PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pendingjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Formal equivalence changes the first-order cochain by a Hochschild coboundary

Statement

Let $⋆$ and $⋆^{'}$ be formally gauge equivalent differential star products on a real Poisson manifold, with first coefficients $B_{1}$ and $B_{1}^{'}$ . If their gauge map is $T = id + ℏ D_{1} + O (ℏ^{2})$ , then $B_{1}^{'} (f, g) = B_{1} (f, g) + (δ D_{1}) (f, g), (δ D_{1}) (f, g) = f D_{1} (g) - D_{1} (f g) + D_{1} (f) g .$ Consequently $B_{1}$ and $B_{1}^{'}$ define the same class in $H H_{d i f f, n o r m}^{2} (C^{\infty} (M), C^{\infty} (M))$ , and their skew parts are identical. This is a first-order invariant only; it does not classify full star products.

Facts & Assumptions

Given: A commutative real algebra $A = C^{\infty} (M, R)$ , two differential star products on $A ⟦ ℏ ⟧$ , and a gauge map $T$ as in [F1].

[F1]

Formal differential star products and first-order normalization: the products are associative, their coefficients are bidifferential, and they preserve the unit.

[F2]

Formal differential gauge equivalence of star products: $T = id + \sum_{r \geq 1} ℏ^{r} D_{r}$ , with differential coefficients vanishing on $1$ , and $f ⋆^{'} g = T^{- 1} (T f ⋆ T g)$ .

[F3]

Hochschild cochains, differential, and Gerstenhaber insertion bracket: for a 1-cochain $D$ , $(δ D) (f, g) = f D (g) - D (f g) + D (f) g$ .

[F4]

The shifted Hochschild cochains form a differential graded Lie algebra: $δ^{2} = 0$ , so the normalized multidifferential cocycles modulo coboundaries define the stated Hochschild cohomology group.

Proof

1.1givenF1F3F4algebra

The coefficient of $ℏ$ in associativity of $⋆$ is $B_{1} (f, g) h + B_{1} (f g, h) - f B_{1} (g, h) - B_{1} (f, g h) = 0$ , which is $- (δ B_{1}) (f, g, h) = 0$ . The same calculation applies to $⋆^{'}$ , so both first coefficients are normalized 2-cocycles in the complex of [F4].

1.2F2givenalgebra

Comparing coefficients of $ℏ$ in $T T^{- 1} = id$ gives $T^{- 1} = id - ℏ D_{1} + O (ℏ^{2})$ .

2.1F2F3step 1.2algebra

Expand the defining transformation from [F2] through first order: $T f ⋆ T g = f g + ℏ (D_{1} (f) g + f D_{1} (g) + B_{1} (f, g)) + O (ℏ^{2}) .$ Applying $T^{- 1}$ subtracts $ℏ D_{1} (f g)$ , hence $B_{1}^{'} = B_{1} + f D_{1} (g) - D_{1} (f g) + D_{1} (f) g = B_{1} + δ D_{1}$ by [F3].

3.1givenF3F4step 1.1step 2.1algebra∎

Since $A$ is commutative, $δ D_{1} (f, g) = f D_{1} (g) - D_{1} (f g) + D_{1} (f) g$ equals $δ D_{1} (g, f)$ . Thus the skew parts of $B_{1}$ and $B_{1}^{'}$ agree, and their difference is the coboundary $δ D_{1}$ . With step 1.1 this proves equality of their classes in the cohomology of [F4].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Maxim Kontsevich, Deformation Quantization of Poisson Manifolds, I (standard reference, not scraped)
Sarah J. Witherspoon, Hochschild Cohomology for Algebras (standard reference, not scraped)
Nima Moshayedi, Lectures on Symplectic Geometry, Poisson Geometry, Deformation Quantization and Quantum Field Theory (standard reference, not scraped)