DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Chevalley–Eilenberg cochains

Definition

For a Lie algebra $g$ over $k$ and a $g$ -module $M$ , the degree- $n$ Chevalley–Eilenberg cochains are

$C^{n} (g, M) = {Hom}_{k} (Λ^{n} g, M) (n \geq 0) .$

Set $C^{n} (g, M) = 0$ for $n < 0$ . Since $Λ^{0} g = k$ , evaluation at $1$ identifies $C^{0} (g, M)$ with $M$ . Cochains may equivalently be viewed as alternating $n$ -linear maps $g^{n} \to M$ . No finite-dimensionality or characteristic hypothesis is needed. If $n > \dim g$ in the finite-dimensional case, the exterior power and hence the cochain space are zero.

Depends on

Dependency tree · 2 levels

Sources

Weibel, Lie Algebra Homology and Cohomology, §7.7