PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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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.

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Zeroth Lie algebra cohomology is invariants

Statement

$H^{0} (g, M) = M^{g} = {m \in M : x m = 0 for every x \in g} .$

Facts & Assumptions

Given: A Lie algebra $g$ and a module $M$ .

[L1]

Degree-zero cochains are $M$ and $(d m) (x) = x m$ (Chevalley–Eilenberg differential).

[L2]

Cohomology is kernel modulo the preceding image (Lie algebra cohomology).

Proof

technique · compute in degree zero

1.1L1given

By [L1], $m$ lies in the degree-zero kernel exactly when every $x$ kills it, so $\ker d^{0} = M^{g}$ .

2.1L2step 1.1∎

The degree-minus-one cochain space is zero, hence $im d^{- 1} = 0$ . Substitution in [L2] proves the displayed equality. If $M = 0$ both sides are zero; if $g = 0$ , the condition is vacuous and both sides are all of $M$ .

Depends on

Used by

Dependency tree · two levels

6 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

Weibel, Lie Algebra Homology and Cohomology, §7.7 (standard reference, not scraped)