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

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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

Used by

Omitting the exterior root-weight shifts gives the wrong dot weight Counterexample
Whitehead vanishing does not apply to the nilpotent radical Counterexample
Chevalley–Eilenberg differential Definition
Classical Lie-action BRST cochain complex Definition
Central actions on nilradical cohomology factor through the Harish–Chandra projection Lemma
Each extremal harmonic space is one-dimensional Lemma
Gan-Ginzburg and Skryabin inputs for the type-A pyramid good grading Lemma
The Chevalley–Eilenberg Laplacian is scalar on weight components Lemma
The extremal weight cochain of a Weyl element is closed and unique Lemma
Chevalley–Eilenberg cohomology computes Ext of the trivial module Proposition
Degree-zero Lie algebra cohomology is the invariant subspace Proposition
The normalizer acts on Lie algebra cohomology Proposition
Kostant's nilradical cohomology theorem Theorem
Skryabin's equivalence for the type-A Whittaker pair Theorem

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

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