False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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$H \mapsto X_{H}$ is a Lie homomorphism under the library Poisson convention

Statement refuted

Under the library convention, $H \mapsto X_{H}$ is a Lie homomorphism.

Facts & Assumptions

Given: The library conventions for Hamiltonian fields and Poisson brackets.

[F1]

The actual identity is $[X_{F}, X_{G}] = - X_{{F, G}}$ . The Hamiltonian vector-field map is a Lie antihomomorphism.

Refutation

technique · direct

1.1

On $(R^{2}, d q \land d p)$ , take $F = q^{2} / 2$ and $G = p^{2} / 2$ . Solving $ι_{X_{F}} ω = d F$ and $ι_{X_{G}} ω = d G$ gives $X_{F} = - q \partial_{p}$ and $X_{G} = p \partial_{q}$ . Hence ${F, G} = ω (X_{F}, X_{G}) = q p$ , whose Hamiltonian vector field is nonzero.

givenalgebra

2.1

By [F1], $[X_{F}, X_{G}] = - X_{q p} \neq X_{q p} = X_{{F, G}}$ . Thus the homomorphism identity fails; the map is an antihomomorphism.

F1step 1.1∎

Depends on

The Hamiltonian vector-field map is a Lie antihomomorphism

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)

H↦XHH↦XH​ is a Lie homomorphism under the library Poisson convention