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

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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

Used by

Omitting the exterior root-weight shifts gives the wrong dot weight Counterexample
Whitehead vanishing does not apply to the nilpotent radical Counterexample
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
First cohomology is derivations modulo inner derivations Proposition
The normalizer acts on Lie algebra cohomology Proposition
Whitehead's second lemma removes the infinitesimal equivariance obstruction for semisimple actions Proposition
Zeroth Lie algebra cohomology is invariants Proposition
Kostant's nilradical cohomology theorem Theorem
Long exact sequence in Lie algebra cohomology Theorem
Second cohomology classifies abelian extensions 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, Corollary 7.7.3 (standard reference, not scraped)