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.

Lie algebra cohomology

Definition

The Lie algebra cohomology of $g$ with coefficients in the $g$ -module $M$ is the cohomology of its Chevalley–Eilenberg cochain complex:

$H^{n} (g, M) = \frac{\ker (d : C^{n} \to C^{n + 1})}{im (d : C^{n - 1} \to C^{n})} .$

The containment of the denominator in the numerator is supplied by The Chevalley–Eilenberg differential squares to zero. For $n < 0$ all cochain groups, and hence all cohomology groups, are zero by convention. This definition is valid without finite-dimensionality or a characteristic restriction.

Depends on

Dependency tree · 2 levels

Sources

Weibel, Lie Algebra Homology and Cohomology, Corollary 7.7.3