lie group $G$ with Lie algebra $g$ on $(M, ω)$ be given. Then there is at most one equivariant moment map for the action: if one equivariant moment map exists, it is the unique one.

Facts & Assumptions

Given: ${A C}_{ω}$ , a connected symplectic $G$ -manifold with $g$ finite-dimensional real semisimple, and an equivariant moment map $μ$ .

[A1]

${A C}_{ω}$ is countable choice; it is used only through [F1].

[F1]

Any two equivariant moment maps for the same action differ by a constant coadjoint-fixed covector $δ \in (g^{*})^{G}$ . Moment maps for one action form an affine space over coadjoint-fixed covectors.

[F2]

If $g$ is finite-dimensional semisimple over a characteristic-zero field then $[g, g] = g$ . Semisimple Lie algebras are centerless and perfect, Simple, semisimple, and reductive Lie algebras.

[F3]

For $α \in g^{*}$ and $ξ, ζ \in g$ , the coadjoint action satisfies ${\frac{d}{d t} ∣}_{0} ⟨ \exp_{G} (t ξ) \cdot α, ζ ⟩ = - ⟨ α, [ξ, ζ] ⟩$ . The coadjoint representation, action and orbits.

Proof

technique · direct

1.1F1given

Let $μ_{1}, μ_{2}$ be two equivariant moment maps. By [F1] there is $δ \in (g^{*})^{G}$ with $μ_{1} - μ_{2} = δ$ . We show $δ = 0$ .

2.1F3step 1.1

Since $g \cdot δ = δ$ for every $g \in G$ , taking $g = \exp_{G} (t ξ)$ and differentiating the constant function $t \mapsto ⟨ \exp_{G} (t ξ) \cdot δ, ζ ⟩$ at $t = 0$ gives $0 = - ⟨ δ, [ξ, ζ] ⟩$ for all $ξ, ζ \in g$ by [F3].

3.1step 2.1F2A1∎

Thus $δ$ vanishes on the linear span of all brackets, that is on $[g, g]$ , which equals $g$ by [F2]. Therefore $δ = 0$ and $μ_{1} = μ_{2}$ ; an equivariant moment map, when it exists, is unique.

Depends on

Used by

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Sources

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